diff --git "a/CMPhysBench.json" "b/CMPhysBench.json" new file mode 100644--- /dev/null +++ "b/CMPhysBench.json" @@ -0,0 +1,7904 @@ +[ + { + "id": 1, + "context": "", + "question": "For an oscillator with charge $q$, its energy operator without an external field is\n\n\\begin{equation*}\nH_{0}=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2} \n\\end{equation*}\n\nIf a uniform electric field $\\mathscr{E}$ is applied, causing an additional force on the oscillator $f= q \\mathscr{E}$, the total energy operator becomes\n\n\\begin{equation*}\nH=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{E} x \n\\end{equation*}\n\nFind the expression for the new energy levels $E_{n}$.", + "answer": "In $H_{0}$ and $H$, $p$ is the momentum operator,\n\n$p=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}$\n\n\nThe potential energy term in equation (2) can be rewritten as\n\n$\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{E} x=\\frac{1}{2} m \\omega^{2}[(x-x_{0})^{2}-x_{0}^{2}]$\n\nwhere\n\n\\begin{equation*}\nx_{0}=\\frac{q \\mathscr{E}}{m \\omega^{2}} \\tag{3}\n\\end{equation*}\n\n\nBy performing a coordinate shift, let\n\n\\begin{equation*}\nx^{\\prime}=x-x_{0} \\tag{4}\n\\end{equation*}\n\n\nBecause\n\n\\begin{equation*}\np=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}=-\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} x}=p^{\\prime} \\tag{5}\n\\end{equation*}\n\n$H$ can be expressed as\n\n\\begin{equation*}\nH=\\frac{p^{\\prime 2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{\\prime 2}-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{6}\n\\end{equation*}\n\n\nComparing equations (1) and (6), it is evident that the difference between $H$ and $H_{0}$ is that the variable changes from $x$ to $x^{\\prime}$, with an added constant term $(-\\frac{1}{2} m \\omega^{2} x_{0}^{2})$. Hence, we get\n\n\\begin{gather*}\nE_{n}=E_{n}^{(0)}-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{7}\\\\\n\\varphi_{n}(x)=\\psi_{n}(x^{\\prime})=\\psi_{n}(x-x_{0}) \\tag{8}\n\\end{gather*}\n\n\nIt is well-known that the energy levels of the free oscillator are\n\n\\begin{equation*}\nE_{n}^{(0)}=(n+\\frac{1}{2}) \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{9}\n\\end{equation*}\n\n\nThus,\n\n\\begin{align*}\nE_{n} & =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\\\\n& =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} \\mathscr{E}^{2}}{2 m \\omega^{2}} \\tag{10}\n\\end{align*}\n\n\nIntroducing the coordinate shift operator\n\n\\begin{equation*}\nD_{x}(x_{0})=\\mathrm{e}^{-\\mathrm{i} x_{0} p / \\hbar}=\\mathrm{e}^{-x_{0} \\frac{d}{d x}} \\tag{11}\n\\end{equation*}\n\n\nIts effect on the wave function is\n\n\\begin{equation*}\nD_{x}(x_{0}) \\psi(x)=\\psi(x-x_{0}) \\tag{11'}\n\\end{equation*}\n\n\nThen the eigenfunctions of $H$ and $H_{0}$ can be related through the shift operator:\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\psi_{n}(x-x_{0})=D_{x}(x_{0}) \\psi_{n}(x) \\tag{12}\n\\end{equation*}\n\n\nConversely,\n\n\\begin{equation*}\n\\psi_{n}(x)=\\varphi_{n}(x+x_{0})=D_{x}(-x_{0}) \\varphi_{n}(x) \\tag{$\\prime$}\n\\end{equation*}\n\n\nSolution two, using the raising and lowering operators of the oscillator\n\n\\begin{equation*}\na=\\sqrt{\\frac{m \\omega}{2 \\hbar}}(x+\\frac{\\mathrm{i}}{m \\omega} p), \\quad a^{+}=\\sqrt{\\frac{m \\omega}{2 \\hbar}}(x-\\frac{\\mathrm{i}}{m \\omega} p) \\tag{11}\n\\end{equation*}\n\n\nExpressing $H_{0}$ and $H$ as\n\n\\begin{gather*}\nH_{0}=(a^{+} a+\\frac{1}{2}) \\hbar \\omega \\tag{14}\\\\\nH=(a^{+} a+\\frac{1}{2}) \\hbar \\omega-q \\mathscr{E} \\sqrt{\\frac{\\hbar}{2 m \\omega}}(a+a^{+}) \\tag{15}\n\\end{gather*}\n\n\nIntroducing\n\n\\begin{equation*}\nx_{0}=q \\varepsilon / m \\omega^{2} \n\\end{equation*}\n\n\nThen\n\n\\begin{align*}\nH & =\\hbar \\omega[a^{+} a+\\frac{1}{2}-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}}(a+a^{+})] \\\\\n& =\\hbar \\omega[(a^{+}-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}})(a-x_{0} \\sqrt{\\frac{m \\omega}{2 \\hbar}})+\\frac{1}{2}-\\frac{m \\omega x_{0}^{2}}{2 \\hbar}] \\\\\n& =\\hbar \\omega[(a^{+}-\\alpha_{0})(a-\\alpha_{0})+\\frac{1}{2}]-\\frac{1}{2} m \\omega^{2} x_{0}^{2} \\tag{16}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\n\\alpha_{0}=x_{0} \\sqrt{m \\omega / 2 \\hbar} \\tag{17}\n\\end{equation*}\n\n\nComparing equations (14), (16), the difference between $H$ and $H_{0}$ is $a \\rightarrow a-\\alpha_{0}, a^{+} \\rightarrow a^{+}-\\alpha_{0}$, and an added constant term $(-\\frac{1}{2} m \\omega^{2} x_{0}^{2})$.\n\nStarting from the fundamental commutation relation\n\n\\begin{equation*}\n[a, a^{+}]=1 \\tag{18}\n\\end{equation*}\n\n\nIt was proved that the energy level formula (9) and the recursion relations between eigenstates\n\n\\begin{equation*}\na \\psi_{n}=\\sqrt{n} \\psi_{n-1}, \\quad a^{-} \\psi_{n}=\\sqrt{n+1} \\psi_{n+1} \\tag{19}\n\\end{equation*}\n\n\nAnd the ground state wave function satisfies\n\n\\begin{equation*}\na \\psi_{0}=0 \\tag{20}\n\\end{equation*}\n\n\nAs\n\n$[a-\\alpha_{0}, a^{+}-\\alpha_{0}]=[a, a^{+}]=1$\n\n\nSo the same reasoning logically leads to similar conclusions for the eigenvalues and eigenfunctions of $H$, simply substituting $(a-\\alpha_{0})$ for $a$ in the entire derivation, $(a^{+}-\\alpha_{0})$ for $a^{+}$. The eigenvalue of $H$ is clearly equation (10). The recursion relations and ground state equation for the eigenfunctions are\n\n\\begin{gather*}\n(a-\\alpha_{0}) \\varphi_{n}(x)=\\sqrt{n} \\varphi_{n-1}(x) \\tag{21}\\\\\n(a^{+}-\\alpha_{0}) \\varphi_{n}(x)=\\sqrt{n+1} \\varphi_{n+1}(x) \\\\\n(a-\\alpha_{0}) \\varphi_{0}(x)=0 \\tag{22}\n\\end{gather*}\n\n$a \\rightarrow(a-\\alpha_{0})$ (and $a^{+} \\rightarrow a^{+}-\\alpha_{0})$ are equivalent to $x \\rightarrow(x-x_{0})$, thus replacing $x$ with $(x-x_{0})$ in $\\psi_{n}(x)$ gives\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\psi_{n}(x-x_{0})=D_{x}(x_{0}) \\psi_{n}(x) \\tag{12}\n\\end{equation*}\n\n$\\varphi_{n}$ and $\\varphi_{0}$ can be related through the operator $(a^{+}-\\alpha_{0})$:\n\n\\begin{equation*}\n\\varphi_{n}(x)=\\frac{1}{\\sqrt{n!}}(a^{+}-\\alpha_{0})^{n} \\varphi_{0}(x) \\tag{23}\n\\end{equation*}", + "final_answer": [ + "E_{n} =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} \\mathscr{E}^{2}}{2 m \\omega^{2}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_n$": "New energy levels of the oscillator in the electric field", + "$n$": "Quantum number, $n=0,1,2, \\cdots$", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega$": "Angular frequency of the oscillator", + "$q$": "Charge of the oscillator", + "$\\mathscr{E}$": "Uniform electric field", + "$m$": "Mass of the oscillator" + } + }, + { + "id": 2, + "context": "", + "question": "A particle of mass $m$ is in the ground state of a one-dimensional harmonic oscillator potential\n\n\\begin{equation*}\nV_{1}(x)=\\frac{1}{2} k x^{2}, \\quad k>0 \n\\end{equation*}\n\n\nWhen the spring constant $k$ suddenly changes to $2k$, the potential then becomes\n\n\\begin{equation*}\nV_{2}(x)=k x^{2} \n\\end{equation*}\n\n\nImmediately measure the energy of the particle, and find the expression for the probability of the particle being in the ground state of the new potential $V_{2}$.", + "answer": "(a) The wave function of the particle $\\psi(x, t)$ should satisfy the time-dependent Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi=-\\frac{\\hbar^{2}}{2 m} \\frac{\\partial^{2}}{\\partial x^{2}} \\psi+V \\psi \\tag{3}\n\\end{equation*}\n\n\nWhen $V$ undergoes a sudden change (from $V_{1} \\rightarrow V_{2}$) but with a finite change quantity, $\\psi$ remains a continuous function of $t$, implying that $\\psi$ does not change when $V$ changes abruptly.\n\nDenote $\\psi_{0}(x)$ and $\\phi_{0}(x)$ as the ground state wave functions of the potential $V_{1}$ and $V_{2}$, respectively. After the potential suddenly changes from $V_{1}$ to $V_{2}$, the wave function of the particle remains $\\psi_{0}$. The probability of measuring the particle in the state $\\phi_{0}$ is $|\\langle\\psi_{0} \\mid \\phi_{0}\\rangle|^{2}$.\n\nRewrite $V_{1}$ and $V_{2}$ in standard form:\n\n\\begin{gather*}\nV_{1}(x)=\\frac{1}{2} k x^{2}=\\frac{1}{2} m \\omega_{1}^{2} x^{2} \\tag{$\\prime$}\\\\\nV_{2}(x)=k x^{2}=\\frac{1}{2} m \\omega_{2}^{2} x^{2} \\tag{$\\prime$}\n\\end{gather*}\n\n\nIt is clear that\n\n\\begin{equation*}\n\\omega_{2}=\\sqrt{2} \\omega_{1} \\tag{4}\n\\end{equation*}\n\n$\\psi_{0}$ and $\\phi_{0}$ can be expressed as in formulas (3) and (5) from problem 3.2, namely\n\n\\begin{align}\n\\psi_{0}(x) &= \\left( \\frac{\\alpha}{\\sqrt{\\pi}} \\right)^{\\frac{1}{2}} \\mathrm{e}^{-\\alpha^2 x^2 / 2}, &\n\\alpha^{2} &= m \\omega_{1} / \\hbar \\notag \\\\\n\\phi_{0}(x) &= \\left( \\frac{\\beta}{\\sqrt{\\pi}} \\right)^{\\frac{1}{2}} \\mathrm{e}^{-\\beta^2 x^2 / 2}, &\n\\beta^{2} &= m \\omega_{2} / \\hbar \\tag{6}\n\\end{align}\n\nwhere\n\n\\begin{equation*}\n\\beta^{2} / \\alpha^{2}=\\omega_{2} / \\omega_{1}=\\sqrt{2} \\tag{7}\n\\end{equation*}\n\n\nThus\n\n\\begin{align*}\n\\langle\\psi_{0} \\mid \\phi_{0}\\rangle & =\\sqrt{\\frac{\\alpha \\beta}{\\pi}} \\int_{-\\infty}^{+\\infty} \\mathrm{e}^{-\\frac{1}{2}(\\alpha^{2}+\\beta^{2}) x^{2}} \\mathrm{~d} x=(\\frac{2 \\alpha \\beta}{\\alpha^{2}+\\beta^{2}})^{\\frac{1}{2}} \\\\\n|\\langle\\psi_{0} \\mid \\phi_{0}\\rangle|^{2} & =\\frac{2 \\alpha \\beta}{\\alpha^{2}+\\beta^{2}}=\\frac{2 \\beta / \\alpha}{1+\\beta^{2} / \\alpha^{2}}=\\frac{2^{5 / 4}}{1+\\sqrt{2}}=0.9852 \\tag{8}\n\\end{align*}\n\n\nThis is the required probability.\n(b) Consider the time when the potential changes for the first time $(V_{1} \\rightarrow V_{2})$ as $t=0$, then the wave function is\n\n\\begin{equation*}\n\\psi(x, 0)=\\psi_{0}(x) \\tag{9}\n\\end{equation*}\n\n\nLet $\\phi_{n}(x)$ denote the energy eigenstates of the potential $V_{2}$, corresponding to energy levels\n\n$E_{n}=(n+\\frac{1}{2}) \\hbar \\omega_{2}$\n\n\nExpand $\\psi_{0}$ as a linear combination of $\\phi_{n}$,\n\n\\begin{equation*}\n\\psi_{0}(x)=\\sum_{n} C_{n} \\phi_{n}(x), \\quad(n \\text { can only take even values }) \\tag{10}\n\\end{equation*}\n\n\nFor $0\\tau$, with the total energy operator given by\n\n\\begin{equation*}\nH_{0}=\\frac{1}{2 m} p^{2}+\\frac{1}{2} m \\omega^{2} x^{2} \n\\end{equation*}\n\n\nThe energy eigenstates are denoted by $\\psi_{n}$, and the energy levels $E_{n}^{(0)}=(n+\\frac{1}{2}) \\hbar \\omega$ . When $0 \\leqslant t \\leqslant \\tau$, a uniform electric field $\\mathscr{E}$ is applied, and the total energy operator becomes\n\n\\begin{equation*}\nH^{\\prime}=\\frac{1}{2 m} p^{2}+\\frac{1}{2} m \\omega^{2} x^{2}-q \\mathscr{\\varepsilon} x \n\\end{equation*}\n\nAssuming the oscillator is in the ground state $\\psi_{0}$ at $t \\leqslant 0$, find the expression for the probability $P_n$ that the system is in the energy eigenstate $\\psi_{n}$ at $t>\\tau$.", + "answer": "At $t>\\tau$, the external electric field has vanished, and the wavefunction satisfies the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi(x, t)=H_{0} \\psi(x, t) \\tag{3}\n\\end{equation*}\n\n\nThe general solution is\n\n\\begin{equation*}\n\\psi(x, t)=\\sum_{n} f_{n} \\psi_{n}(x) \\mathrm{e}^{-\\mathrm{i} E_{n}^{(0)}(t-\\tau) / \\hbar} \\tag{4}\n\\end{equation*}\n\n\nwhere the components of the $\\psi_{n}$ states are\n\n\\begin{equation*}\n|\\langle\\psi_{n} \\mid \\psi\\rangle|^{2}=|f_{n}|^{2} \\tag{5}\n\\end{equation*}\n\n\nEach coefficient $f_{n}$ depends on the wavefunction at $t=\\tau$\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\sum_{n} f_{n} \\psi_{n}(x) \\tag{6}\n\\end{equation*}\n\n\nSo the key is to find $\\psi(x, \\tau)$.\nIn the interval $0 \\leqslant t \\leqslant \\tau$, the Schrödinger equation is\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\partial}{\\partial t} \\psi(x, t)=H \\psi(x, t) \\tag{7}\n\\end{equation*}\n\n\nThe solution is\n\n\\begin{equation*}\n\\psi(x, t)=\\sum_{n} C_{n} \\varphi_{n}(x) \\mathrm{e}^{-\\mathrm{i} \\mathrm{E}_{n} t / \\hbar} \\tag{8}\n\\end{equation*}\n\n\nwhere $C_{n}$ depends on the initial wavefunction\n\n\\begin{equation*}\n\\psi(x, 0)=\\psi_{0}(x)=\\sum_{n} C_{n} \\varphi_{n}(x) \\tag{9}\n\\end{equation*}\n\n\nIt has been proven that\n\n\\begin{equation*}\n\\psi_{0}(x)=\\varphi_{0}(x+x_{0})=D_{x}(-x_{0}) \\varphi_{0}(x) \\tag{10}\n\\end{equation*}\n\n\nwhere $x_{0}=q \\mathscr{E} / m \\omega^{2}$. If we express the displacement operator $D_{x}(-x_{0})$ in terms of the ladder operators,\n\n\\begin{equation*}\nD_{x}(-x_{0})=\\mathrm{e}^{\\mathrm{i} x_{0} p / \\hbar}=\\mathrm{e}^{-a_{0}(a^{+}-a)} \\tag{11}\n\\end{equation*}\n\n\nUtilizing Glauber's formula\n\n$\\mathrm{e}^{A+B}=\\mathrm{e}^{A} \\mathrm{e}^{B} \\mathrm{e}^{-\\frac{1}{2}[A, B]}$\n\n\nWe obtain\n\n\\begin{align*}\nD_{x}(-x_{0}) & =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0} a^{+}} \\mathrm{e}^{\\alpha_{0} a} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\tag{$\\prime$}\n\\end{align*}\n\n\nwhere $\\alpha_{0}=x_{0} \\sqrt{m \\omega / 2 \\hbar}$. Substituting equation ( $11^{\\prime}$ ) into equation (10), we obtain\n\n\\begin{equation*}\n\\psi_{0}(x)=\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\varphi_{0}(x) \\tag{12}\n\\end{equation*}\n\n\nIt has been shown in item 3.4\n\n\\begin{gather*}\n(a-\\alpha_{0}) \\varphi_{0}=0 \\tag{13}\\\\\n(a^{+}-\\alpha_{0})^{n} \\varphi_{0}=\\sqrt{n!} \\varphi_{n} \\tag{14}\n\\end{gather*}\n\n\nThus,\n\n\\begin{align*}\n& \\mathrm{e}^{\\alpha_{0}(a-\\alpha_{0})} \\varphi_{0}=\\sum_{n=0}^{\\infty} \\frac{\\alpha_{0}^{n}}{n!}(a-\\alpha_{0})^{n} \\varphi_{0}=\\varphi_{0} \\tag{$\\prime$}\\\\\n& \\begin{aligned}\n\\psi_{0}(x) & =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha_{0}(a^{+}-\\alpha_{0})} \\varphi_{0} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\sum_{n} \\frac{(-\\alpha_{0})^{n}}{n!}(a^{+}-\\alpha_{0})^{n} \\varphi_{0} \\\\\n& =\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\sum_{n} \\frac{(-\\alpha_{0})^{n}}{\\sqrt{n!}} \\varphi_{n}(x)\n\\end{aligned}\n\\end{align*}\n\n\nnamely, $\\psi_{0}(x)$ is a coherent state wavefunction composed of the basis vectors $\\varphi_{n}$. Comparing equations (9) and (15), we obtain\n\n\\begin{equation*}\nC_{n}=\\frac{(-\\alpha_{0})^{n}}{\\sqrt{n!}} \\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\tag{16}\n\\end{equation*}\n\n\nSubstituting equation (16) into equation (8), and using the energy formula derived in item 3.4\n\n$E_{n}=(n+\\frac{1}{2}) \\hbar \\omega-\\frac{1}{2} m \\omega^{2} x_{0}^{2}$\n\nwe obtain\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{-\\frac{1}{2} a_{0}^{2}} \\sum_{n} \\frac{[-\\alpha(\\tau)]^{n}}{\\sqrt{n!}} \\varphi_{n}(x) \\tag{17}\n\\end{equation*}\n\n\nwhere\n\n\\begin{gather*}\n\\alpha(\\tau)=\\alpha_{0} \\mathrm{e}^{-\\mathrm{i} \\omega \\tau} \\tag{18}\\\\\n\\delta=\\frac{m \\omega^{2} x_{0}^{2} \\tau}{2 \\hbar}-\\frac{\\omega \\tau}{2} \\tag{19}\n\\end{gather*}\n\n$\\psi(x, \\tau)$ is also a coherent state wavefunction composed of the $\\varphi_{n}$ states.\nUsing equation (14), equation (17) can be expressed as\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\mathrm{e}^{-\\alpha(\\tau)(a^{+}-\\alpha_{0})} \\varphi_{0}(x) \\tag{20}\n\\end{equation*}\n\n\nFrom equation (12 ${ }^{\\prime}$ ), it follows that\n\n$\\mathrm{e}^{-\\frac{1}{2} \\alpha_{0}^{2}} \\varphi_{0}=\\mathrm{e}^{\\alpha_{0}(a^{+}-\\alpha_{0})} \\psi_{0}$\n\n\nSubstituting into equation (20), we obtain\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{[\\alpha_{0}-\\alpha(\\tau)](a^{+}-a_{0})} \\psi_{0}=\\mathrm{e}^{\\wp \\delta} \\mathrm{e}^{-\\alpha_{0}^{2}+\\alpha_{0} \\alpha(\\tau)} \\mathrm{e}^{[\\alpha_{0}-\\alpha(\\tau)] a^{+}} \\psi_{0} \\tag{21}\n\\end{equation*}\n\n\nTo express $\\psi(x, \\tau)$ as a linear superposition of the $\\psi_{n}$, we use the formula\n\n\\begin{equation*}\n(a^{+})^{n} \\psi_{0}=\\sqrt{n!} \\psi_{n} \\tag{22}\n\\end{equation*}\n\n\nThus, we have\n\n\\begin{equation*}\n\\psi(x, \\tau)=\\mathrm{e}^{\\mathrm{i} \\delta} \\mathrm{e}^{\\alpha_{0} \\alpha(\\tau)-\\alpha_{0}^{2}} \\sum_{n} \\frac{[\\alpha_{0}-\\alpha(\\tau)]^{n}}{\\sqrt{n!}} \\psi_{n}(x) \\tag{23}\n\\end{equation*}\n\n\nwhere the component fraction of the $\\psi_{n}$ state is\n\n\\begin{align*}\n|\\langle\\psi_{n} \\mid \\psi(x, \\tau)\\rangle|^{2} & =|\\mathrm{e}^{\\alpha_{0} \\alpha(\\tau)-\\alpha_{0}^{2}}|^{2} \\frac{|\\alpha_{0}-\\alpha(\\tau)|^{2 n}}{n!} \\\\\n& =\\frac{1}{n!}(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2 n} \\mathrm{e}^{-(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2}} \\tag{24}\n\\end{align*}\n\n\nIt can be easily verified that\n\n\\begin{equation*}\n\\sum_{n}|\\langle\\psi_{n} \\mid \\psi(x, \\tau)\\rangle|^{2}=1 \\tag{25}\n\\end{equation*}\n\n\nThis is a specific manifestation of the conservation of total probability.\nBy changing the external electric field duration $\\tau$, each time\n\n$\\tau=k 2 \\pi / \\omega, \\quad k=1,2,3, \\cdots$\n\n\nIn $\\psi(x, \\tau)$, the components of each excited state $(n \\geqslant 1)$ become 0, and the ground state $(\\psi_{0})$ component is 1. Each time\n\n$\\tau=(2 k+1) \\pi / \\omega, \\quad k=0,1,2, \\cdots$\n\nthe ground state component in $\\psi(x, \\tau)$ reaches a minimum value of $\\mathrm{e}^{-4 \\alpha_{0}^{2}}, \\psi_{n}$ state components are\n\n$$|f_{n}|^{2}=\\frac{(4 \\alpha_{0}^{2})^{n}}{n!} \\mathrm{e}^{-4 \\alpha_{0}^{2}}$$", + "final_answer": [ + "P_n = \\frac{1}{n!}(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2 n} \\mathrm{e}^{-(2 \\alpha_{0} \\sin \\frac{\\omega \\tau}{2})^{2}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$P_n$": "Probability that the system is in the energy eigenstate $\\psi_n$ at $t>\\tau$.", + "$n$": "Quantum number, representing the energy level.", + "$\\alpha_0$": "Dimensionless parameter related to the displacement, $\\alpha_{0}=x_{0} \\sqrt{m \\omega / 2 \\hbar}$.", + "$\\omega$": "Angular frequency of the harmonic oscillator.", + "$\\tau$": "Duration for which the uniform electric field is applied.", + "$\\mathrm{e}$": "Base of the natural logarithm." + } + }, + { + "id": 4, + "context": "", + "question": "Calculate the result of the commutator $[\\boldsymbol{p}, \\frac{1}{r}]$.", + "answer": "Using the commutator\n\n\\begin{equation*}\n[\\boldsymbol{p}, F(\\boldsymbol{r})]=-\\mathrm{i} \\hbar \\nabla F \\tag{1}\n\\end{equation*}\n\n\nWe obtain\n\n\\begin{equation*}\n[p, \\frac{1}{r}]=-\\mathrm{i} \\hbar(\\nabla \\frac{1}{r})=\\mathrm{i} \\hbar \\frac{r}{r^{3}} \\tag{2}\n\\end{equation*}\n\n\nUsing the formula (see question 4.2)\n\n$[\\boldsymbol{A} \\cdot \\boldsymbol{B}, F]=[\\boldsymbol{A}, F] \\cdot \\boldsymbol{B}+\\boldsymbol{A} \\cdot[\\boldsymbol{B}, F]$\n\n\nWe obtain\n\n\\[\n\\begin{aligned}\n{[\\boldsymbol{p}^{2}, \\frac{1}{r}] } & =[\\boldsymbol{p}, \\frac{1}{r}] \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot[\\boldsymbol{p}, \\frac{1}{r}]=\\mathrm{i} \\hbar(\\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\frac{\\boldsymbol{r}}{r^{3}}) \\\\\n& =2 \\mathrm{i} \\hbar \\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}+\\hbar^{2}(\\nabla \\cdot \\frac{\\boldsymbol{r}}{r^{3}})\n\\end{aligned}\n\\]\n\nHowever, since\n\n$\\nabla \\cdot \\frac{\\boldsymbol{r}}{r^{3}}=\\frac{1}{r^{3}} \\nabla \\cdot \\boldsymbol{r}+\\boldsymbol{r} \\cdot(\\nabla \\frac{1}{r^{3}})=\\frac{3}{r^{3}}-\\boldsymbol{r} \\cdot \\frac{3 \\boldsymbol{r}}{r^{5}}=0$\n\n\nSo\n\n\\begin{equation*}\n[\\boldsymbol{p}^{2}, \\frac{1}{r}]=2 \\mathrm{i} \\hbar \\frac{1}{r^{3}} \\boldsymbol{r} \\cdot \\boldsymbol{p}=2 \\hbar^{2} \\frac{1}{r^{2}} \\frac{\\partial}{\\partial r} \\tag{3}\n\\end{equation*}\n\n\nSubsequently, using equation (1), we get\n\n\\begin{align*}\n{[\\boldsymbol{p}, r^{2}] } & =-i \\hbar \\nabla(r^{2})=-2 i \\hbar \\boldsymbol{r} \\tag{4}\\\\\n{[\\boldsymbol{p}^{2}, r^{2}] } & =[\\boldsymbol{p}, r^{2}] \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot[\\boldsymbol{p}, r^{2}]=-2 i \\hbar(\\boldsymbol{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\boldsymbol{r}) \\\\\n& =-4 i \\hbar \\boldsymbol{r} \\cdot \\boldsymbol{p}-2 \\hbar^{2} \\nabla \\cdot \\boldsymbol{r}=-4 i \\hbar \\boldsymbol{r} \\cdot \\boldsymbol{p}-6 \\hbar^{2} \\\\\n& =-4 \\hbar^{2} r \\frac{\\partial}{\\partial r}-6 \\hbar^{2} \\tag{5}\n\\end{align*}\n\n\nFinally, using the commutator\n\n\\begin{equation*}\n[\\boldsymbol{p}^{2}, \\boldsymbol{r}]=-\\mathrm{i} \\hbar \\frac{\\partial p^{2}}{\\partial \\boldsymbol{p}}=-2 \\mathrm{i} \\hbar \\boldsymbol{p} \\tag{6}\n\\end{equation*}\n\n\nAnd equation (3), we get\n\n\\begin{align*}\n{[\\boldsymbol{p}^{2}, \\frac{\\boldsymbol{r}}{r}] } & =[\\boldsymbol{p}^{2}, \\boldsymbol{r}] \\frac{1}{r}+\\boldsymbol{r}[\\boldsymbol{p}^{2}, \\frac{1}{r}] \\\\\n& =-2 \\mathrm{i} \\hbar \\boldsymbol{p} \\frac{1}{r}+2 \\mathrm{i} \\hbar \\boldsymbol{r} \\frac{1}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p}) \\\\\n& =-2 \\mathrm{i} \\hbar[\\frac{1}{r} \\boldsymbol{p}-\\mathrm{i} \\hbar(\\nabla \\frac{1}{r})]+2 \\mathrm{i} \\hbar \\frac{\\boldsymbol{r}}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p}) \\\\\n& =2 \\hbar^{2} \\frac{\\boldsymbol{r}}{r^{3}}+2 \\mathrm{i} \\hbar[\\frac{\\boldsymbol{r}}{r^{3}}(\\boldsymbol{r} \\cdot \\boldsymbol{p})-\\frac{1}{r} \\boldsymbol{p}] \\\\\n& =2 \\hbar^{2}(\\frac{\\boldsymbol{r}}{r^{3}}+\\boldsymbol{r} \\frac{1}{r^{2}} \\frac{\\partial}{\\partial r}-\\frac{1}{r} \\nabla) \\tag{7}\n\\end{align*}", + "final_answer": [ + "\\mathrm{i} \\hbar \\frac{\\boldsymbol{r}}{r^{3}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mathrm{i}$": "Imaginary unit.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\boldsymbol{r}$": "Position vector.", + "$r$": "Magnitude of the position vector $\\boldsymbol{r}$." + } + }, + { + "id": 5, + "context": "", + "question": "For the hydrogen-like ion (nuclear charge $Z e$) with the $(H, l^{2}, l_{z})$ common eigenstate $\\psi_{n l m}$, it is known that the various $\\langle r^{\\lambda}\\rangle$ satisfy the following recursion relation (Kramers' formula):\n\n\\begin{equation*}\n\\frac{\\lambda+1}{n^{2}}\\langle r^{\\lambda}\\rangle-(2 \\lambda+1) \\frac{a_{0}}{Z}\\langle r^{\\lambda-1}\\rangle+\\frac{\\lambda}{4}[(2 l+1)^{2}-\\lambda^{2}] \\frac{a_{0}^{2}}{Z^{2}}\\langle r^{\\lambda-2}\\rangle=0 \n\\end{equation*}\n\n\nIt is known that $\\langle r^{0}\\rangle=1$. Use this formula to calculate the expression for $\\langle r\\rangle_{n l m}$.", + "answer": "The spherical coordinate expression of $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\psi_{n l m}=R_{n l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{1}{r} u_{n l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi) \\tag{2}\n\\end{equation*}\n\nThe expectation value of $r^{\\lambda}$ is\n\n\\begin{equation*}\n\\langle r^{\\lambda}\\rangle_{n l m}=\\int r^{\\lambda}|\\psi_{n l m}|^{2} \\mathrm{~d}^{3} x=\\int_{0}^{\\infty} r^{\\lambda}(u_{n l})^{2} \\mathrm{~d} r \\tag{3}\n\\end{equation*}\n\n$u_{n l}$ satisfies the radial equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r}] u=E_{n} u \\tag{4}\n\\end{equation*}\n\n\nBecause\n\n\\begin{equation*}\nE_{n}=-\\frac{Z^{2} e^{2}}{2 n^{2} a_{0}}, \\quad a_{0}=\\frac{\\hbar^{2}}{\\mu e^{2}} \\tag{5}\n\\end{equation*}\n\n\nEquation (4) can be rewritten as\n\n\\begin{equation*}\nu^{\\prime \\prime}+[\\frac{2 Z}{a_{0} r}-\\frac{l(l+1)}{r^{2}}-(\\frac{Z}{n a_{0}})^{2}] u=0 \\tag{$\\prime$}\n\\end{equation*}\n\n\nMultiply each term of equation ($4^{\\prime}$) by $r^{\\lambda} u$ and integrate $\\int_{0}^{\\infty} \\cdots \\mathrm{d} r$, the last three terms clearly yield the expectation values of $r^{\\lambda-1}, ~ r^{\\lambda-2}$ and $r^{\\lambda}$, while\n\nThe first term gives\n\n\\begin{align*}\n\\int_{0}^{\\infty} r^{\\lambda} u u^{\\prime \\prime} \\mathrm{d} r & =r^{\\lambda} u u^{\\prime}|_{0} ^{\\infty}-\\int_{0}^{\\infty}(r^{\\lambda} u^{\\prime}+\\lambda r^{\\lambda-1} u) u^{\\prime} \\mathrm{d} r \\\\\n& =(r^{\\lambda} u u^{\\prime}-\\frac{\\lambda}{2} r^{\\lambda-1} u^{2})|_{0} ^{\\infty}+\\frac{\\lambda(\\lambda-1)}{2}\\langle r^{\\lambda-2}\\rangle-\\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{6}\n\\end{align*}\n\n\nIf the value of $\\lambda$ ensures that\n\n\\begin{equation*}\nr^{\\lambda} u u^{\\prime}|_{0} ^{\\infty}=0,\\quad r^{\\lambda-1} u^{2}|_{0} ^{\\infty}=0 \\tag{7}\n\\end{equation*}\n\n\nWe obtain the following preliminary result:\n\n\\begin{equation*}\n[\\frac{\\lambda(\\lambda-1)}{2}-l(l+1)]\\langle r^{\\lambda-2}\\rangle+\\frac{2 Z}{a_{0}}\\langle r^{\\lambda-1}\\rangle-(\\frac{Z}{n a_{0}})^{2}\\langle r^{\\lambda}\\rangle=\\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{8}\n\\end{equation*}\n\n\nMoreover, multiply each term of equation ($4^{\\prime}$) by $2 r^{\\lambda+1} u^{\\prime}$ and integrate, successively obtaining\n\n\\[\n\\begin{aligned}\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} u^{\\prime \\prime} \\mathrm{d} r=r^{\\lambda+1}(u^{\\prime})^{2}|_{0} ^{\\infty}-\\int_{0}^{\\infty}(\\lambda+1) r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} u \\mathrm{~d} r=r^{\\lambda+1} u^{2}|_{0} ^{\\infty}-(\\lambda+1)\\langle r^{\\lambda}\\rangle \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} \\frac{u}{r} \\mathrm{~d} r=r^{\\lambda} u^{2}|_{0} ^{\\infty}-\\lambda\\langle r^{\\lambda-1}\\rangle \\\\\n& \\int_{0}^{\\infty} 2 r^{\\lambda+1} u^{\\prime} \\frac{u}{r^{2}} \\mathrm{~d} r=r^{\\lambda-1} u^{2}|_{0} ^{\\infty}-(\\lambda-1)\\langle r^{\\lambda-2}\\rangle\n\\end{aligned}\n\\]\n\n\nUnder the conditions guaranteed by equation (7), all first terms in the above equations are zero, thus\n\n\\begin{equation*}\n(\\lambda-1) l(l+1)\\langle r^{\\lambda-2}\\rangle-2 \\lambda \\frac{Z}{a_{0}}\\langle r^{\\lambda-1}\\rangle+(\\lambda+1)(\\frac{Z}{n a_{0}})^{2}\\langle r^{\\lambda}\\rangle=(\\lambda+1) \\int_{0}^{\\infty} r^{\\lambda}(u^{\\prime})^{2} \\mathrm{~d} r \\tag{9}\n\\end{equation*}\n\n\nCombine equations (8) and (9), eliminating the integrals on the right to obtain equation (1). The condition for the validity of equation (1) is equation (7). Given ${ }^{(1)}$\n\n\\begin{align*}\n& r \\rightarrow 0, \\quad u \\sim r^{l+1} \\\\\n& r \\rightarrow \\infty, \\quad u \\sim r^{n} \\mathrm{e}^{-Z r / n a_{0}} \\tag{10}\n\\end{align*}\n\n\nIt is evident that the necessary and sufficient condition for equation (7) to hold is\n\n\\begin{equation*}\n\\lambda>-(2 l+1) \\tag{11}\n\\end{equation*}\n\n\nIn equation (1), taking $\\lambda=0$ and noting $\\langle r^{0}\\rangle=1$ immediately yields\n\n\\begin{equation*}\n\\langle\\frac{1}{r}\\rangle_{n l m}=\\frac{Z}{n^{2} a_{0}} \\tag{12}\n\\end{equation*}\n\n\n\\footnotetext{\n(1) Refer to Zeng Jin-Yan. Quantum Mechanics Volume I. Beijing: Science Press, 1997. §6.3.\n}\n\nThis result has been obtained using the virial theorem.\nSequentially taking $\\lambda=1, 2$, leads to\n\n\\begin{gather*}\n\\langle r\\rangle_{n l m}=\\frac{1}{2}[3 n^{2}-l(l+1)] \\frac{a_{0}}{Z} \\tag{13}\\\\\n\\langle r^{2}\\rangle_{n l m}=\\frac{n^{2}}{2}[1+5 n^{2}-3 l(l+1)](\\frac{a_{0}}{Z})^{2} \\tag{14}\n\\end{gather*}\n\n\nFor example\n\\[\n\\begin{array}{rc}\n1 \\mathrm{~s} \\text { state (ground state), } & \\langle r\\rangle_{100}=\\frac{3}{2},\\langle r^{2}\\rangle_{100}=3 \\\\\n2 \\mathrm{~s} \\text { state, } & \\langle r\\rangle_{200}=6,\\langle r^{2}\\rangle_{200}=42 \\tag{15}\\\\\n2 \\mathrm{p} \\text { state, } & \\langle r\\rangle_{21 m}=5,\\langle r^{2}\\rangle_{21 m}=30\n\\end{array}\n\\]\n\nIn equation (15), $\\langle r\\rangle$ is measured in units of $a_{0} / Z$ and $\\langle r^{2}\\rangle$ is measured in units of $a_{0}^{2} / Z^{2}$.\nNote that in the context of this problem, equation (1) cannot be used to calculate $\\langle r^{-2}\\rangle$, but if the result from the previous problem on $\\langle r^{-2}\\rangle$ is used, then by substituting $\\lambda=-1$ into equation (1), $\\langle r^{-3}\\rangle$ can be calculated, with results consistent with the previous problem. Furthermore, by taking $\\lambda=-2(l \\geqslant 1)$, it is possible to calculate\n\n\\begin{equation*}\n\\langle r^{-4}\\rangle=(\\frac{Z}{a_{0}})^{4} \\frac{3 n^{2}-l(l+1)}{2 n^{5}(l-\\frac{1}{2}) l(l+\\frac{1}{2})(l+1)(l+\\frac{3}{2})} \\tag{16}\n\\end{equation*}\n\n\nCalculating other expectations $\\langle r^{\\lambda}\\rangle$ can follow this analogy ${ }^{(1)}$.", + "final_answer": [ + "\\langle r\\rangle_{n l m}=\\frac{1}{2}[3 n^{2}-l(l+1)] \\frac{a_{0}}{Z}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\langle r\\rangle_{nlm}$": "Expectation value of the radial coordinate $r$ for the state $\\psi_{nlm}$.", + "$n$": "Principal quantum number.", + "$l$": "Azimuthal (orbital) quantum number.", + "$a_0$": "Bohr radius, defined as $a_0 = \\frac{\\hbar^2}{\\mu e^2}$.", + "$Z$": "Nuclear charge number of the hydrogen-like ion." + } + }, + { + "id": 6, + "context": "", + "question": "Three-dimensional isotropic harmonic oscillator, the total energy operator is\n\n\\begin{equation*}\nH=\\frac{\\boldsymbol{p}^{2}}{2 \\mu}+\\frac{1}{2} \\mu \\omega^{2} r^{2}=-\\frac{\\hbar^{2}}{2 \\mu} \\nabla^{2}+\\frac{1}{2} \\mu \\omega^{2} r^{2} \n\\end{equation*}\n\n\nFor the common eigenstates of $(H, l^{2}, l_{z})$\n\n\\begin{equation*}\n\\psi_{n_{r} l m}=R_{n_{r} l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{1}{r} u_{n_{r} l}(r) \\mathrm{Y}_{l m}(\\theta, \\varphi) \n\\end{equation*}\n\nCalculate the expression for $\\langle r^{-2}\\rangle_{n_{r} l m}$.", + "answer": "The energy levels of the three-dimensional isotropic harmonic oscillator are\n\n\\begin{equation*}\nE_{n, l m}=E_{N}=\\left(N+\\frac{3}{2}\\right) \\hbar \\omega, \\quad N=l+2 n_{r} . \\tag{3}\n\\end{equation*}\n\n\nFor $\\psi_{n_{r}, l m}, H$ is equivalent to\n\n\\footnotetext{\n(1) Refer to H. A. Kramers. Quantum Mechanics. Amsterdam: North-Holland, 1958. § 59.\n}\n\n\\begin{equation*}\nH \\rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r+l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}+\\frac{1}{2} \\mu \\omega^{2} r^{2} \\tag{4}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, there should be\n\n\\begin{equation*}\n\\frac{\\partial E_{n_{l} l m}}{\\partial l}=\\langle\\frac{\\partial H}{\\partial l}\\rangle_{n_{r} l m}=(l+\\frac{1}{2}) \\frac{\\hbar^{2}}{\\mu}\\langle\\frac{1}{r^{2}}\\rangle_{n_{r}, l m} \\tag{5}\n\\end{equation*}\n\n\nFrom equation (3) it is evident\n\n\\begin{equation*}\n\\frac{\\partial E_{n, l m}}{\\partial l}=\\frac{\\partial E_{N}}{\\partial N}=\\hbar \\omega \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (5), we get\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}=\\frac{1}{l+\\frac{1}{2}} \\frac{\\mu \\omega}{\\hbar}=\\frac{\\alpha^{2}}{l+\\frac{1}{2}}, \\quad \\alpha=\\sqrt{\\frac{\\mu \\omega}{\\hbar}} \\tag{7}\n\\end{equation*}\n\n\nThe average value of the centrifugal potential energy is\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r}, m}=\\frac{l(l+1)}{2 l+1} \\hbar \\omega \\tag{8}\n\\end{equation*}\n\n\nNote that the average value of the centrifugal potential energy is directly determined by the angular quantum number $l$ and is independent of the principal quantum number $N$. Among the states with the same energy level $E_{N}$, the centrifugal potential energy is highest in the state $l=N$.\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{l=N}=\\frac{N(N+1)}{2 N+1} \\hbar \\omega \\tag{9}\n\\end{equation*}\n\n\nThe corresponding radial kinetic energy is only\n\n\\begin{align*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{l=N} & =\\langle\\frac{\\boldsymbol{p}^{2}}{2 \\mu}-\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{l=N}=\\frac{E_{N}}{2}-\\frac{N(N+1)}{2 N+1} \\hbar \\omega \\\\\n& =(1+\\frac{1}{4 N+2}) \\frac{\\hbar \\omega}{2} \\tag{10}\n\\end{align*}\n\n\nWhen $N \\gg 1$, it follows\n\n\\begin{equation*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{l=N} \\approx \\frac{1}{2} \\hbar \\omega \\tag{$\\prime$}\n\\end{equation*}\n\n\nThis situation corresponds to the circular orbit in Bohr's quantum theory. The general formula for the average value of radial kinetic energy is\n\n\\begin{equation*}\n\\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{n_{r} l m}=\\frac{E_{N}}{2}-\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r} l m}=[N+\\frac{3}{2}-\\frac{l(l+1)}{l+1 / 2}] \\frac{\\hbar \\omega}{2} \\tag{11}\n\\end{equation*}\n\n\nIf $n_{r} \\ggg 1$ (quasi-classical case), the approximate treatment can be made as follows:\n\n$$\\frac{l(l+1)}{l+\\frac{1}{2}} \\approx l+\\frac{1}{2}$$\n\n\nEquations (8) and (11) become\n\n\\begin{align*}\n& \\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n_{r} l m} \\approx(l+\\frac{1}{2}) \\frac{\\hbar \\omega}{2}=(\\frac{l}{2}+\\frac{1}{4}) \\hbar \\omega \\tag{$\\prime$}\\\\\n& \\langle\\frac{p_{r}^{2}}{2 \\mu}\\rangle_{n_{r} l m} \\approx(N+1-l) \\frac{\\hbar \\omega}{2}=(n_{r}+\\frac{1}{2}) \\hbar \\omega \\tag{11'}\n\\end{align*}", + "final_answer": [ + "\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}=\\frac{1}{l+\\frac{1}{2}} \\frac{\\mu \\omega}{\\hbar}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}$": "Expectation value of $1/r^2$ for the state $\\psi_{n_r l m}$ (where `n` is the radial quantum number)", + "$l$": "Orbital angular momentum quantum number", + "$\\mu$": "Reduced mass", + "$\\omega$": "Angular frequency of the harmonic oscillator", + "$\\hbar$": "Reduced Planck's constant" + } + }, + { + "id": 7, + "context": "", + "question": "A particle with mass $\\mu$ moves in a \"spherical square well\" potential, $$V(r)= \\begin{cases}0, & r0, & r \\geqslant a.\\end{cases}$$ Consider only the bound state $(0a(\\text { outside the well }) \\tag{4b} \\end{align*} \nThe equation inside the well (4a) is precisely the spherical Bessel equation, and the physically allowed solution is the spherical Bessel function \n\\begin{equation*} R(r)=j_{l}(k_{0} r), \\quad k_{0}=\\sqrt{2 \\mu V_{0}} / \\hbar \\tag{5a} \\end{equation*} \nThe solution outside the well (4b) is \n\\begin{equation*} R(r)=C / r^{l+1} \\tag{5b} \\end{equation*} \n(The other solution $r^{l}$ does not satisfy the bound state boundary condition $R \\rightarrow 0$ as $r \\rightarrow \\infty$, so it is discarded.) For the wave function outside the well (5b), it is evident that \n\\begin{equation*} \\frac{\\mathrm{d}}{\\mathrm{~d} r}[r^{l+1} R(r)]=0, \\quad r \\geqslant a \\tag{6} \\end{equation*} \nAt $r=a$, $R$ and $R^{\\prime}$ should both be continuous. Therefore, for the wave function inside the well (5a), as $r \\rightarrow a$, it should also satisfy condition (6), that is, \n\\begin{equation*} \\frac{\\mathrm{d}}{\\mathrm{~d} r}[r^{l+1} j_{l}(k_{0} r)]_{r=a}=0 \\tag{$\\prime$} \\end{equation*} \nUsing the formula \n$\\frac{\\mathrm{d}}{\\mathrm{~d} x}[x^{l+1} j_{l}(x)]=x^{l+1} j_{l-1}(x)$ Equation $(6^{\\prime})$ \ncan be transformed into \n\\begin{equation*} j_{l-1}(k_{0} a)=0 \\tag{7} \\end{equation*} \nThis is the condition for a new bound state (angular quantum number $l$, energy level $E_{n l} \\approx V_{0}$) to appear. For the first bound state $l=0$, considering that $j_{-1}(k_{0} a)=\\frac{\\cos k_{0} a}{k_{0} a}$ The condition for the appearance of a new s-state $(l=0)$ energy level $(E \\approx V_{0})$ is \\begin{equation*} \\cos k_{0} a=0, \\quad k_{0} a=\\frac{\\pi}{2}, \\frac{3 \\pi}{2}, \\frac{5 \\pi}{2}, \\cdots \\tag{8} \\end{equation*} When the first bound state appears, $k_{0} a=\\pi / 2$, which means \\begin{equation*} V_{0} a^{2}=\\frac{\\pi^{2} \\hbar^{2}}{8 \\mu} \\tag{9} \\end{equation*} As $V_{0}$ gradually increases, whenever condition (7) is satisfied, a new energy level $E_{n l} \\approx V_{0}$ appears. The order of appearance of each energy level can be determined based on the zeros of the spherical Bessel function $j_{l}(x)$:\n\n\\[\n\\begin{aligned}\n&1~\\mathrm{s},\\quad 1~\\mathrm{p},\\quad 1~\\mathrm{d},\\quad 2~\\mathrm{s},\\quad 1~\\mathrm{f},\\quad 2~\\mathrm{p},\\quad 1~\\mathrm{g},\\quad 2~\\mathrm{d},\\quad 3~\\mathrm{s},\\\\\n&1~\\mathrm{h},\\quad 2~\\mathrm{f},\\quad 1~\\mathrm{i},\\quad 3~\\mathrm{p},\\quad 2~\\mathrm{g},\\quad 1~\\mathrm{k},\\quad 3~\\mathrm{d},\\quad 4~\\mathrm{s},\\ \\cdots\n\\end{aligned}\n\\]\n\nThe $l$ values corresponding to each spectral notation are:\n\n\\[\n\\begin{array}{ccccccccc}\nl & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\n\\text{Letter} & \\mathrm{s} & \\mathrm{p} & \\mathrm{d} & \\mathrm{f} & \\mathrm{g} & \\mathrm{h} & \\mathrm{i} & \\mathrm{j}\n\\end{array}\n\\]\n\nWhen $V_0$ is large, the total number of bound states can be approximated by “one bound state per $h^3$ volume in phase space.” The maximum momentum for a particle in a bound state inside the well is:\n\\begin{equation*} p_{0}=\\hbar k_{0}=\\sqrt{2 \\mu V_{0}} \\tag{10} \\end{equation*} \nTherefore, the total phase space volume occupied by the bound states is $\\frac{4 \\pi}{3} a^{3} \\cdot \\frac{4 \\pi}{3} p_{0}^{3}=\\frac{16}{9} \\pi^{2}(a \\hbar k_{0})^{3}=\\Omega$ The total number of bound states is \\begin{align*} N & \\approx \\Omega /(2 \\pi \\hbar)^{3}=\\frac{2 \\pi^{2}}{9}(\\frac{a k_{0}}{\\pi})^{3} \\\\ & =\\frac{2 \\pi^{2}}{9}(\\frac{a}{\\pi \\hbar})^{3}(2 \\mu V_{0})^{3 / 2} \\tag{11} \\end{align*} For example, when $a k_{0}=7 \\pi / 2$, the highest energy level is 4 s, and counting all the states from 1 s to 4 s (consider the degeneracy of energy levels $E_{n l}$ as $2 l+1$) gives 99, while equation (11) yields $N \\approx 94$, which is indeed very close ${ }^{(1)}$.", + "final_answer": [ + "V_{0} a^{2}=\\frac{\\pi^{2} \\hbar^{2}}{8 \\mu}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$V_0$": "Height of the potential barrier", + "$a$": "Radius of the spherical square well", + "$\\hbar$": "Reduced Planck's constant", + "$\\mu$": "Mass of the particle" + } + }, + { + "id": 8, + "context": "", + "question": "For the common eigenstate $|l m\\rangle$ of $l^{2}$ and $l_{z}$, calculate the expectation value $\\overline{l_{n}}$ of $l_{n}=\\boldsymbol{n} \\cdot \\boldsymbol{l}$. Here, $\\boldsymbol{n}$ is a unit vector in an arbitrary direction, and its angle with the $z$-axis is $\\gamma$.", + "answer": "Using the basic commutation relation $\\boldsymbol{l} \\times \\boldsymbol{l}=\\mathrm{i} \\hbar \\boldsymbol{l}$, we have\n\\[\n\\begin{aligned}\n& \\mathrm{i} \\hbar l_{x} l_{y}=(l_{y} l_{z}-l_{z} l_{y}) l_{y}=l_{y} l_{z} l_{y}-l_{z} l_{y}^{2} y \\\\\n& \\mathrm{i} \\hbar l_{y} l_{x}=l_{y}(l_{y} l_{z}-l_{z} l_{y})=l_{y}^{2} l_{z}-l_{y} l_{z} l_{y}\n\\end{aligned}\n\\]\n\nCalculating the expectation value in the state $|l m\\rangle$, since\n\n$\\overline{l_{z} l_{y}^{2}}=\\overline{l_{y}^{2} l_{z}}=m \\hbar \\overline{l_{y}^{2}}$\n\n\nTherefore\n\n$\\overline{l_{y} l_{x}}=-\\overline{l_{x} l_{y}}$\n\n\nThus\n\n\\begin{align*}\n& \\overline{l_{x} l_{y}}-\\overline{l_{y} l_{x}}=2 \\overline{l_{x} l_{y}}=\\mathrm{i} \\hbar \\overline{l_{z}}=\\mathrm{i} \\hbar^{2} m \\\\\n& \\overline{l_{x} l_{y}}=\\mathrm{i}^{2} m / 2, \\quad \\overline{l_{y} l_{x}}=-\\mathrm{i}^{2} m / 2 \\tag{1}\n\\end{align*}\n\nThe projection operator of $\\boldsymbol{l}$ in the direction of $\\boldsymbol{n}$ is\n\n$l_{n}=\\boldsymbol{n} \\cdot \\boldsymbol{l}=l_{x} \\cos \\alpha+l_{y} \\cos \\beta+l_{z} \\cos \\gamma$\n\n\nFor the state $|l m\\rangle$, since $\\overline{l_{x}}=0, \\overline{l_{y}}=0$, we have\n\n\\begin{gather*}\n\\overline{l_{n}}=m \\hbar \\cos \\gamma \\tag{2}\\\\\nl_{n}^{2}=l_{x}^{2} \\cos ^{2} \\alpha+l_{y}^{2} \\cos ^{2} \\beta+l_{z}^{2} \\cos ^{2} \\gamma \\\\\n\\\\\n+(l_{x} l_{y}+l_{y} l_{x}) \\cos \\alpha \\cos \\beta+\\cdots \\text { (similar terms with cyclic permutations) }\n\\end{gather*}\n\n\nCalculating the expectation value in the state $|l m\\rangle$, since\n\n\\begin{gather*}\n\\overline{l_{x} l_{y}}+\\overline{l_{y} l_{x}}=0, \\quad \\overline{l_{y} l_{z}}+\\overline{l_{z} l_{y}}=2 m \\hbar \\overline{l_{y}}=0, \\cdots \\\\\n\\overline{l_{x}^{2}}=\\overline{l_{y}^{2}}=\\frac{1}{2}(\\overline{l^{2}}-\\overline{l_{z}^{2}})=\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}} \\tag{3}\n\\end{gather*}\n\n\nThus\n\n\\begin{align*}\n\\overline{l_{n}^{2}} & =m^{2} \\hbar^{2} \\cos ^{2} \\gamma+\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}}(\\cos ^{2} \\alpha+\\cos ^{2} \\beta) \\\\\n& =m^{2} \\hbar^{2} \\cos ^{2} \\gamma+\\frac{\\hbar^{2}}{2}{l(l+1)-m^{2}}(1-\\cos ^{2} \\gamma) \\\\\n& =\\frac{\\hbar^{2}}{2}{l(l+1)(1-\\cos ^{2} \\gamma)+m^{2}(3 \\cos ^{2} \\gamma-1)} \\tag{4}\n\\end{align*}\n\n\nIf $\\boldsymbol{n}$ is orthogonal to the $z$-axis, that is, $\\cos \\gamma=0$, then $\\overline{l_{n}^{2}}=\\overline{l_{x}^{2}}=\\overline{l_{y}^{2}}, \\overline{l_{n}}=0$.", + "final_answer": [ + "m \\hbar \\cos \\gamma" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$m$": "Magnetic quantum number, associated with the z-component of angular momentum.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\gamma$": "Angle of the unit vector $\\boldsymbol{n}$ with the $z$-axis." + } + }, + { + "id": 9, + "context": "", + "question": "For an electron's spin state $\\chi_{\\frac{1}{2}}(\\sigma_{z}=1)$ (i.e., the state where the Pauli matrix $\\sigma_z$ has an eigenvalue of $+1$), if we measure its spin projection in an arbitrary direction $\\boldsymbol{n}$, $\\sigma_n = \\boldsymbol{\\sigma} \\cdot \\boldsymbol{n}$, where $\\boldsymbol{n}$ is a unit vector and $n_z$ is its component along the $z$-axis. Find the expression for the probability of measuring $\\sigma_n = +1$ (expressed in terms of $n_z$).", + "answer": "One Using the eigenfunctions of $\\sigma_{n}$ obtained from the previous problem, it is easy to find\n(a) In the spin state $\\chi_{\\frac{1}{2}}=[\\begin{array}{l}1 \\\\ 0\\end{array}]$,\nthe probability of $\\sigma_{n}=1$ is\n\n\\begin{equation*}\n|\\langle\\phi_{1} \\lvert\\, \\chi_{\\frac{1}{2}}\\rangle|^{2}=\\cos ^{2} \\frac{\\theta}{2}=\\frac{1}{2}(1+n_{z}) \\tag{1}\n\\end{equation*}\n\nthe probability of $\\sigma_{n}=-1$ is\n\n\\begin{equation*}\n|\\langle\\phi_{-1} \\lvert\\, \\chi_{\\frac{1}{2}}\\rangle|^{2}=\\sin ^{2} \\frac{\\theta}{2}=\\frac{1}{2}(1-n_{z}) \\tag{2}\n\\end{equation*}\n\n(b) In the spin state $\\phi_{1}(\\sigma_{n}=1)$,\nthe probability of $\\sigma_{z}=1$ is\n\n\\begin{equation*}\n|\\langle\\chi_{\\frac{1}{2}} \\rvert\\, \\phi_{1}\\rangle|^{2}=\\frac{1}{2}(1+n_{z}) \\tag{3}\n\\end{equation*}\n\nthe probability of $\\sigma_{z}=-1$ is\n\n\\begin{equation*}\n1-\\frac{1}{2}(1+n_{z})=\\frac{1}{2}(1-n_{z}) \\tag{4}\n\\end{equation*}\n\n\n\\begin{equation*}\n\\langle\\sigma_{z}\\rangle=\\frac{1}{2}(1+n_{z})-\\frac{1}{2}(1-n_{z})=n_{z} \\tag{5}\n\\end{equation*}\n\nConsidering\n\n$\\sigma_{n}=\\sigma_{x} n_{x}+\\sigma_{y} n_{y}+\\sigma_{z} n_{z}$\n\nThe components of $\\boldsymbol{\\sigma}$ and $\\boldsymbol{n}$ have symmetric roles in the construction of $\\sigma_{n}$, so using formulas (3), (4), (5), and cyclically permuting $x, ~ y$, $z$, the following can be deduced:\n\n\\begin{gather*}\n\\sigma_{x}= \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{x}) \\tag{6}\\\\\n\\langle\\sigma_{x}\\rangle=n_{x} \\tag{7}\\\\\n\\sigma_{y}= \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{y}) \\tag{8}\\\\\n\\langle\\sigma_{y}\\rangle=n_{y} \\tag{9}\n\\end{gather*}\n\nBy combining equations (5), (7), (9) in vector form as follows:\nIn the spin state $\\phi_{1}(\\sigma_{n}=1)$,\n\n\\begin{equation*}\n\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{10}\n\\end{equation*}\n\nSimilarly, it is easy to calculate\nIn the spin state $\\phi_{-1}(\\sigma_{n}=-1)$,\n\n\\begin{equation*}\n\\langle\\boldsymbol{\\sigma}\\rangle=-\\boldsymbol{n} \\tag{11}\n\\end{equation*}\n\nSolution two (a) In the spin state $\\chi_{\\frac{1}{2}}$ where $\\sigma_{z}=1$, the possible measured values of $\\sigma_{n}$ are the eigenvalues $\\pm 1$; let the corresponding probabilities be $w_{+}$ and $w_{-}$, then\n\n\\begin{equation*}\n\\langle\\sigma_{n}\\rangle=w_{+} \\times 1+w_{-} \\times(-1)=w_{+}-w_{-} \\tag{12}\n\\end{equation*}\n\nSince\n\n\\begin{equation*}\n\\sigma_{n}=\\sigma_{x} n_{x}+\\sigma_{y} n_{y}+\\sigma_{z} n_{z} \\tag{13}\n\\end{equation*}\n\nConsidering that in the eigenstate of $\\sigma_{z}$ the average value of $\\sigma_{x}$ and $\\sigma_{y}$ is zero, and the average value of $\\sigma_{z}$ is the eigenvalue, hence in the state $\\chi_{\\frac{1}{2}}$,\n\n\\begin{equation*}\n\\langle\\sigma_{n}\\rangle=\\langle\\sigma_{z}\\rangle n_{z}=n_{z}=\\cos \\theta \\tag{14}\n\\end{equation*}\n\nFrom equations (12), (14), and using $w_{+}+w_{-}=1$, it can be found that\n\n\\begin{equation*}\nw_{+}=\\frac{1}{2}(1+n_{z}), \\quad w_{-}=\\frac{1}{2}(1-n_{z}) \\tag{15}\n\\end{equation*}\n\nThese are the equations (1), (2) in solution one.\n(b) In equation (14), $\\theta$ is the angle parameter in the $z$-axis and $\\boldsymbol{n}$. The choices of the $z$-axis and $\\boldsymbol{n}$ are arbitrary, and the original $z$-axis can be taken as the new $\\boldsymbol{n}$, while the original $\\boldsymbol{n}$ is taken as the new $z$-axis. Thus, it can be known that in the spin state where $\\sigma_{n}=1$\n\nThe average value of $\\sigma_{z}$ remains $\\cos \\theta$, which is $n_{z}$. By letting $x, ~ y, ~ z$ permute, we obtain\n\n\\begin{equation*}\n\\text { In the spin state } \\phi_{1}(\\sigma_{n}=1) \\text {, }\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{10}\n\\end{equation*}\n\n\nIn the state $\\phi_{1}$, the values of each component of $\\boldsymbol{\\sigma}$ are of course all $\\pm 1$, and their probabilities can be written similarly to those in (a), thus\n\n\\begin{align*}\n\\sigma_{x} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{x}) \\tag{6}\\\\\n\\sigma_{y} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{y}) \\tag{8}\\\\\n\\sigma_{z} & = \\pm 1 \\text { with probability } \\frac{1}{2}(1 \\pm n_{z}) \\tag{3,4}\n\\end{align*}", + "final_answer": [ + "\\frac{1}{2}(1+n_z)" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$n_z$": "Component of the unit vector $\\boldsymbol{n}$ along the $z$-axis." + } + }, + { + "id": 10, + "context": "", + "question": "Express the operator $(I+\\sigma_{x})^{1 / 2}$ (where $\\sigma_x$ is the Pauli matrix) as a linear combination of the $2 \\times 2$ identity matrix (denoted as $1$ in the expression) and $\\sigma_x$. The operation takes the principal square root.", + "answer": "(a) $(I+\\sigma_{x})^{I / 2}$. The eigenvalues of $\\sigma_{x}$ are $\\pm 1$, and for each eigenvalue, $(I+\\sigma_{x})^{1 / 2}$ gives a clear value (principal root is taken), so it can be concluded that $(I+\\sigma_{x})^{1 / 2}$ exists, and it is a function of $\\sigma_{x}$. According to the argument in problem 6.14, we can set\n\n\\begin{equation*}\n(I+\\sigma_{x})^{1 / 2}=C_{0}+C_{1} \\sigma_{x} \\tag{1}\n\\end{equation*}\n\n\nSquaring this equation, we get\n\n$I+\\sigma_{x}=(C_{0}+C_{1} \\sigma_{x})^{2}=C_{0}^{2}+C_{1}^{2}+2 C_{0} C_{1} \\sigma_{x}$\n\n\nTherefore\n\\[\n\\begin{gathered}\nC_{0}^{2}+C_{1}^{2}=1 \\\\\n2 C_{0} C_{1}=1\n\\end{gathered}\n\\]\n\nAdding and subtracting the two equations, we get\n\\[\n\\begin{aligned}\n& (C_{0}+C_{1})^{2}=2 \\\\\n& (C_{0}-C_{1})^{2}=0\n\\end{aligned}\n\\]\n\nIf we set $(C_{0}+C_{1})$ to take positive value, we can solve\n\n$C_{0}=C_{1}=1 / \\sqrt{2}$\n\n\nSubstituting into equation (1), we obtain\n\n\\begin{equation*}\n(1+\\sigma_{z})^{1 / 2}=\\frac{1}{\\sqrt{2}}(I+\\sigma_{x}) \\tag{2}\n\\end{equation*}\n\n\nIt is easy to verify that for any eigenvalue $( \\pm 1)$ of $\\sigma_{x}$, this equation holds true.", + "final_answer": [ + "\\frac{1}{\\sqrt{2}}(1+\\sigma_x)" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$1$": "The $2 \\times 2$ identity matrix", + "$\\sigma_x$": "Pauli matrix" + } + }, + { + "id": 11, + "context": "", + "question": "For a spin $1/2$ particle, $\\langle\\boldsymbol{\\sigma}\\rangle$ is often called the polarization vector, denoted as $\\boldsymbol{P}$, which is the spatial orientation of the spin angular momentum. Given the initial spin wave function at $t=0$ (in the $\\sigma_{z}$ representation) as:\n\n\\chi(0)=[\\begin{array}{c}\n\\cos \\delta \\mathrm{e}^{-\\mathrm{i} \\alpha} \\\\\n\\sin \\delta \\mathrm{e}^{\\mathrm{i} \\alpha}\n\\end{array}]\n\n\nwhere $\\delta, ~ \\alpha$ are positive real numbers (or 0), $\\delta \\leqslant \\pi / 2, \\alpha \\leqslant \\pi$. Determine the azimuthal angle $\\theta_{0}$ (polar angle) of the initial polarization vector $\\boldsymbol{P}(t=0)$.", + "answer": "Any definite spin state is an eigenstate (with eigenvalue 1) of the projection $\\sigma_{n}$ of $\\boldsymbol{\\sigma}$ in some direction $(\\theta, \\varphi)$, and\n\n\\begin{equation*}\n\\boldsymbol{P}=\\langle\\boldsymbol{\\sigma}\\rangle=\\boldsymbol{n} \\tag{3}\n\\end{equation*}\n\n\nwhere $\\boldsymbol{n}$ is the unit vector in the direction $(\\theta, \\varphi)$, the eigenfunction of $\\sigma_{n}=1$ is\n\\begin{equation}\n \\phi_{1}(\\theta, \\varphi) =[\\begin{array}{l}\n\\cos \\frac{\\theta}{2} \\mathrm{e}^{-\\mathrm{i} \\varphi / 2} \\tag{4}\\\\\n\\sin \\frac{\\theta}{2} \\mathrm{e}^{\\mathrm{i} \\varphi / 2}\n\\end{array}]\n\\end{equation}\n\n\n\nBy comparing Equation (1) and (4), we find the azimuthal angle of the initial polarization vector $\\boldsymbol{P}(t=0)$\n\n\\begin{equation*}\n\\theta_{0}=2 \\delta, \\quad \\varphi_{0}=2 \\alpha \\tag{5}\n\\end{equation*}", + "final_answer": [ + "2 \\delta" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\delta$": "Angle parameter in the initial spin wave function $\\chi(0)$, a positive real number or 0." + } + }, + { + "id": 12, + "context": "", + "question": "For a system composed of two spin $1 / 2$ particles, let $s_{1}$ and $s_{2}$ denote their spin angular momentum operators. Calculate and simplify the triple product $s_{1} \\cdot (s_{1} \\times s_{2})$ (take $\\hbar=1$ ).", + "answer": "The basic relationships are as follows (the single particle formula is only written for particle 1)\n(a) $s_{1}^{2}=\\frac{3}{4}, \\boldsymbol{\\sigma}_{1}^{2}=3 ;(s_{1 x})^{2}=\\frac{1}{4},(\\sigma_{1 x})^{2}=1$, and so on.\n(b) $s_{1} \\times s_{1}=\\mathrm{i} s_{1}, \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{1}=2 \\mathrm{i} \\boldsymbol{\\sigma}_{1}$\n(c) $\\sigma_{1 x} \\sigma_{1 y}=-\\sigma_{1}, \\sigma_{1 x}=\\mathrm{i} \\sigma_{1 z}$, and so on.\n(d) $s_{1}$ commutes with $s_{2}$, $s_{1} \\cdot s_{2}=s_{2} \\cdot s_{1}, s_{1} \\times s_{2}=-s_{2} \\times s_{1}$\nFor the triple product, there are the following types:\n\n\\begin{equation*}\n1^{\\circ} \\tag{5}\n\\end{equation*}\n\n\n\\begin{align*}\ns_{1} \\cdot(s_{1} \\times s_{2}) & =(s_{1} \\times s_{1}) \\cdot s_{2}=\\mathrm{i} s_{1} \\cdot s_{2} \\\\\ns_{2} \\cdot(s_{1} \\times s_{2}) & =-s_{2} \\cdot(s_{2} \\times s_{1}) \\\\\n& =-(s_{2} \\times s_{2}) \\cdot s_{1}=-\\mathrm{i} s_{1} \\cdot s_{2} \\tag{6}\\\\\n\\boldsymbol{\\sigma}_{1} \\cdot(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =2 \\mathrm{i} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{7}\\\\\n\\boldsymbol{\\sigma}_{2} \\cdot(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =-2 \\mathrm{i} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{8}\n\\end{align*}\n\n$\\mathbf{2}^{\\circ} \\boldsymbol{\\sigma}_{1}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})$ type. Since $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ commute, using the formula proven in question 6.21\n\n\\begin{aligned}\n\\boldsymbol{\\sigma}(\\boldsymbol{\\sigma} \\cdot \\boldsymbol{A})-\\boldsymbol{A} & =\\boldsymbol{A}-(\\boldsymbol{\\sigma} \\cdot \\boldsymbol{A}) \\boldsymbol{\\sigma} \\\\\n& =\\mathrm{i} \\boldsymbol{A} \\times \\boldsymbol{\\sigma}, \\quad(\\boldsymbol{\\sigma}, \\boldsymbol{A} \\text { commute })\n\\end{aligned}\n\n\nLet $\\boldsymbol{\\sigma}$ and $\\boldsymbol{A}$ be equal to $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ respectively, and obtain\n\n\\begin{gather*}\n\\boldsymbol{\\sigma}_{1}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})=\\boldsymbol{\\sigma}_{2}-\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{9}\\\\\n\\boldsymbol{\\sigma}_{2}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})=\\boldsymbol{\\sigma}_{1}+\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{10}\\\\\n(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}) \\boldsymbol{\\sigma}_{1}=\\boldsymbol{\\sigma}_{2}+\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{11}\\\\\n(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}) \\boldsymbol{\\sigma}_{2}=\\boldsymbol{\\sigma}_{1}-\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2} \\tag{12}\\\\\ns_{1}(s_{1} \\cdot s_{2})=\\frac{1}{4} s_{2}-\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}, \\text { etc. } \\tag{13}\\\\\n{[s_{1} \\cdot s_{2}, s_{1}]=\\mathrm{i} s_{1} \\times s_{2}} \\tag{14}\\\\\n{[s_{1} \\cdot s_{2}, s_{2}]=-\\mathrm{i} s_{1} \\times s_{2}} \\tag{15}\n\\end{gather*}\n\n$3^{\\circ} s_{1} \\times(s_{1} \\times s_{2})$ type. Using the vector operator formula (see question 4.1)\n\n\\boldsymbol{A} \\times(\\boldsymbol{B} \\times \\boldsymbol{C})=\\overparen{\\boldsymbol{A} \\cdot(\\boldsymbol{B C})}-(\\boldsymbol{A} \\cdot \\boldsymbol{B}) \\boldsymbol{C}\n\n\n(\\boldsymbol{A} \\times \\boldsymbol{B}) \\times \\boldsymbol{C}=\\boldsymbol{A \\cdot ( B C )}-\\boldsymbol{A}(\\boldsymbol{B} \\cdot \\boldsymbol{C})\n\n\nHence, we obtain\n\n\\begin{align*}\ns_{1} \\times(s_{1} \\times s_{2}) & =(s_{1} \\cdot s_{2}) s_{1}-\\frac{3}{4} s_{2}=\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}-\\frac{1}{2} s_{2} \\tag{16}\\\\\n(s_{1} \\times s_{2}) \\times s_{1} & =\\frac{3}{4} s_{2}-s_{1}(s_{1} \\cdot s_{2})=\\frac{1}{2} s_{2}+\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2} \\tag{17}\\\\\ns_{2} \\times(s_{1} \\times s_{2}) & =\\frac{1}{2} s_{1}+\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2} \\tag{18}\\\\\n(s_{1} \\times s_{2}) \\times s_{2} & =\\frac{\\mathrm{i}}{2} s_{1} \\times s_{2}-\\frac{1}{2} s_{1} \\tag{19}\\\\\n\\boldsymbol{\\sigma}_{1} \\times(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}-2 \\boldsymbol{\\sigma}_{2} \\tag{20}\\\\\n(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) \\times \\boldsymbol{\\sigma}_{1} & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}+2 \\boldsymbol{\\sigma}_{2} \\tag{21}\\\\\n\\boldsymbol{\\sigma}_{2} \\times(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}+2 \\boldsymbol{\\sigma}_{1} \\tag{22}\\\\\n(\\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}) \\times \\boldsymbol{\\sigma}_{2} & =\\mathrm{i} \\boldsymbol{\\sigma}_{1} \\times \\boldsymbol{\\sigma}_{2}-2 \\boldsymbol{\\sigma}_{1} \\tag{23}\n\\end{align*}\n\n$4^{\\circ}$ Total spin $\\boldsymbol{S}=s_{1}+s_{2}$,\n\n\\begin{align*}\n& \\boldsymbol{S}^{2}=2 s_{1} \\cdot s_{2}+\\frac{3}{2} \\tag{24}\\\\\n& {[\\boldsymbol{S}^{2}, s_{1}]=2[s_{1} \\cdot s_{2}, s_{1}]=2 \\mathrm{i} s_{1} \\times s_{2}} \\tag{25}\\\\\n& {[\\boldsymbol{S}^{2}, s_{2}]=-2 \\mathrm{i} s_{1} \\times s_{2}} \\tag{26}\\\\\n& \\boldsymbol{S} \\cdot(s_{1} \\times s_{2})=(s_{1} \\times s_{2}) \\cdot \\boldsymbol{S}=0 \\tag{27}\\\\\n& \\boldsymbol{S}(s_{1} \\cdot s_{2})=(s_{1} \\cdot s_{2}) \\boldsymbol{S}=\\frac{1}{4} \\boldsymbol{S} \\tag{28}\\\\\n& \\boldsymbol{S S}^{2}=\\boldsymbol{S}^{2} \\boldsymbol{S}=2 \\boldsymbol{S} \\tag{29}\n\\end{align*}", + "final_answer": [ + "\\mathrm{i} s_{1} \\cdot s_{2}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mathrm{i}$": "Imaginary unit", + "$s_1$": "Spin angular momentum operator for particle 1", + "$s_2$": "Spin angular momentum operator for particle 2" + } + }, + { + "id": 13, + "context": "", + "question": "Two localized non-identical particles with spin $1/2$ (ignoring orbital motion) have an interaction energy given by (setting $\\hbar=1)$\n\n\\begin{equation*}\nH=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} \n\\end{equation*}\n\nAt $t=0$, particle 1 has spin 'up' $(s_{1 z}=1 / 2)$, and particle 2 has spin 'down' $(s_{2 z}=-\\frac{1}{2})$. Find the probability that particle 1 has spin 'up' $(s_{1 z}=1 / 2)$ at any time $t>0$.", + "answer": "Start by finding the spin wave function of the system. Since\n\n\\begin{equation*}\nH=A s_{1} \\cdot s_{2}=\\frac{A}{2}(\\boldsymbol{S}^{2}-\\frac{3}{2}) \\tag{$\\prime$}\n\\end{equation*}\n\n\nIt is evident that the total spin $\\boldsymbol{S}$ is a conserved quantity, so the stationary wave function can be chosen as a common eigenfunction of $\\boldsymbol{S}^{2}, ~ S_{z}$. According to different values of the total spin quantum number $S$, the eigenfunctions and energy levels are\n\n\\begin{array}{ll}\nS=1, & \\chi_{1 M_{s}}, \\quad E_{1}=A / 4 \\tag{2}\\\\\nS=0, & \\chi_{00}, \\quad E_{0}=-3 A / 4\n\\end{array}}\n\nAt $t=0$, the spin state of the system is\n\n\\begin{equation*}\n\\chi(0)=\\alpha(1) \\beta(2)=\\frac{1}{\\sqrt{2}}(\\chi_{10}+\\chi_{00}) \\tag{3}\n\\end{equation*}\n\n\nTherefore, the wave function at $t>0$ is\n\n\\begin{equation*}\n\\chi(t)=\\frac{1}{\\sqrt{2}} \\chi_{10} \\mathrm{e}^{-\\mathrm{i} E_{1} t}+\\frac{1}{\\sqrt{2}} \\chi_{00} \\mathrm{e}^{-\\mathrm{i} E_{0} t} \\tag{4}\n\\end{equation*}\n\n\nThat is\n\n\\begin{align*}\n\\chi(t) & =\\frac{1}{2}[\\alpha(1) \\beta(2)+\\beta(1) \\alpha(2)] \\mathrm{e}^{-\\mathrm{i} A / 4}+\\frac{1}{2}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)] \\mathrm{e}^{3 \\mathrm{i} t / 4} \\\\\n& =[\\alpha(1) \\beta(2) \\cos \\frac{A t}{2}-\\mathrm{i} \\beta(1) \\alpha(2) \\sin \\frac{A t}{2}] \\mathrm{e}^{\\mathrm{i} A / 4} \\tag{$4^\\prime$}\n\\end{align*}\n\nFrom formula ($4^{\\prime}$), it can be seen that at time $t$, the probability of particle 1 having spin 'up' [while particle 2 has spin 'down', corresponding to the $\\alpha(1) \\beta(2)$ term] is $\\cos ^{2}(\\frac{A t}{2})$.", + "final_answer": [ + "\\cos^{2}(\\frac{A t}{2})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$A$": "Constant in the interaction energy.", + "$t$": "Time." + } + }, + { + "id": 14, + "context": "", + "question": "Consider a system consisting of three distinguishable particles each with spin $1/2$, with the Hamiltonian given by\n\n\\begin{equation*}\nH=A(s_{1} \\cdot s_{2}+s_{2} \\cdot s_{3}+s_{3} \\cdot s_{1}) \\quad \\text { ( } A \\text { is real) } \n\\end{equation*}\n\nLet $S$ denote the total spin quantum number of the system. Determine the expression for the energy level $E_{3/2}$ when $S=3/2$ (taking $\\hbar=1$).", + "answer": "The Hamiltonian $H$ can be written as\n\n\\begin{equation*}\nH=\\frac{A}{2}(\\boldsymbol{S}_{123}^{2}-3 \\times \\frac{3}{4}) \\tag{$\\prime$}\n\\end{equation*}\n\n\nTherefore, the energy levels (taking $\\hbar=1$) are\n\n\\begin{equation*}\nE_{S}=\\frac{A}{2}[S(S+1)-\\frac{9}{4}] \\tag{2}\n\\end{equation*}\n\n\nA complete set of conserved quantities can be taken as $[S_{123}^{2}, ~(S_{123})_{z}, S_{12}^{2}]$, with eigenvalues\n\n\\begin{array}{l}\n\\boldsymbol{S}_{12}^{2}=S^{\\prime}(S^{\\prime}+1), \\quad S^{\\prime}=1,0 \\tag{3}\\\\\n\\boldsymbol{S}_{123}^{2}=S(S+1), \\quad S=\\frac{3}{2}, \\frac{1}{2}, \\frac{1}{2} \\\\\n(S_{123})_{z}=M=S, S-1, \\cdots,(-S)\n\\end{array}}\n\n\nThe possible combinations for the quantum numbers $S^{\\prime}, ~ S$ are $S=S^{\\prime} \\pm \\frac{1}{2}>0$, i.e.,\n\n\\begin{array}{l}\nS=3 / 2, \\quad S^{\\prime}=1 \\tag{4}\\\\\nS=1 / 2, \\quad S^{\\prime}=1,0\n\\end{array}}\n\n\nFor each pair $(S, S^{\\prime})$, the degeneracy of the energy level is $(2 S+1)$, so\n\n\\begin{align*}\n& E_{3 / 2}=\\frac{3}{4} A, \\quad S=\\frac{3}{2}, \\quad S^{\\prime}=1, \\quad \\text { Degeneracy }=4 \\\\\n& E_{1 / 2}=-\\frac{3}{4} A, \\quad S=\\frac{1}{2}, \\quad S^{\\prime}=1,0, \\quad \\text { Degeneracy }=4 \\tag{5}\n\\end{align*}\n\n\nWe now aim to determine the common eigenstates of $[\\boldsymbol{S}_{123}^{2},(S_{123})_{z}, \\boldsymbol{S}_{12}^{2}]$. The spin \"up\" and \"down\" states of the $k$-th particle $(k=1,2,3)$ are denoted as $\\alpha(k), ~ \\beta(k)$, corresponding to $s_{k z}=\\frac{1}{2}, ~-\\frac{1}{2}$. As is known, the common eigenstates of $S_{12}^{2}$ and $(S_{12})_{z}$ are given by:\n\n\\begin{aligned}\n& \\chi_{11}(1,2)=\\alpha(1)_{\\alpha}(2), \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=1 \\\\\n& \\chi_{10}(1,2)=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)+\\beta(1)_{\\alpha}(2)], \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=0 \\\\\n& \\chi_{1,-1}(1,2)=\\beta(1) \\beta(2), \\quad S^{\\prime}=1, \\quad(S_{12})_{z}=-1 \\\\\n& \\chi_{00}(1,2)=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)], \\quad S^{\\prime}=0, \\quad(S_{12})_{z}=0\n\\end{aligned}\n\nThe common eigenstates of $\\boldsymbol{S}_{12}^{2}, ~ \\boldsymbol{S}_{123}^{2}, ~(S_{123})_{z}$ are denoted as $\\chi_{S_{S M}}(1,2,3)$. When all three quantum numbers take their maximum values, the eigenstate is clearly\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2} \\frac{3}{2}}=\\chi_{11}(1,2)_{\\alpha}(3)=\\alpha(1)_{\\alpha}(2) \\alpha(3) \\tag{6a}\n\\end{equation*}\n\nWhen $S=3 / 2$, $M$ has 4 possible values; corresponding eigenstates for $M=\\frac{1}{2},-\\frac{1}{2},-\\frac{3}{2}$ can be obtained by repeatedly applying the ladder operator $[(S_{123})_{x}-\\mathrm{i}(S_{123})_{y}]$ to $\\chi_{1 \\frac{3}{2}} \\frac{3}{2}$. Since the ladder operator and $\\chi_{1 \\frac{3}{2} \\frac{3}{2}}$ are symmetric under permutation of the particles, each $\\chi_{1 \\frac{3}{2} M}$ obtained from this is a symmetric function. Based on this observation and considering the values of $M$, the only possible structure of these functions can be immediately written:\n\n\\begin{align*}\n\\chi_{1 \\frac{3}{2} \\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\alpha(1) \\alpha(2) \\beta(3)+\\beta(1) \\alpha(2) \\alpha(3)+\\alpha(1) \\beta(2) \\alpha(3)] \\tag{6b}\\\\\n\\chi_{1 \\frac{3}{2},-\\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\alpha(1) \\beta(2) \\beta(3)+\\beta(1) \\alpha(2) \\beta(3)+\\beta(1) \\beta(2) \\alpha(3)] \\tag{6c}\\\\\n\\chi_{1 \\frac{3}{2},-\\frac{3}{2}} & =\\beta(1) \\beta(2) \\beta(3) \\tag{6d}\n\\end{align*}\n\nWhen $S^{\\prime}=0$, the part of the system wave function concerning particles $1, ~ 2$ can only be $\\chi_{00}$, and considering the values of $M$, it can be immediately concluded that\n\n\\begin{align*}\n\\chi_{0 \\frac{1}{2} \\frac{1}{2}} & =\\chi_{00}(1,2) \\alpha(3) \\\\\n& =\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2) \\alpha(3)-\\beta(1) \\alpha(2) \\alpha(3)] \\tag{7a}\\\\\n\\chi_{0 \\frac{1}{2},-\\frac{1}{2}} & =\\chi_{00}(1,2) \\beta(3) \\\\\n& =\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2) \\beta(3)-\\beta(1) \\alpha(2) \\beta(3)] \\tag{7b}\n\\end{align*}\n\n\nNow only $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$ and $\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}$ remain to be determined. In the construction of $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$, each term should include two $\\alpha$ states and one $\\beta$ state. It must be a linear combination of $\\chi_{11}(1,2) \\beta(3)$ and $\\chi_{10}(1,2) \\alpha(3)$, and should be orthogonal to $\\chi_{1 \\frac{3}{2} \\frac{1}{2}}$ since they have different $S$ values. Expression (6b) for $\\chi_{1 \\frac{3}{2} \\frac{1}{2}}$ can be written as\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2} \\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\chi_{11}(1,2) \\beta(3)+\\sqrt{2} \\chi_{10}(1,2)_{\\alpha}(3)] \\tag{$\\prime$}\n\\end{equation*}\n\n\nThus, the construction of $\\chi_{1 \\frac{1}{2} \\frac{1}{2}}$ can only be\n\n\\begin{align*}\n\\chi_{1 \\frac{1}{2} \\frac{1}{2}} & =\\frac{1}{\\sqrt{3}}[\\sqrt{2} \\chi_{11}(1,2) \\beta(3)-\\chi_{10}(1,2) \\alpha(3)] \\\\\n& =\\frac{1}{\\sqrt{6}}[2 \\alpha(1) \\alpha(2) \\beta(3)-\\alpha(1) \\beta(2) \\alpha(3)-\\beta(1) \\alpha(2) \\beta(3)] \\tag{8a}\n\\end{align*}\n\n\nSimilarly, expression (6c) can be written as\n\n\\begin{equation*}\n\\chi_{1 \\frac{3}{2},-\\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\sqrt{2} \\chi_{10}(1,2) \\beta(3)+\\chi_{1-1}(1,2)_{\\alpha}(3)] \\tag{$\\prime$}\n\\end{equation*}\n\n$\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}$ should be formed with the same terms but orthogonal to the above expression, so\n\n\n$$\\chi_{1 \\frac{1}{2},-\\frac{1}{2}}=\\frac{1}{\\sqrt{3}}[\\chi_{\\mathrm{i} 0}(1,2) \\beta(3)-\\sqrt{2} \\chi_{1,-1}(1,2) \\alpha(3)]$$\n\n\n\\begin{equation*}\n=\\frac{1}{\\sqrt{6}}[\\alpha(1) \\beta(2) \\beta(3)+\\beta(1) \\alpha(2) \\beta(3)-2 \\beta(1) \\beta(2) \\alpha(3)] \\tag{8b}\n\\end{equation*}\n\n\nExpressions (6), (7), and (8) above are the complete set of common eigenfunctions for $\\boldsymbol{S}_{12}^{2}, ~ \\boldsymbol{S}_{123}^{2}, ~(S_{123})_{z}$. It is easy to verify that they are indeed orthogonal to each other. Since each particle has two spin states $\\alpha$ and $\\beta$, the system of three particles has a total of 8 independent states, thus expressions (6) to (8) form the sought orthogonal and complete set of energy eigenstates.\n\nThe reader can easily verify that according to the theory of angular momentum coupling, similar results would be obtained using the C.G. coefficient table.", + "final_answer": [ + "E_{3/2} = \\frac{3}{4}A" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_{3/2}$": "Energy level of the system when the total spin quantum number $S=3/2$.", + "$A$": "Real constant in the Hamiltonian." + } + }, + { + "id": 15, + "context": "", + "question": "An electron moves freely in a one-dimensional region $-L/2 \\leqslant x \\leqslant L/2$, with the wave function satisfying periodic boundary conditions $\\psi(x)=\\psi(x+L)$. Its unperturbed energy eigenvalue is $E_n^{(0)}$. A perturbation $H^{\\prime}=\\varepsilon \\cos q x$ is applied to this system where $L q=4 \\pi N$ ($N$ is a large positive integer). Consider the degenerate state where the electron's momentum is $|p|=q \\hbar / 2$ (corresponding to energy $E_N^{(0)}$) without perturbation. Find the expression for the second-order energy correction $E_N^{(2)}$ of these degenerate states caused by the perturbation $H^{\\prime}$.", + "answer": "(a) For a free particle, the common eigenfunction of the energy $(H_{0})$ and momentum $(p)$ satisfying the periodic condition is\n\n\\begin{equation*}\n\\psi_{n}^{(0)}=\\frac{1}{\\sqrt{L}} \\mathrm{e}^{\\mathrm{i} 2 \\pi x / L}, \\quad n=0, \\pm 1, \\pm 2, \\cdots \\tag{1}\n\\end{equation*}\n\n\nThe eigenvalue is\n\n\\begin{align*}\np_{n} & =\\hbar k_{n}=2 \\pi n \\hbar / L \\\\\nE_{n}^{(0)} & =\\frac{p_{n}^{2}}{2 m}=\\frac{(2 \\pi n \\hbar)^{2}}{2 m L^{2}} \\tag{2}\n\\end{align*}\n\n(b) The perturbation operator is\n\n\\begin{equation*}\nH^{\\prime}=\\varepsilon \\cos q x=\\frac{\\varepsilon}{2}(\\mathrm{e}^{\\mathrm{i} \\pi \\lambda N_{x} / L}+\\mathrm{e}^{-\\mathrm{i} 4 \\pi N_{x} / L}) \\tag{3}\n\\end{equation*}\n\n\nFor $|p|=q \\hbar / 2=2 \\pi N \\hbar / L$ , the corresponding zeroth-order wave function is\n\n\\begin{equation*}\n\\psi^{(0)}=C_{N} \\psi_{N}^{(0)}+C_{-, ~} \\psi_{-, ~}^{(0)} \\tag{4}\n\\end{equation*}\n\n\nIt can be easily calculated that\n\n\\begin{equation*}\nH_{\\mathrm{vv}}^{\\prime}=H_{-\\mathrm{v},-\\mathrm{N}}^{\\prime}=0, \\quad H_{v,-\\mathrm{v}}^{\\prime}=H_{-\\mathrm{N}, \\mathrm{v}}^{\\prime}=\\frac{\\varepsilon}{2} \\tag{5}\n\\end{equation*}\n\n\nTherefore, in the subspace ${\\psi_{N}^{(0)}, \\psi_{-N}^{(0)}}$, the matrix representation of $H^{\\prime}$ is\n\nH^{\\prime}=\\frac{\\varepsilon}{2}[\\begin{array}{ll}\n0 & 1 \\tag{$\\prime$}\\\\\n1 & 0\n\\end{array}]\n\n\nIn this subspace, $\\psi^{(0)}$ should satisfy the eigenvalue equation\n\n$H^{\\prime} \\psi^{(0)}=E_{N}^{(1)} \\psi^{(0)}\n\n\nThat is\n\\begin{equation}\n\\frac{\\varepsilon}{2}\\left[\\begin{array}{ll}\n0 & 1 \\tag{6}\\\\\n1 & 0\n\\end{array}\\right]\\left[\\begin{array}{l}\nC_{\\mathrm{N}} \\\\\nC_{-\\mathrm{N}}\n\\end{array}\\right]=E_{N}^{(1)}\\left[\\begin{array}{l}\nC_{\\mathrm{N}} \\\\\nC_{-N}\n\\end{array}\\right]\n\\end{equation}\n\n\n\nIt is easy to solve\n\n\\begin{align*}\n& E_{\\mathrm{N}+}^{(1)}=\\varepsilon / 2 \\tag{7}\\\\\n& \\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{N}^{(0)}+\\psi_{-\\mathrm{N}}^{(0)}]=\\psi_{N+}^{(0)} \\\\\n& E_{\\mathrm{N}-}^{(1)}=-\\varepsilon / 2 \\\\\n& \\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{\\mathrm{N}}^{(0)}-\\psi_{-\\mathrm{N}}^{(0)}]=\\psi_{N-}^{(0)}\n\\end{align*}\n\n\nThe first-order correction to the wave function is\n\n\\begin{equation*}\n\\psi^{(1)}=\\sum_{n}^{\\prime} \\frac{1}{E_{N}^{(0)}-E_{n}^{(0)}}\\langle\\psi_{n}^{(0)}| H^{\\prime}|\\psi^{(0)}\\rangle \\psi_{n}^{(0)} \\quad(n \\neq \\pm N) \\tag{8}\n\\end{equation*}\n\n\nThe matrix elements contributing to $\\psi^{(1)}$ are (corresponding to $n= \\pm 3 N$)\n\n$$ H_{3, \\mathrm{~V} . \\mathrm{N}}^{\\prime}=H_{-3, \\mathrm{~V},-\\mathrm{N}}^{\\prime}=\\varepsilon / 2$$\n\n\nThe corresponding energy difference is\n\n$$ E_{N}^{(0)}-E_{3 N}^{(0)}=-8 E_{N}^{(0)}=-\\frac{8(2 \\pi N \\hbar)^{2}}{2 m L^{2}}$$ \n\n\nThus, the first-order correction to the wave function is\n\n\\begin{align}\nE^{(1)} & =E_{N+}^{(1)}=\\frac{\\varepsilon}{2} \\\\\n\\psi^{(1)} & =-\\frac{\\varepsilon}{2} \\frac{2 m L^{2}}{8(2 \\pi N \\hbar)^{2}} \\frac{1}{\\sqrt{2}}[\\psi_{3 N}^{(0)}+\\psi_{-3 N}^{(0)}]\n\\end{align}\n\n\n\\begin{align*}\nE^{(1)} & =E_{N-}^{(1)}=-\\frac{\\varepsilon}{2} \\\\\n\\psi^{(1)} & =-\\frac{\\varepsilon}{2} \\frac{2 m L^{2}}{8(2 \\pi N \\hbar)^{2}} \\frac{1}{\\sqrt{2}}[\\psi_{3 N}^{(0)}-\\psi_{-3 N}^{(0)}] \\tag{9}\n\\end{align*}\n\n(c) The second-order correction to the energy is\n\n\\begin{equation*}\nE_{N}^{(2)}=\\langle\\psi^{(0)}| H^{\\prime}|\\psi^{(1)}\\rangle \\tag{10}\n\\end{equation*}\n\n\nSubstituting equations (7) and (9) into the above formula, we obtain\n\n\\begin{equation*}\nE_{N}^{(2)}=-\\frac{\\varepsilon^{2}}{32} \\frac{2 m L^{2}}{(2 \\pi N \\hbar)^{2}}=-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}} \\tag{11}\n\\end{equation*}\n\n\nCombining equations (7) and (11), the conclusion for the energy level up to $\\varepsilon^{2}$ order is:\n\n\\begin{equation*}\nE_{N}=E_{N}^{(0)}+E_{N}^{(1)}+E_{N}^{(2)}=E_{\\mathrm{N}}^{(0)} \\pm \\frac{\\varepsilon}{2}-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}} \\tag{12}\n\\end{equation*}\n\n\nCorresponding to\n\n$$\\psi^{(0)}=\\frac{1}{\\sqrt{2}}[\\psi_{N}^{(0)} \\pm \\psi_{-, .}^{(0)}] $$", + "final_answer": [ + "-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\varepsilon$": "Strength of the perturbation.", + "$E_N^{(0)}$": "Unperturbed energy eigenvalue for the $n$-th state, $E_{n}^{(0)} = \\frac{(2 \\pi n \\hbar)^{2}}{2 m L^{2}}$.", + "$m$": "Mass of the electron.", + "$\\hbar$": "Reduced Planck's constant.", + "$q$": "Wave number parameter of the perturbation, defined by $L q=4 \\pi N$." + } + }, + { + "id": 16, + "context": "", + "question": "For the $n$-th bound state $\\psi_{n}, ~ E_{n}$ of a square well (depth $V_{0}$, width $a$), under the condition $V_{0} \\gg E_{n}$, calculate the probability of the particle appearing outside the well.", + "answer": "Take the even parity state as an example. The energy eigenvalue equation can be written as\n\n\\begin{equation}\n \\begin{array}{lll}\n\\psi^{\\prime \\prime}+k^{2} \\psi=0, & |x| \\leqslant a / 2 & \\text { (inside the well) } \\tag{1}\\\\\n\\psi^{\\prime \\prime}-\\beta^{2} \\psi=0, & |x| \\geqslant a / 2 & \\text { (outside the well) }\n\\end{array}\n\\end{equation}\n\n\n\nwhere\n\n\\begin{equation*}\nk=\\sqrt{2 m E} / \\hbar, \\quad \\beta=\\sqrt{2 m(V_{0}-E)} / \\hbar \\tag{2}\n\\end{equation*}\n\n\nNote that under the condition $V_{0} \\gg E$, $\\beta \\gg k$.\nThe even parity solution of equation (1) is\n\n\\begin{array}{ll}\n\\psi=\\cos k x, & |x| \\leqslant a / 2 \\\\\n\\psi=C \\mathrm{e}^{-\\beta|x|}, & |x| \\geqslant a / 2 \\tag{3}\n\\end{array}\n\nAt $x=a / 2$, $\\psi$ should be continuous, thus yielding\n\n\\begin{equation*}\nC=\\mathrm{e}^{\\beta a / 2} \\cos \\frac{k a}{2} \\tag{4}\n\\end{equation*}\n\nAt $x=a / 2$, $\\psi^{\\prime}$ should also be continuous, thus yielding\n\n$$ C=\\frac{k}{\\beta} \\mathrm{e}^{\\beta a / 2} \\sin \\frac{k a}{2} $$\n\n\nDividing by equation (4), we obtain the energy level formula\n\n\\begin{equation*}\n\\tan \\frac{k a}{2}=\\frac{\\beta}{k} \\tag{5}\n\\end{equation*}\n\n\nUnder the condition $\\beta \\gg k$, the solution of equation (5) is\n\n\\begin{equation*}\nk a \\approx n \\pi, \\quad n=1,3,5, \\cdots \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (2), the energy levels are\n\n\\begin{equation*}\nE_{n}=\\frac{\\hbar^{2} k^{2}}{2 m} \\approx \\frac{1}{2 m}(\\frac{n \\pi \\hbar}{a})^{2} \\tag{7}\n\\end{equation*}\n\n\nThis is precisely the energy level formula for an infinitely deep potential well.\nNow calculate the probability of the particle appearing inside and outside the well. From equations (3) and (4), it is easy to find\n\n\\begin{align*}\n& \\int_{\\text {outside }}|\\psi|^{2} \\mathrm{~d} x=2 C^{2} \\int_{a / 2}^{\\infty} \\mathrm{e}^{-2 \\beta x} \\mathrm{~d} x=\\frac{C^{2}}{\\beta} \\mathrm{e}^{-\\beta a}=\\frac{1}{\\beta} \\cos ^{2} \\frac{k a}{2} \\tag{8}\\\\\n& \\int_{\\text {inside }}|\\psi|^{2} \\mathrm{~d} x=2 \\int_{0}^{a / 2} \\cos ^{2} k x \\mathrm{~d} x=\\frac{a}{2}(1+\\frac{\\sin k a}{k a}) \\tag{9}\n\\end{align*}\n\n\nConsidering $k a \\approx n \\pi(n=1,3,5, \\cdots), \\sin k a$ and $\\cos (k a / 2)$ are both close to zero, it is understood that the probability of the particle appearing outside the well is much smaller than that inside the well. Additionally,\n\n\\begin{gather*}\n\\int_{-\\infty}^{+\\infty}|\\psi|^{2} \\mathrm{~d} x \\approx \\int_{\\text {inside }}|\\psi|^{2} \\mathrm{~d} x \\approx \\frac{a}{2} \\\\\n\\text { Outside probability }=\\frac{\\int_{\\text {outside }}|\\psi|^{2} \\mathrm{~d} x}{\\int_{-\\infty}^{+\\infty}|\\psi|^{2} \\mathrm{~d} x} \\approx \\frac{2}{\\beta a} \\cos ^{2} \\cdot \\frac{k a}{2} \\tag{10}\n\\end{gather*}\n\n\nUsing the energy level formula (5), it is easy to obtain\n\n\\begin{gather*}\n1+\\tan ^{2} \\frac{k a}{2}=\\frac{k^{2}+\\beta^{2}}{k^{2}}=\\frac{V_{0}}{E} \\\\\n\\cos ^{2} \\frac{k a}{2}=\\frac{E}{V_{0}} \\tag{11}\n\\end{gather*}\n\n\nSubstituting into equation (10), we get\n\n\\begin{equation*}\n\\text { Outside probability }=\\frac{2 E}{a \\beta V_{0}} \\approx \\frac{2 \\hbar}{a \\sqrt{2 m V_{0}}} \\frac{E}{V_{0}} \\tag{12}\n\\end{equation*}\n\n\nConsidering $V_{0} \\gg E$ and the energy level formula (7), it is easily seen\n\n\\begin{equation*}\n\\sqrt{2 m V_{0}} \\gg n \\pi \\hbar / a \\tag{13}\n\\end{equation*}\n\n\nThus,\n\n\\begin{equation*}\n\\text { Outside probability } \\ll \\frac{2 E}{n \\pi V_{0}} \\tag{14}\n\\end{equation*}", + "final_answer": [ + "\\frac{2 \\hbar E_n}{a V_0 \\sqrt{2 m V_0}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\hbar$": "Reduced Planck's constant.", + "$E_n$": "Energy of the $n$-th bound state.", + "$a$": "Width of the square well.", + "$V_0$": "Depth of the square well potential.", + "$m$": "Mass of the particle." + } + }, + { + "id": 17, + "context": "", + "question": "A particle moves freely, and the initial wave function at $t=0$ is given as\n$$ \\psi(x, 0)=(2 \\pi a^{2})^{-1 / 4} \\exp [\\mathrm{i} k_{0}(x-x_{0})-(\\frac{x-x_{0}}{2 a})^{2}], \\quad a>0 $$\n\nFind the wave function $\\varphi(p)$ in the $p$ representation at $t=0$;", + "answer": "First, consider the shape of the wave packet at $t=0$\n\\begin{equation*}\n|\\psi(x, 0)|^{2}=(2 \\pi a^{2})^{-1 / 2} \\mathrm{e}^{-(x-x_{0})^{2} / 2 a^{2}} \\tag{1}\n\\end{equation*}\nwhich is a Gaussian distribution. According to the mean value formula\n\\begin{equation}\n \\overline{f(x)}=\\int_{-\\infty}^{+\\infty}|\\psi|^{2} f(x) \\mathrm{d} x\n\\end{equation}\nit is easy to calculate that at $t=0$\n\\begin{align*}\n& \\bar{x}=x_{0}, \\quad \\bar{x}^{2}=a^{2}+x_{0}^{2} \\\\\n& \\Delta x=(\\bar{x}^{2}-\\bar{x}^{2})^{1 / 2}=a \\tag{2}\n\\end{align*}\nwhich means at $t=0$, the center of the wave packet is at $x_{0}$, and the width of the wave packet is $a$. We first calculate the wave function in terms of the wave number $k$. Let\n\\begin{equation*}\n\\psi(x, 0)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\mathrm{e}^{\\mathrm{i} k x} \\mathrm{~d} k \\tag{3}\n\\end{equation*}\nwhere $\\varphi(k)$ is the initial wave function in the $k$ representation. According to the Fourier transform formula\n\\begin{align*}\n \\varphi(k)&=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\psi(x, 0) \\mathrm{e}^{-\\mathrm{i} k x} \\mathrm{~d} x\\\\\n & =(\\frac{1}{2 \\pi})^{3 / 4} \\frac{1}{\\sqrt{a}} \\mathrm{e}^{-\\mathrm{i} k x_{0}} \\int_{-\\infty}^{+\\infty} \\mathrm{d} x \\exp [-(\\frac{x-x_{0}}{2 a})^{2}-\\mathrm{i}(k-k_{0})(x-x_{0})] \\\\\n& =(\\frac{2 a^{2}}{\\pi})^{\\frac{1}{4}} \\exp [-\\mathrm{i} k x_{0}-a^{2}(k-k_{0})^{2}] \\tag{4}\n\\end{align*}\n\nThe integration formula is used in the calculation (refer to the note at the end of the question). From equation (4)\n\\begin{equation*}\n|\\varphi(k)|^{2}=\\sqrt{\\frac{2}{\\pi}} a \\mathrm{e}^{-2 a^{2}(k-k_{0})^{2}} \\tag{5}\n\\end{equation*}\nwhich indicates that the probability distribution of $k$ is also a Gaussian distribution. According to the mean value formula\n\\begin{equation}\n \\overline{f(k)}=\\int_{-\\infty}^{+\\infty}|\\varphi(k)|^{2} f(k) \\mathrm{d} k \n\\end{equation}\nit is readily calculated that\n\\begin{equation*}\n\\bar{k}=k_{0}, \\quad \\overline{k^{2}}=k_{0}^{2}+\\frac{1}{4 a^{2}} \\tag{6}\n\\end{equation*}\n\nSince $p=\\hbar k$, thus\n\\begin{align*}\n& \\bar{p}=\\hbar k_{0}, \\quad \\bar{p}^{2}=\\hbar^{2} k_{0}^{2}+\\frac{\\hbar^{2}}{4 a^{2}} \\tag{7}\\\\\n& \\Delta p=(\\overline{p^{2}}-\\bar{p}^{2})^{1 / 2}=\\hbar / 2 a\n\\end{align*}\n\nThese are the characteristics of the momentum distribution at $t=0$. Since it is a free particle, momentum conservation implies that the probability distribution of momentum is also conserved. Therefore, equation (7) applies to any time. From equations (2) and (7), it follows that\n\\begin{equation*}\n\\Delta x \\cdot \\Delta p=\\hbar / 2 \\quad(t=0 \\text{ instant}) \\tag{8}\n\\end{equation*}\n\nFinally, the wave function in the momentum representation $\\varphi(p)$ is related to $\\varphi(k)$ by the normalization condition $\\int |\\varphi(p)|^2 dp = \\int |\\varphi(k)|^2 dk = 1$. Since $dp = \\hbar dk$, we have $\\varphi(p) = \\frac{1}{\\sqrt{\\hbar}}\\varphi(k)|_{k=p/\\hbar}$.", + "final_answer": [ + "\\varphi(p) = (\\frac{2 a^{2}}{\\pi \\hbar^2})^{\\frac{1}{4}} \\exp [-\\mathrm{i} \\frac{p x_{0}}{\\hbar}-a^{2}(\\frac{p}{\\hbar}-k_{0})^{2}]" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\varphi(p)$": "Wave function in the momentum representation at $t=0$.", + "$a$": "Parameter defining the width of the initial wave packet, $a>0$. It also represents the uncertainty in position at $t=0$, $\\Delta x = a$.", + "$\\mathrm{i}$": "Imaginary unit.", + "$p$": "Momentum.", + "$x_0$": "Initial position of the center of the wave packet. It also represents the mean position at $t=0$, $\\bar{x} = x_0$.", + "$\\hbar$": "Reduced Planck's constant.", + "$k_0$": "Initial wave number, representing the center of the wave number distribution." + } + }, + { + "id": 18, + "context": "", + "question": "The particle moves freely, with the initial wave function at $t=0$ given as\n\n$$ \\psi(x, 0)=(2 \\pi a^{2})^{-1 / 4} \\exp [\\mathrm{i} k_{0}(x-x_{0})-(\\frac{x-x_{0}}{2 a})^{2}], \\quad a>0 $$\n\nFind the wave function $\\psi(x, t)$ for $t>0$", + "answer": "In equation \\begin{equation*}\n\\psi(x, 0)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\mathrm{e}^{\\mathrm{i} k x} \\mathrm{~d} k, \\tag{3}\n\\end{equation*} let\n\n$$ \\mathrm{e}^{\\mathrm{i} k x} \\rightarrow \\mathrm{e}^{\\mathrm{i}(k x-\\omega t)}, \\quad \\omega=\\frac{\\hbar k^{2}}{2 m}=\\frac{k^{2}}{2 m} \\quad(\\hbar=1) $$\n\n\nthus obtaining the wave function $\\psi(x, t)$ for $t>0$, i.e.,\n\n\\begin{equation*}\n\\psi(x, t)=\\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty} \\varphi(k) \\exp (\\mathrm{i} k x-\\frac{\\mathrm{i} k^{2} t}{2 m}) \\mathrm{d} k \\tag{9}\n\\end{equation*}\n\n\nSubstitute equation \\begin{align*}\n \\varphi(k)&= (\\frac{2 a^{2}}{\\pi})^{\\frac{1}{4}} \\exp [-\\mathrm{i} k x_{0}-a^{2}(k-k_{0})^{2}] \n\\end{align*} into the above expression and we obtain\n\\begin{align*}\n& \\psi(x, t) \\\\\n= & \\frac{\\exp [\\mathrm{i} k_{0}(x-x_{0})-\\mathrm{i} t k_{0}^{2} / 2 m]}{(2 \\pi)^{1 / 4}(a+\\mathrm{i} t / 2 m a)^{1 / 2}} \\exp [-\\frac{1}{4}(x-x_{0}-\\frac{k_{0} t}{m})^{2} \\frac{1-\\mathrm{i} t / 2 m a^{2}}{a^{2}+(t / 2 m a)^{2}}] \\tag{10}\\\\\n\\end{align*}", + "final_answer": [ + "\\psi(x, t) = \\frac{\\exp [\\mathrm{i} k_{0}(x-x_{0})-\\mathrm{i} t k_{0}^{2} / 2 m]}{(2 \\pi)^{1 / 4}(a+\\mathrm{i} t / 2 m a)^{1 / 2}} \\exp [-\\frac{1}{4}(x-x_{0}-\\frac{k_{0} t}{m})^{2} \\frac{1-\\mathrm{i} t / 2 m a^{2}}{a^{2}+(t / 2 m a)^{2}}]" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\psi(x, t)$": "Wave function of the particle at time $t$", + "$\\mathrm{i}$": "Imaginary unit", + "$k_0$": "Initial wave number", + "$x$": "Position coordinate", + "$x_0$": "Initial position offset", + "$t$": "Time", + "$m$": "Mass of the particle", + "$\\pi$": "Mathematical constant pi", + "$a$": "A positive constant related to the width of the initial wave packet" + } + }, + { + "id": 19, + "context": "", + "question": "Using the raising and lowering operators $a^{+}, ~ a$, find the energy eigenfunctions of the harmonic oscillator (in the $x$ representation), and briefly discuss their mathematical properties.", + "answer": "Start with the ground state wave function $\\psi_{0}(x)$. We have,\n\n\\begin{equation*}\na|0\\rangle=0 \\tag{1}\n\\end{equation*}\n\n\nIn the $x$ representation this reads as\n\n\\begin{equation*}\n(\\mathrm{i} \\hat{p}+m \\omega x) \\psi_{0}(x)=(\\hbar \\frac{\\mathrm{d}}{\\mathrm{~d} x}+m \\omega x) \\psi_{0}(x)=0 \\tag{2}\n\\end{equation*}\n\n\nLet $\\alpha=\\sqrt{m \\omega / \\hbar}$, the above equation can be written as\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} x} \\psi_{0}+\\alpha^{2} x \\psi_{0}=0 \\tag{2'}\n\\end{equation*}\n\n\nIt is easy to solve\n\n\\begin{equation*}\n\\psi_{0}(x)=N_{0} \\mathrm{e}^{-\\alpha^{2} x^{2} / 2} \\tag{3}\n\\end{equation*}\n\n\nwhere $N_{0}$ is the normalization constant. From the normalization condition\n\n\\begin{equation*}\n\\int_{-\\infty}^{+\\infty}|\\psi_{n}(x)|^{2} \\mathrm{~d} x=1 \\tag{4}\n\\end{equation*}\n\n\nFind (taking as real)\n\n\\begin{equation*}\nN_{0}=\\sqrt{\\alpha} / \\pi^{1 / 4} \\tag{5}\n\\end{equation*}\n\n\nSecondly, we have\n\n$$ a^{+}|0\\rangle=|1\\rangle $$\n\nThat is\n\n\\begin{equation*}\n\\psi_{1}(x)=a^{+} \\psi_{0}(x)=\\frac{1}{\\sqrt{2}}(\\alpha x-\\frac{1}{\\alpha} \\frac{\\mathrm{~d}}{\\mathrm{~d} x}) \\psi_{0}(x) \\tag{6}\n\\end{equation*}\n\n\nSubstituting equation (3) and (5) into equation (6), it's easy to find\n\n\\begin{equation*}\n\\psi_{1}=\\sqrt{2} \\alpha x \\psi_{0}=\\frac{\\sqrt{2 \\alpha}}{\\pi^{1 / 4}} \\alpha x \\mathrm{e}^{-\\alpha^{2} x^{2} / 2} \\tag{7}\n\\end{equation*}\n\n\nGenerally, we have\n\n\\begin{equation*}\n\\psi_{n}=\\frac{1}{\\sqrt{n}} a^{+} \\psi_{n-1}=\\frac{1}{\\sqrt{2 n}}(\\alpha x-\\frac{1}{\\alpha} \\frac{\\mathrm{~d}}{\\mathrm{~d} x}) \\psi_{n-1} \\tag{8}\n\\end{equation*}\n\n\nIntroducing the dimensionless variable\n\n\\begin{equation*}\n\\xi=\\alpha x, \\tag{9}\n\\end{equation*}\n\nwe thus obtain\n\n\\begin{equation*}\n\\psi_{n}=\\frac{1}{\\sqrt{2 n}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}) \\psi_{n-1} \\tag{10}.\n\\end{equation*}\n\n\nRecursively, we get\n\n\\begin{equation*}\n\\psi_{n}=(\\frac{1}{2^{n} n!})^{\\frac{1}{2}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\psi_{0}=(\\frac{\\alpha}{\\sqrt{\\pi} 2^{n} n!})^{\\frac{1}{2}}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\mathrm{e}^{-\\xi^{2} / 2}=N_{n} H_{n}(\\xi) \\mathrm{e}^{-\\xi^{2} / 2} \\tag{11}\n\\end{equation*}\n\n\nWhere\n\n\\begin{equation*}\nN_{n}=(\\frac{\\alpha}{\\sqrt{\\pi} 2^{n} n!})^{\\frac{1}{2}} \\tag{12}\n\\end{equation*}\n\n\nis the normalization constant. Notice\n\n\\begin{equation*}\nN_{n}=N_{n-1} / \\sqrt{2 n} \\tag{13}\n\\end{equation*}\n\n\nIn equation (11),\n\n\\begin{equation*}\nH_{n}(\\xi)=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{n} \\mathrm{e}^{-\\xi^{2} / 2} \\tag{1}\n\\end{equation*}\n\nIt is obvious that $H_{n}(\\xi)$ is an $n$-th degree polynomial in $\\xi$, called the Hermite polynomial.\nBelow is a brief discussion of the mathematical properties of $H_{n}(\\xi)$.\n\n{Parity}\n\nIt is obvious from equation (14)\n\n\\begin{equation*}\nH_{n}(-\\xi)=(-1)^{n} H_{n}(\\xi) \\tag{15}\n\\end{equation*}\n\n\nWhen $n$ is even, $H_{n}(\\xi)$ has even parity; when $n$ is odd, $H_{n}(\\xi)$ has odd parity.\n\n{Recursion relations}\n\nWe also have,\n\n$$ a \\psi_{n}=\\frac{1}{\\sqrt{2}}(\\xi+\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}) \\psi_{n}=\\sqrt{n} \\psi_{n-1}$$\n\nSubstituting equation (11) into this equation, and using equation (13) to eliminate the normalization constant, we get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi} H_{n}(\\xi)=2 n H_{n-1}(\\xi) \\tag{16},\n\\end{equation*}\n\nWe also have:\n$$ \\xi \\psi_{n}=\\sqrt{\\frac{n+1}{2}} \\psi_{n+1}+\\sqrt{\\frac{n}{2}} \\psi_{n-1} $$\n\nSubstituting equations (11), (13) into this equation, we get\n\n\\begin{equation*}\n2 \\xi H_{n}(\\xi)=H_{n+1}(\\xi)+2 n H_{n-1}(\\xi) \\tag{17}\n\\end{equation*}\n\n\n{Differential equation}\n\nCombining equations (16) and (17), eliminate $H_{n-1}$, then differentiate once more, and using equation (16) to eliminate $\\mathrm{d} H_{n+1} / \\mathrm{d} \\xi$, we find that $H_{n}$ satisfies the following differential equation:\n\n\\begin{equation*}\n\\frac{\\mathrm{d}^{2}}{\\mathrm{~d} \\xi^{2}} H_{n}(\\xi)-2 \\xi \\frac{\\mathrm{~d}}{\\mathrm{~d} \\xi} H_{n}(\\xi)+2 n H_{n}(\\xi)=0 \\tag{18}\n\\end{equation*}\n\n\nThis equation is known as the Hermite equation. Equation (14) is the unique polynomial solution to this equation.\n$4{ }^{\\circ}$ Differential expression\nEquation (14) is equivalent to the following:\n\n\\begin{equation*}\nH_{n}(\\xi)=(-1)^{n} e^{\\xi^{2}}(\\frac{d}{d \\xi})^{n} e^{-\\xi^{2}} \\tag{$\\prime$}\n\\end{equation*}\n\n\nThis can be proved by mathematical induction. First, for $n=0,1$, both equations (14) and ($14'$) give\n\n$$H_{0}(\\xi)=1, \\quad H_{1}(\\xi)=2 \\xi $$\n\n\nAssume the proposition holds for $n=k$, i.e., assume\n\n\\mathrm{e}^{\\xi^{2}}(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k} \\mathrm{e}^{-\\xi^{2}}=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k} \\mathrm{e}^{-\\xi^{2} / 2}=H_{k}(\\xi)\n\n\nThen\n\n\\begin{gathered}\n(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2}}=-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi}(\\mathrm{e}^{-\\xi^{2}} H_{k})=(2 \\xi H_{k}-H_{k}^{\\prime}) \\mathrm{e}^{-\\xi^{2}} \\\\\n(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2} / 2}=(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})(\\mathrm{e}^{-\\xi^{2} / 2} H_{k})=(2 \\xi H_{k}-H_{k}^{\\prime}) \\mathrm{e}^{-\\xi^{2} / 2}\n\\end{gathered}\n\n\nThus\n\n\\mathrm{e}^{\\xi^{2}}(-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2}}=2 \\xi H_{k}-H_{k}^{\\prime}=\\mathrm{e}^{\\xi^{2} / 2}(\\xi-\\frac{\\mathrm{d}}{\\mathrm{~d} \\xi})^{k+1} \\mathrm{e}^{-\\xi^{2} / 2}\n\n\nThe proposition holds for $n=k+1$. Proof complete.\n$5^{\\circ} H_{n}(0)$ and $\\psi_{n}(0)$\nWhen $n=2 k+1$ (odd), from equation (15), we have\n\nH_{2 k+1}(-\\xi)=-H_{2 k+1}(\\xi)\n\n\nLet $\\xi \\rightarrow 0$, we get\n\n\\begin{equation*}\nH_{2 k+1}(0)=0, \\quad k=1,2,3, \\cdots \\tag{19}\n\\end{equation*}\n\n\nWhen $n=2 k$ (even), in equation (17) let $\\xi \\rightarrow 0$, and change $n$ to ( $n-1$ ), we get\n\nH_{2 k}(0)=-2(2 k-1) H_{2 k-2}(0)\n\n\nUsing this repeatedly, and noting that $H_{0}=1$, we get\n\n\\begin{equation*}\nH_{2 k}(0)=(-1)^{k} 2^{k}(2 k-1)!!=(-1)^{k}(2 k)!/ k! \\tag{20}\n\\end{equation*}\n\n\nUsing equations (11), (12) again, we get\n\n\\begin{gather*}\n\\psi_{2 k+1}(0)=0 \\tag{21}\\\\\n\\psi_{2 k}(0)=(-1)^{k} \\frac{\\sqrt{\\alpha}}{\\pi^{1 / 4}} \\frac{\\sqrt{(2 k)!}}{(2 k)!!} \\tag{22}\n\\end{gather*}\n\n\n\\begin{equation*}\n[\\psi_{2 k}(0)]^{2}=\\frac{\\alpha}{\\sqrt{\\pi}} \\frac{(2 k)!}{[(2 k)!!]^{2}}=\\frac{\\alpha}{\\sqrt{\\pi}} \\frac{(2 k-1)!!}{(2 k)!!} \\tag{23}\n\\end{equation*}", + "final_answer": [ + "\\psi_n(x) = N_n H_n(\\alpha x) e^{-\\frac{1}{2}(\\alpha x)^2}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\psi_n(x)$": "Energy eigenfunction for the $n$-th state in the x representation.", + "$N_n$": "Normalization constant for the $n$-th energy eigenfunction.", + "$H_n(\\alpha x)$": "Hermite polynomial of degree $n$ evaluated at $\\alpha x$.", + "$\\alpha$": "Dimensionless constant defined as $\\sqrt{m \\omega / \\hbar}$.", + "$x$": "Position variable in the x representation.", + "$\\pi$": "Pi, a mathematical constant.", + "$n$": "Quantum number representing the energy level or state index." + } + }, + { + "id": 20, + "context": "", + "question": "The emission of recoil-free $\\gamma$ radiation by nuclei bound in a lattice is a necessary condition for the Mössbauer effect. The potential acting on the nuclei in the lattice can be approximated as a harmonic oscillator potential\n\n$$ V(x)=\\frac{1}{2} M \\omega^{2} x^{2} $$\n\nwhere $M$ is the mass of the nucleus, $x$ is the coordinate of the nucleus's center of mass, and $\\omega$ is the vibration frequency. Assume that initially, the nucleus's center of mass motion (harmonic vibration) is in its ground state, and at $t=0$, due to a transition of energy levels within the nucleus, a photon is emitted along the $x$-axis with energy $E_{\\gamma}$, and momentum $E_{\\gamma} / c$. Since the $\\gamma$ radiation occurs suddenly, the only effect on the nucleus's center of mass motion is that its momentum eigenvalue changes from $p$ to $(p-E_{\\gamma} / c)$. Determine the expression for the probability that the nucleus's center of mass motion remains in the ground state after the photon is emitted.", + "answer": "Due to the harmonic oscillator potential, the center of mass motion of the nucleus is harmonic, initially (for $t<0$) in the ground state $\\psi_{0}(x)$. Expanding $\\psi_{0}(x)$ using momentum eigenfunctions gives:\n\n\\begin{equation*}\n\\psi_{0}(x)=(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\tag{1}\n\\end{equation*}\n\n\nDue to the emission of $\\gamma$ photon, momentum changes (total momentum is conserved!) from $p$ to $p-p_{0},(p_{0}=E_{\\gamma} / c)$, which means the wave function $\\varphi(p)$ transforms as follows:\n\n\\begin{equation*}\n\\varphi(p) \\rightarrow \\varphi(p+p_{0}) \\tag{2}\n\\end{equation*}\n\n\nTherefore, after $\\gamma$ emission, the wave function of the nucleus's center of mass motion becomes\n\n\\begin{align*}\n\\psi(x) & =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p+p_{0}) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\\\\n& =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p^{\\prime}) \\mathrm{e}^{\\mathrm{i}(p^{\\prime}-p_{0}) x / \\hbar} \\mathrm{d} p^{\\prime} \\\\\n& =\\psi_{0}(x) \\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\tag{3}\n\\end{align*}\n\n\nHere, the exponential factor indicates that the nucleus's center of mass momentum is reduced by $p_{0}$. The probability that the nucleus's center of mass vibration remains in the ground state after photon emission is\n\n\\begin{equation*}\nP=|\\langle\\psi_{0} \\mid \\psi\\rangle|^{2}=|\\int_{-\\infty}^{+\\infty} \\psi_{0}^{2}(x) \\mathrm{e}^{-i p_{0} x / \\hbar} \\mathrm{d} x|^{2} \\tag{4}\n\\end{equation*}\n\n\nUsing the explicit form of the ground state wave function $\\psi_{0}$\n\n\\begin{equation*}\n\\psi_{0}(x)=\\frac{\\sqrt{\\alpha}}{\\pi^{1 / 4}} \\mathrm{e}^{-\\alpha^{2} x^{2} / 2}, \\quad \\alpha=\\sqrt{\\frac{M_{\\omega}}{\\hbar}} \\tag{5}\n\\end{equation*}\n\n\nIt is straightforward to calculate\n\n\\begin{align*}\n\\langle\\psi_{0} \\mid \\psi\\rangle & =\\frac{\\alpha}{\\sqrt{\\pi}} \\int_{-\\infty}^{+\\infty} \\mathrm{e}^{-\\alpha^{2} x^{2}-i p_{0} x / \\hbar} \\mathrm{d} x \\\\\n& =\\frac{\\alpha}{\\sqrt{\\pi}} \\int_{-\\infty}^{+\\infty} \\cos (\\frac{p_{0} x}{\\hbar}) \\mathrm{e}^{-\\alpha^{2} x^{2}} \\mathrm{~d} x \\\\\n& =\\exp (-\\frac{p_{0}^{2}}{4 \\hbar^{2} \\alpha^{2}})=\\exp (-\\frac{E_{\\gamma}^{2}}{4 \\hbar \\omega M c^{2}}) \\tag{6}\n\\end{align*}\n\n\nSubstituting into equation (4), we obtain the probability, which is\n\n\\begin{equation*}\nP=|\\langle\\psi_{0} \\mid \\psi\\rangle|^{2}=\\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}}) \\tag{7}\n\\end{equation*}\n\n\nCalculated numerical values are as follows:\n\n\\begin{gathered}\n{ }^{57} \\mathrm{Fe} \\text { nucleus, } M c^{2} \\approx 57 m_{\\mathrm{p}} c^{2} \\approx 57 \\times 938 \\mathrm{MeV}=5.35 \\times 10^{10} \\mathrm{eV} \\\\\nE_{\\gamma}=18 \\mathrm{keV}=1.8 \\times 10^{4} \\mathrm{eV} \\\\\n\\hbar \\omega=\\hbar c \\cdot \\frac{\\omega}{c} \\approx 1.97 \\times 10^{-5} \\mathrm{eV} \\cdot \\mathrm{~cm} \\times \\frac{2 \\pi \\times 10^{12}}{3 \\times 10^{10}} \\mathrm{~cm}^{-1} \\\\\n=4.13 \\times 10^{-3} \\mathrm{eV} \\\\\nP=\\exp [-\\frac{(1.8 \\times 10^{4})^{2}}{2 \\times 4.13 \\times 10^{-3} \\times 5.35 \\times 10^{10}}] \\\\\n=\\mathrm{e}^{-0.733}=0.48\n\\end{gathered}", + "final_answer": [ + "P=\\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$P$": "Probability that the nucleus's center of mass motion remains in the ground state after photon emission.", + "$\\mathrm{e}$": "Base of the natural logarithm.", + "$E_{\\gamma}$": "Energy of the emitted photon.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega$": "Vibration frequency of the nucleus in the lattice.", + "$M$": "Mass of the nucleus.", + "$c$": "Speed of light." + } + }, + { + "id": 21, + "context": "", + "question": "Calculate the probability that the nucleus is in various energy eigenstates after $\\gamma$ radiation, as in the previous problem.", + "answer": "From the previous problem, the center-of-mass wave function of the nucleus after $\\gamma$ radiation is\n\n\\begin{equation*}\n\\psi(x)=\\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\psi_{0}(x), \\quad p_{0}=E_{\\gamma} / c \\tag{1}\n\\end{equation*}\n\n\nThe probability of being in the ${ }^{\\prime} n$-th vibrational excited state $\\psi_{n}(x)$ is\n\n\\begin{equation*}\nP_{n}=|\\langle\\psi_{n} \\mid \\psi\\rangle|^{2}=|\\langle\\psi_{n} \\mid \\mathrm{e}^{-\\mathrm{i} p_{0} x / \\hbar} \\psi_{0}\\rangle|^{2} \\tag{2}\n\\end{equation*}\n\n\nLet $\\lambda=-p_{0} / \\hbar$, calculate $\\langle\\psi_{n} \\mid \\mathrm{e}^{\\mathrm{i} \\lambda x} \\psi_{0}\\rangle$ where $\\mathrm{e}^{\\mathrm{i} \\lambda x}$ can be viewed as an operator, expressed in terms of creation and annihilation operators, with the following relations:\n\n\\begin{align*}\n& \\mathrm{i} \\lambda x=\\alpha a^{+}-\\alpha^{*} a, \\quad \\alpha=\\mathrm{i} \\lambda \\sqrt{\\frac{\\hbar}{2 M \\omega}} \\tag{3}\\\\\n& \\mathrm{e}^{\\mathrm{i} \\lambda x}=\\mathrm{e}^{\\alpha a^{+}-\\alpha^{*} a}=\\mathrm{e}^{\\alpha a^{+}} \\mathrm{e}^{-\\alpha^{*} a} \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\tag{4}\\\\\n& \\mathrm{e}^{\\mathrm{i} \\lambda x}|\\psi_{0}\\rangle=\\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\mathrm{e}^{\\alpha a^{+}} \\mathrm{e}^{-\\alpha^{*} a}|\\psi_{0}\\rangle=|\\psi_{\\alpha}\\rangle \\\\\n&= \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\sum_{n} \\frac{\\alpha^{n}}{\\sqrt{n!}}|\\psi_{n}\\rangle \\tag{5}\n\\end{align*}\nwhere $\\psi_{\\alpha}$ is the coherent state of the harmonic oscillator. Utilizing equation (5), we get\n\n\\begin{gather*}\n\\langle\\psi_{n} \\mid \\mathrm{e}^{\\mathrm{i} \\lambda x} \\psi_{0}\\rangle=\\frac{\\alpha^{n}}{\\sqrt{n!}} \\mathrm{e}^{-\\frac{1}{2}|\\alpha|^{2}} \\tag{6}\\\\\nP_{n}=\\frac{|\\alpha|^{2 n}}{n!} \\mathrm{e}^{-|\\alpha|^{2}}=\\frac{1}{n!}(\\frac{\\lambda^{2} \\hbar}{2 M \\omega})^{n} \\mathrm{e}^{-\\frac{\\lambda^{2} \\hbar}{2 M \\omega}} \\\\\n= \\frac{1}{n!}(\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})^{n} \\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}}) \\tag{7}\n\\end{gather*}\n\n\nIt is easy to verify $\\sum_{n} P_{n}=1$. For the example of the ${ }^{57} \\mathrm{Fe}$ nucleus, the probabilities for the first few states are\n\n$$ P_{0}=0.48, \\quad P_{1}=0.35, \\quad P_{2}=0.13 $$", + "final_answer": [ + "P_n = \\frac{1}{n!}(\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})^{n} \\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$P_n$": "Probability of the nucleus being in the $n$-th vibrational excited state.", + "$n$": "Quantum number for the $n$-th vibrational excited state.", + "$E_{\\gamma}$": "Energy of the $\\gamma$ radiation (gamma ray photon).", + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega$": "Angular frequency of the vibrational excited states (harmonic oscillator).", + "$M$": "Mass of the nucleus.", + "$c$": "Speed of light in vacuum.", + "$\\mathrm{e}$": "Base of the natural logarithm." + } + }, + { + "id": 22, + "context": "", + "question": "Suppose the operator $\\hat{H}$ has continuous eigenvalues $\\omega$, and its eigenfunctions $u_{\\omega}(\\boldsymbol{x})$ form an orthonormal complete system, i.e.\n\n\\begin{gather*}\n\\hat{H} u_{\\omega}(\\boldsymbol{x})=\\omega u_{\\omega}(\\boldsymbol{x}) \\tag{1}\\\\\n\\int u_{\\omega^{*}}^{*}(\\boldsymbol{x}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d}^{3} x=\\delta(\\omega-\\omega^{\\prime}) \\tag{2}\\\\\n\\int u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega=\\delta(\\boldsymbol{x}-\\boldsymbol{x}^{\\prime}) \\tag{3}\n\\end{gather*}\n\n\nSolve the equation\n\n\\begin{equation*}\n(\\hat{H}-\\omega_{0}) u(\\boldsymbol{x})=F(\\boldsymbol{x}) \\tag{4}\n\\end{equation*}\n\n\nwhere $F(\\boldsymbol{x})$ is a known function, and $\\omega_{0}$ is a specific eigenvalue of $H$.\n\nIf the answer exists in an integral, then find the integrand", + "answer": "Since $u_{\\omega}(\\boldsymbol{x})$ is a complete system, the solution of equation (4) can always be expressed as\n\n\\begin{equation*}\\nu(x)=\\int C_{\\omega} u_{\\omega}(x) \\mathrm{d} \\omega \\tag{5}\n\\end{equation*}\n\n\nSubstitute into equation (4), we obtain\n\n\\begin{equation*}\n\\int(\\omega-\\omega_{0}) C_{\\omega} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega=F(\\boldsymbol{x}) \\tag{6}\n\\end{equation*}\n\n\nExpand $F(\\boldsymbol{x})$ as\n\n\\begin{align*}\nF(\\boldsymbol{x}) & =\\int F(\\boldsymbol{x}^{\\prime}) \\delta(\\boldsymbol{x}-\\boldsymbol{x}^{\\prime}) \\mathrm{d}^{3} x^{\\prime} \\\\\n& =\\iint F(\\boldsymbol{x}^{\\prime}) u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) u_{\\omega}(\\boldsymbol{x}) \\mathrm{d}^{3} x^{\\prime} \\mathrm{d} \\omega \\\\\n& =\\int F_{\\omega} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega \\tag{7}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\nF_{\\omega}=\\int F(\\boldsymbol{x}^{\\prime}) u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) \\mathrm{d}^{3} x^{\\prime}=\\int F(\\boldsymbol{x}) u_{\\omega}^{*}(\\boldsymbol{x}) \\mathrm{d}^{3} x \\tag{8}\n\\end{equation*}\n\n\nSubstitute equation (7) into equation (6), we obtain\n\n\\begin{equation*}\nC_{\\omega}=F_{\\omega} /(\\omega-\\omega_{0}) \\tag{9}\n\\end{equation*}\n\n\nSubstitute back into equation (5), the solution to equation (4) is\n\n\\begin{equation*}\\nu(\\boldsymbol{x})=\\int \\frac{F_{\\omega}}{\\omega-\\omega_{0}} u_{\\omega}(\\boldsymbol{x}) \\mathrm{d} \\omega \\tag{10}\n\\end{equation*}", + "final_answer": [ + "\\frac{F_{\\omega}}{\\omega-\\omega_{0}} u_{\\omega}(\\boldsymbol{x})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$F_{\\omega}$": "Expansion coefficient for the known function $F(\\boldsymbol{x})$ in the basis of eigenfunctions $u_{\\omega}(\\boldsymbol{x})$, defined as $F_{\\omega}=\\int F(\\boldsymbol{x}^{\\prime}) u_{\\omega}^{*}(\\boldsymbol{x}^{\\prime}) \\mathrm{d}^{3} x^{\\prime}$.", + "$\\omega$": "Continuous eigenvalue of the operator $\\hat{H}$.", + "$\\omega_{0}$": "A specific, constant eigenvalue of the operator $\\hat{H}$.", + "$u_{\\omega}(\\boldsymbol{x})$": "Eigenfunction of the operator $\\hat{H}$ corresponding to eigenvalue $\\omega$ at position $\\boldsymbol{x}$." + } + }, + { + "id": 23, + "context": "", + "question": "For a hydrogen-like ion (nuclear charge $Z e$ ), an electron is in the bound state $\\psi_{n l m}$, calculate $\\langle r^{\\lambda}\\rangle, \\lambda=-3．", + "answer": "Known energy levels of a hydrogen-like ion are\n\n\\begin{equation*}\nE_{n l m}=E_{n}=-\\frac{Z^{2} e^{2}}{2 n^{2} a_{0}}, \\quad n=n_{r}+l+1 \\tag{1}\n\\end{equation*}\n\n\nwhere $a_{0}=\\hbar^{2} / \\mu e^{2}$ is the Bohr radius. According to the virial theorem,\n\n\\begin{equation*}\n\\langle\\frac{p^{2}}{2 \\mu}\\rangle_{n l m}=\\langle\\frac{r}{2} \\frac{\\mathrm{~d} V}{\\mathrm{~d} r}\\rangle_{n l m}=\\frac{Z e^{2}}{2}\\langle\\frac{1}{r}\\rangle_{n l m}=-\\frac{1}{2}\\langle V\\rangle_{n l m} \\tag{2}\n\\end{equation*}\n\n\nso\n\nE_{n}=\\frac{1}{2}\\langle V\\rangle_{n l m}=-\\frac{Z e^{2}}{2}\\langle\\frac{1}{r}\\rangle_{n l m}\n\n\n\\begin{equation*}\n\\langle\\frac{1}{r}\\rangle_{n l m}=-\\frac{2 E_{n}}{Z e^{2}}=\\frac{Z}{n^{2} a_{0}} . \\quad n=1,2,3, \\cdots \\tag{3}\n\\end{equation*}\n\n\nFurther, the spherical coordinate representation of $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\psi_{n l m}(r, \\theta, \\varphi)=R_{n l}(r) Y_{l m}(\\theta, \\varphi) \\tag{4}\n\\end{equation*}\n\n\nIt is a common eigenfunction of $(H, l^{2}, l_{z})$, satisfying the energy eigenvalue equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r \\psi_{n l m}+[\\frac{l(l+1) \\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r}] \\psi_{n l m}=E_{n} \\psi_{n l m} \\tag{5}\n\\end{equation*}\n\n\nThe total energy operator is equivalent to\n\n\\begin{equation*}\nH \\rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{1}{r} \\frac{\\partial^{2}}{\\partial r^{2}} r+l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{Z e^{2}}{r} \\tag{6}\n\\end{equation*}\n\n\nConsidering $l$ as a parametric variable, differentiate equation (6) with respect to $l$, using the Hellmann theorem (refer to Chapter 8), we get\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial l}=\\langle\\frac{\\partial H}{\\partial l}\\rangle_{n l m}=(l+\\frac{1}{2}) \\frac{\\hbar^{2}}{\\mu}\\langle\\frac{1}{r^{2}}\\rangle_{n l m} \\tag{7}\n\\end{equation*}\n\n\nSince $n=n_{r}+l+1$, we find\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial l}=\\frac{\\partial E_{n}}{\\partial n}=\\frac{Z^{2} e^{2}}{n^{3} a_{0}} \\tag{8}\n\\end{equation*}\n\n\nSubstituting into equation (7) and using $a_{0}=\\hbar^{2} / \\mu e^{2}$, we obtain\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{2}}\\rangle_{n l m}=\\frac{1}{(l+\\frac{1}{2}) n^{3}} \\frac{Z^{2}}{a_{0}^{2}} \\tag{9}\n\\end{equation*}\n\n\nFinally, compute $(r^{-3})$．\nFor the s state $(l=0), r \\rightarrow 0$, $\\psi \\rightarrow C$ (constant), therefore\n\n\\begin{equation*}\n\\langle r^{-3}\\rangle_{n 00} \\rightarrow \\infty \\tag{10}\n\\end{equation*}\n\n\nWhen $l \\neq 0$, using formula (7b) in problem (5.7), we get\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{Z}{l(l+1) a_{0}}\\langle\\frac{1}{r^{2}}\\rangle_{n l m} \\tag{11}\n\\end{equation*}\n\n\nThus\n\n\\begin{equation*}\n\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{1}{n^{3} l(l+\\frac{1}{2})(l+1)}(\\frac{Z}{a_{0}})^{3} \\tag{12}\n\\end{equation*}\n\n\nAs $l \\rightarrow 0$, the right side of the equation $\\rightarrow \\infty$, so this formula is actually applicable for all $l$ values.\nDiscussion: Since both the total energy operator and radial equation are independent of the magnetic quantum number $m$, $\\langle r^{\\lambda}\\rangle$ is independent of $m$. However, $\\langle r^{-1}\\rangle$ is also independent of the angular quantum number $l$, depending only on the principal quantum number $n.$ $\\langle r^{-2}\\rangle$ and $\\langle r^{-3}\\rangle$ depend on both $n$ and $l$, meaning for states with the same energy level but different \"orbital shapes\" (different $l$), $\\langle r^{-2}\\rangle$ or $\\langle r^{-3}\\rangle$ have different values.\n\nUsing formula (9), it can be easily obtained that the average value of the centrifugal potential energy in the state $\\psi_{n l m}$ is\n\n\\begin{equation*}\n\\langle\\frac{l^{2}}{2 \\mu r^{2}}\\rangle_{n l m}=\\frac{l(l+1) Z^{2} e^{2}}{(2 l+1) n^{3} a_{0}}=-\\frac{l(l+1)}{(l+\\frac{1}{2}) n} E_{n} \\tag{13}\n\\end{equation*}\n\n\nSince $(-E_{n})$ is the average kinetic energy, the proportion of centrifugal potential energy within kinetic energy is $l(l+1) /(l+\\frac{1}{2}) n$. When $n$ is fixed, as $l$ increases, this proportion grows. When $l$ takes the maximum value $(l=n-1)$, this proportion is $(n-1) /$ $(n-\\frac{1}{2})$, and the radial kinetic energy occupies only $1 /(2 n-1)$ of the kinetic energy in this case. Therefore, if $n \\gg 1,(n, n-1, m)$ implies small radial kinetic energy, corresponding to circular orbits in Bohr's quantum theory.", + "final_answer": [ + "\\langle\\frac{1}{r^{3}}\\rangle_{n l m}=\\frac{1}{n^{3} l(l+\\frac{1}{2})(l+1)}(\\frac{Z}{a_{0}})^{3}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\langle\\frac{1}{r^{3}}\\rangle_{n l m}$": "Expectation value of $1/r^3$ for the state $\\psi_{n l m}$.", + "$n$": "Principal quantum number, $n=n_{r}+l+1$.", + "$l$": "Azimuthal (orbital angular momentum) quantum number.", + "$Z$": "Nuclear charge number (atomic number) of the hydrogen-like ion.", + "$a_{0}$": "Bohr radius, $a_{0}=\\hbar^{2} / \\mu e^{2}$." + } + }, + { + "id": 24, + "context": "", + "question": "The potential acting on the valence electron (outermost electron) of a monovalent atom by the atomic nucleus (atomic nucleus and inner electrons) can be approximately expressed as\n\n\\begin{equation*}\nV(r)=-\\frac{e^{2}}{r}-\\lambda \\frac{e^{2} a_{0}}{r^{2}}, \\quad 0<\\lambda \\ll 1 \\tag{1}\n\\end{equation*}\nwhere $a_{0}$ is the Bohr radius. Find the energy level of the valence electron and compare it with the energy level of the hydrogen atom.", + "answer": "Take the complete set of conserved quantities as $(H, l^{2}, l_{z})$, whose common eigenfunctions are\n\n\\begin{equation*}\n\\psi(r, \\theta, \\varphi)=R(r) \\mathrm{Y}_{l m}(\\theta, \\varphi)=\\frac{u(r)}{r} \\mathrm{Y}_{l m}(\\theta, \\varphi) \\tag{2}\n\\end{equation*}\n\n$u(r)$ satisfies the radial equation\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{e^{2}}{r}-\\lambda \\frac{e^{2} a_{0}}{r^{2}}] u=E u \\tag{3}\n\\end{equation*}\n\n\nLet\n\n\\begin{equation*}\nl(l+1)-2 \\lambda=l^{\\prime}(l^{\\prime}+1) \\tag{4}\n\\end{equation*}\n\n\nEquation (3) can then be transformed into\n\n\\begin{equation*}\n-\\frac{\\hbar^{2}}{2 \\mu} u^{\\prime \\prime}+[l^{\\prime}(l^{\\prime}+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}-\\frac{e^{2}}{r}] u=E u \\tag{3'}\n\\end{equation*}\n\n\nThis is equivalent to the radial equation of the hydrogen atom with $l$ replaced by $l^{\\prime}$. Therefore, the solution process for equation ($3^{\\prime}$) is entirely analogous to the hydrogen atom problem. The latter's energy levels are\n\n\\begin{equation*}\nE_{n}=-\\frac{e^{2}}{2 n^{2} a_{0}}, \\quad n=n_{r}+l+1, \\quad n_{r}=0,1,2, \\cdots \\tag{5}\n\\end{equation*}\n\n\nReplacing $l$ with $l^{\\prime}$ gives the energy levels of the valence electron:\n\n\\begin{equation*}\nE_{n l}=-\\frac{e^{2}}{2(n^{\\prime})^{2} a_{0}}, \\quad n^{\\prime}=n_{r}+l^{\\prime}+1 \\tag{6}\n\\end{equation*}\n\n\nIt is generally assumed that\n\n\\begin{equation*}\nl^{\\prime}=l+\\Delta_{l} \\tag{7}\n\\end{equation*}\n\n\n\\begin{equation*}\nn^{\\prime}=n_{r}+l+\\Delta_{l}+1=n+\\Delta_{l} \\tag{8}\n\\end{equation*}\n\n$\\Delta_{l}$ is referred to as the 'correction number' of the quantum numbers $l$ and $n$. Since $\\lambda \\ll 1$, equation (4) can be approximated as follows:\n\n\\begin{aligned}\nl(l+1)-2 \\lambda & =l^{\\prime}(l^{\\prime}+1)=(l+\\Delta_{l})(l+\\Delta_{l}+1) \\\\\n& =l(l+1)+(2 l+1) \\Delta_{l}+(\\Delta_{l})^{2}\n\\end{aligned}\n\n\nNeglecting $(\\Delta_{l})^{2}$, we get\n\n\\begin{equation*}\n\\Delta_{l} \\approx-\\lambda /(l+\\frac{1}{2}) \\tag{9}\n\\end{equation*}\n\n\nSince $\\lambda \\ll 1$, $|\\Delta_{l}| \\ll 1$. Thus, the energy level $E_{n l}$ obtained in this problem has only a small difference from the hydrogen atomic energy level, but the 'degeneracy in $l$' of the energy levels has been removed. Equation (6) is broadly consistent with experimental data on alkali metal spectra; in particular, the correction number $|\\Delta_{l}|$ decreases as $l$ increases, which agrees well with experiments.\n\nThe exact solution for equation (4) is\n\n\\begin{equation*}\nl^{\\prime}=-\\frac{1}{2}+(l+\\frac{1}{2})[1-\\frac{8 \\lambda}{(2 l+1)^{2}}]^{\\frac{1}{2}} \\tag{10}\n\\end{equation*}\n\n\nExpanding the above equation as a binomial series and retaining the $\\lambda$ term while neglecting terms of order $\\lambda^{2}$ and higher, yields equation (9).", + "final_answer": [ + "E_{n l}=-\\frac{e^{2}}{2(n^{\\prime})^{2} a_{0}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_{nl}$": "Energy levels of the valence electron.", + "$e$": "Elementary charge.", + "$n'$": "Modified principal quantum number for the valence electron.", + "$a_0$": "Bohr radius." + } + }, + { + "id": 25, + "context": "", + "question": "For a particle of mass $\\mu$ moving in a spherical shell $\\delta$ potential well\n\n\\begin{equation*}\nV(r)=-V_{0} \\delta(r-a), \\quad V_{0}>0, a>0 \\tag{1}\n\\end{equation*}\nfind the minimum value of $V_{0}$ required for bound states to exist.", + "answer": "The ground state is an s-state $(l=0)$, and the wave function can be expressed as\n\n\\begin{equation*}\n\\psi(r)=u(r) / r \\tag{2}\n\\end{equation*}\n\n$u(r)$ satisfies the radial equation\n\n\\begin{equation*}\\nu^{\\prime \\prime}+\\frac{2 \\mu E}{\\hbar^{2}} u+\\frac{2 \\mu V_{0}}{\\hbar^{2}} \\delta(r-a) u=0 \\tag{3}\n\\end{equation*}\n\n\nAs $r \\rightarrow \\infty$, $V(r) \\rightarrow 0$, so for a bound state $E<0$. Define\n\n\\begin{equation*}\n\\beta=\\sqrt{-2 \\mu E} / \\hbar \\tag{4}\n\\end{equation*}\n\n\nEquation (3) can be rewritten as\n\n\\begin{equation*}\n\\nu^{\\prime \\prime} - \\beta^{2} u + \\frac{2 \\mu V_{0}}{\\hbar^{2}} \\delta(r-a) u=0 \\tag{3^{\\prime}}\n\\end{equation*}\n\n\nThe boundary conditions are\n\n$$r \\rightarrow 0, \\infty \\text {, } u \\rightarrow 0$$\n\n\nIntegrating equation ($3^{\\prime}$) around $r \\sim a$, we obtain the jump condition for $u^{\\prime}$ (see Problem 2.1)\n\n\\begin{equation*}\n \\nu^{\\prime}(a+0)-u^{\\prime}(a-0)=-\\frac{2 \\mu V_{0}}{\\hbar^{2}} u(a) \\tag{5}\n\\end{equation*}\n\n\nThis means\n\n\\begin{equation*}\n\\frac{u^{\\prime}}{u}|_{r=a-0} ^{r=a+0}=-\\frac{2 \\mu V_{0}}{\\hbar^{2}} \\tag{$\\prime$}\n\\end{equation*}\n\n\nFor $r \\neq a$, equation ($3^{\\prime}$) becomes\n\n\\begin{equation*}\\nu^{\\prime \\prime}-\\beta^{2} u=0 \\tag{\\prime\\prime}\n\\end{equation*}\n\n\nIn the region $r>a$, the solution satisfying the boundary condition at infinity is\n\n\\begin{equation*}\\nu=C \\mathrm{e}^{-\\beta r}, \\quad r>a \\tag{6}\n\\end{equation*}\n\n\nThus\n\n\\begin{equation*}\n(\\frac{u^{\\prime}}{u})_{r=a+0}=-\\beta \\tag{7}\n\\end{equation*}\n\n\nIf the value of $V_{0}$ is just sufficient to form the first bound state, the energy level must be $E=0^{-}$, at which point $\\beta=0$, and equation ($3^{\\prime\\prime}$) becomes\n\n\\begin{equation*}\\nu^{\\prime \\prime}=0 \\quad(E \\rightarrow 0^{-}) \\tag{8}\n\\end{equation*}\n\n\nEquation (7) becomes\n\n\\begin{equation*}\n(\\frac{u^{\\prime}}{u})_{r=a+0}=0 \\quad(E \\rightarrow 0^{-}) \\tag{7'}\n\\end{equation*}\n\nWhen $E \\rightarrow 0^{-}$, the solution of equation (8) in the region $r0$, i.e., $\\langle\\sigma_{z}\\rangle_{t}$.", + "answer": "First, determine the spin wave function, then compute $\\langle\\boldsymbol{\\sigma}\\rangle$. Using $\\boldsymbol{n}$ to denote the unit vector in the $(\\theta, \\varphi)$ direction, $\\boldsymbol{e}_{1}, ~ \\boldsymbol{e}_{2}, ~ \\boldsymbol{e}_{3}$ denote the unit vectors in the $x, ~ y, ~ z$ directions, respectively:\n\n\\begin{align*}\n\\boldsymbol{n} & =n_{1} \\boldsymbol{e}_{1}+n_{2} \\boldsymbol{e}_{2}+n_{3} \\boldsymbol{e}_{3} \\\\\n& =\\sin \\theta \\cos \\varphi \\boldsymbol{e}_{1}+\\sin \\theta \\sin \\varphi \\boldsymbol{e}_{2}+\\cos \\theta \\boldsymbol{e}_{3} \\tag{1}\n\\end{align*}\n\nThe Hamiltonian associated with the spin motion is\n\n\\begin{equation*}\nH=-\\boldsymbol{\\mu} \\cdot \\boldsymbol{B}=-\\mu_{0} B \\boldsymbol{\\sigma} \\cdot \\boldsymbol{n}=-\\mu_{0} B_{\\sigma_{n}} \\tag{2}\n\\end{equation*}\n\nIn the $\\sigma_{z}$ representation, the matrix representation of $\\sigma_{n}$ is\n\n\\sigma_{n}=\\sigma_{x} n_{1}+\\sigma_{y} n_{2}+\\sigma_{z} n_{3}=[\\begin{array}{cc}\nn_{3} & n_{1}-\\mathrm{i} n_{2} \\tag{3}\\\\\nn_{1}+\\mathrm{i} n_{2} & -n_{3}\n\\end{array}]\n\nAssume the spin wave function of the particle is\n\n\\chi(t)=[\\begin{array}{l}\na(t) \\tag{4}\\\\\nb(t)\n\\end{array}]\n\n$\\chi(t)$ satisfies the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\chi(t)=H \\chi(t)=-\\mu_{0} B \\sigma_{n} \\chi(t) \\tag{5}\n\\end{equation*}\n\nThe eigenvalues of $\\sigma_{n}$ are $\\pm 1$, the eigenvalues of $H$ (i.e., stationary energy levels) are\n\n\\begin{gather*}\nE=\\mp \\mu_{0} B=\\mp \\hbar \\omega \\\\\n\\omega=\\mu_{0} B / \\hbar \\tag{6}\n\\end{gather*}\n\n\nSince $\\omega$ and $\\mu_{0}$ have the same sign, if $\\mu_{0}<0$ then $\\omega<0$. The eigenfunctions of $\\sigma_{n}$ and $H$ are\n\n\\begin{gather*}\n\\phi_{1}=\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{c}\n1+n_{3} \\\\\nn_{1}+\\mathrm{i} n_{2}\n\\end{array}] \\quad(\\sigma_{n}=1, E=-\\hbar \\omega) \\\\\n\\phi_{-1}=\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{c}\nn_{1}-\\mathrm{i} n_{2} \\\\\n-1-n_{3}\n\\end{array}] \\quad(\\sigma_{n}=-1, E=\\hbar \\omega) \\tag{7}\n\\end{gather*}\n\nThe general solution of equation (5) is\n\n\\begin{equation*}\n\\chi(t)=C_{1} \\phi_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}+C_{-1} \\phi_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t} \\tag{8}\n\\end{equation*}\n\n$C_{1}, ~ C_{-1}$ are determined by initial conditions:\n\n\\begin{equation*}\nC_{1}=\\phi_{1}^{+} \\chi(0), \\quad C_{-1}=\\phi_{-1}^{+} \\chi(0) \\tag{9}\n\\end{equation*}\n\n\nThe initial wave function for this problem is the eigenfunction of $\\sigma_{z}=1$, that is\n\n\\chi(0)=\\chi_{\\frac{1}{2}}=[\\begin{array}{l}\n1 \\tag{10}\\\\\n0\n\\end{array}]\n\n\nThus\n\n\\begin{equation*}\nC_{1}=\\sqrt{\\frac{1+n_{3}}{2}}, \\quad C_{-1}=\\frac{n_{1}+\\mathrm{i} n_{2}}{\\sqrt{2(1+n_{3})}} \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (8), we get\n\n\\begin{align*}\n\\chi(t) & =\\frac{1}{\\sqrt{2(1+n_{3})}}[\\begin{array}{l}\n(1+n_{3}) C_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}+(n_{1}-\\mathrm{i} n_{2}) C_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t} \\\\\n(n_{1}+\\mathrm{i} n_{2}) C_{1} \\mathrm{e}^{\\mathrm{i} \\omega t}-(1+n_{3}) C_{-1} \\mathrm{e}^{-\\mathrm{i} \\omega t}\n\\end{array}] \\\\\n& =[\\begin{array}{l}\n\\cos \\omega t+\\mathrm{i} n_{3} \\sin \\omega t \\\\\n(\\mathrm{i} n_{1}-n_{2}) \\sin \\omega t\n\\end{array}]=[\\begin{array}{l}\na(t) \\\\\nb(t)\n\\end{array}] \\tag{12}\n\\end{align*}\n\n\nThe expectation values of $\\boldsymbol{\\sigma}$ in the $\\chi(t)$ state are\n\n\\begin{align*}\n\\langle\\sigma_{x}\\rangle & =\\chi^{+} \\sigma_{x} \\chi=[a^{*} b^{*}][\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array}][\\begin{array}{l}\na \\\\\nb\n\\end{array}]=a^{*} b+b^{*} a \\\\\n& =n_{1} n_{3}(1-\\cos 2 \\omega t)-n_{2} \\sin 2 \\omega t \\tag{13a}\\\\\n\\langle\\sigma_{y}\\rangle & =\\chi^{+} \\sigma_{y} \\chi=\\mathrm{i}(b^{*} a-a^{*} b) \\\\\n& =n_{2} n_{3}(1-\\cos 2 \\omega t)+n_{1} \\sin 2 \\omega t \\tag{1;b}\\\\\n\\langle\\sigma_{z}\\rangle & =\\chi^{-} \\sigma_{*} \\chi=a^{*} a-b^{*} b \\\\\n& =n_{3}^{2}+(1-n_{3}^{2}) \\cos 2 \\omega t \\tag{13c}\n\\end{align*}\n\n\nIf the magnetic field $\\boldsymbol{B}$ points in the positive $x$ direction, then\n\nn_{1}=1, \\quad n_{2}=n_{3}=0\n\n\nThen equation (13) becomes\n\n\\langle\\sigma_{x}\\rangle=0,\\langle\\sigma_{y}\\rangle=\\sin 2 \\omega t, \\quad\\langle\\sigma_{z}\\rangle=\\cos 2 \\omega t\n\n\nThis is precisely the result obtained in the previous problem (note that $\\omega$ in the previous problem corresponds to $-\\omega$ in this problem).", + "final_answer": [ + "n_{3}^{2}+(1-n_{3}^{2}) \\cos 2 \\omega t" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$n_3$": "z-component of the unit vector in the magnetic field direction, $n_3 = \\cos \\theta$.", + "$\\omega$": "Larmor frequency, $\\omega=\\mu_{0} B / \\hbar$.", + "$t$": "Time.", + "$\\theta$": "Polar angle of the magnetic field direction." + } + }, + { + "id": 28, + "context": "", + "question": "In a system with a spin of $\\hbar / 2$, the magnetic moment $\\boldsymbol{\\mu}=\\mu_{0} \\boldsymbol{\\sigma}$ is placed in a uniform magnetic field $\\boldsymbol{B}_{0}$ directed along the positive $z$ direction for $t<0$. At $t \\geqslant 0$, an additional rotating magnetic field $\\boldsymbol{B}_{1}(t)$, perpendicular to the $z$ axis, is applied: \n\n$$ \\boldsymbol{B}_{1}(t)=B_{1} \\cos 2 \\omega_{0} t e_{1}-B_{1} \\sin 2 \\omega_{0} t e_{2},$$\n\nwhere $\\omega_{0}=\\mu_{0} B_{0} / \\hbar$. It is known that for $t \\leqslant 0$, the system is in the eigenstate $\\chi_{\\frac{1}{2}}$ with $s_{z}=\\hbar / 2$. Find the expression for the time $\\Delta t$ it takes for the system's spin to first reverse from $s_z = \\hbar/2$ (along the positive $z$ direction) to $s_z = -\\hbar/2$ (along the negative $z$ direction) starting from $t=0$. Express this in terms of $\\omega_1 = \\mu_0 B_1 / \\hbar$ and relevant constants.", + "answer": "The Hamiltonian related to the spin motion of the system is\n\n\\begin{equation*}\nH=-\\mu \\cdot[\\boldsymbol{B}_{0}+\\boldsymbol{B}_{1}(t)], \\quad t \\geqslant 0 \\tag{1}\n\\end{equation*}\n\n\nIn the $s_{z}$ representation, the matrix form of $H$ is\n\n\\begin{align}\nH & =-\\mu_{0} B_{1}(\\sigma_{x} \\cos 2 \\omega_{0} t-\\sigma_{y} \\sin 2 \\omega_{0} t)-\\mu_{0} B_{0} \\sigma_{z} \\\\\n& =-\\mu_{0} \\begin{pmatrix}\nB_{0} & B_{1} \\mathrm{e}^{2 i \\omega_{0} t} \\\\\nB_{1} \\mathrm{e}^{-2 i \\omega_{0} t} & -B_{0}\n\\end{pmatrix}\n\\tag{\\prime}\n\\end{align}\n\n\nAssuming the wave function for $t \\geqslant 0$ is\n\n$$\\chi(t)=\\begin{pmatrix}\na(t) \\tag{2}\\\\\nb(t)\n\\end{pmatrix}.$$\n\nSubstituting into the Schrödinger equation\n\n\\begin{equation*}\n\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\chi(t)=H \\chi(t) \\tag{3},\n\\end{equation*}\nwe have \n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} a(t)=\\mathrm{i} \\omega_{0} a(t)+\\mathrm{i} \\omega_{1} b(t) \\mathrm{e}^{2 \\mathrm{i} \\omega_{0} t} \\\\\n\\frac{\\mathrm{~d}}{\\mathrm{~d} t} b(t)=-\\mathrm{i} \\omega_{0} b(t)+\\mathrm{i} \\omega_{1} a(t) \\mathrm{e}^{-2 \\mathrm{i} \\omega_{0} t}. \\tag{4}\n\\end{gather*}\n\n\nwhere\n\n\\begin{equation*}\n\\omega_{0}=\\frac{\\mu_{0} B_{0}}{\\hbar}, \\quad \\omega_{1}=\\frac{\\mu_{0} B_{1}}{\\hbar} \\tag{5}\n\\end{equation*}\n\n\nLet\n\n\\begin{equation*}\na(t)=c_{1}(t) \\mathrm{e}^{\\mathrm{i} \\omega_{0} t}, \\quad b(t)=c_{2}(t) \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (4), yields equations for $c_{1}$ and $c_{2}$\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} c_{1}(t)=\\mathrm{i} \\omega_{1} c_{2}(t) \\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} c_{2}(t)=\\mathrm{i} \\omega_{1} c_{1}(t) \\tag{7}\n\\end{align*}\n\n\nThe initial conditions are\n\n\\begin{equation*}\n\\chi(0)=\\begin{pmatrix}{l}\na(0) \\\\\nb(0)\n\\end{pmatrix}=\\begin{pmatrix}\n1 \\\\\n0\n\\end{pmatrix}=\\chi_{\\frac{1}{2}}, \\quad \\text { i.e., } \\quad \\left\\{\\begin{array}{l}\nc_{1}(0)=1 \\\\\nc_{2}(0)=0\n\\end{array}\\right.\n\\tag{8}\n\\end{equation*}\n\nBy adding and subtracting equations in (7), we obtain\n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(c_{1}+c_{2})=\\mathrm{i} \\omega_{1}(c_{1}+c_{2}) \\\\\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(c_{1}-c_{2})=-\\mathrm{i} \\omega_{1}(c_{1}-c_{2}) \\tag{9}\n\\end{gather*}\n\n\nThe solution is\n\n\\begin{align*}\nc_{1}(t)+c_{2}(t) & =[c_{1}(0)+c_{2}(0)] \\mathrm{e}^{\\mathrm{i} \\omega_{1} t} = \\mathrm{e}^{\\mathrm{i} \\omega_{1} t}\\\\\nc_{1}(t)-c_{2}(t) &= (c_{1}(0)-c_{2}(0)) \\mathrm{e}^{-\\mathrm{i} \\omega_{1} t}=\\mathrm{e}^{-\\mathrm{i} \\omega_{1} t} .\n\\end{align*}\n\n\nAdding and subtracting these, we find\n\n\\begin{equation*}\nc_{1}(t)=\\cos \\omega_{1} t, \\quad c_{2}(t)=\\mathrm{i} \\sin \\omega_{1} t \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (6), we obtain\n\n\\begin{equation*}\na(t)=\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t}, \\quad b(t)=\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\tag{12}\n\\end{equation*}\n\n\nSubstituting into equation (2), we find\n\\begin{equation}\n\\begin{split}\n\\chi(t)&=\\begin{pmatrix}\n\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\\\\n\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t}\n\\end{pmatrix} \\\\\n&=\\cos \\omega_{1} t \\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\chi_{\\frac{1}{2}}+\\mathrm{i} \\sin \\omega_{1} t \\mathrm{e}^{-\\mathrm{i} \\omega_{0} t} \\chi_{-\\frac{1}{2}} \\tag{13}\n\\end{split}\n\\end{equation}\n\nClearly\n\\begin{gathered}\nt=0, \\quad \\chi=\\chi_{\\frac{1}{2}}=[\\begin{array}{l}\n1 \\\\\n0\n\\end{array}] \\\\\ns_{z}=\\frac{\\hbar}{2}, \\quad\\langle s\\rangle=\\frac{\\hbar}{2} e_{3} \\\\\nt=\\frac{\\pi}{2 \\omega_{1}}, \\quad \\chi=\\mathrm{ie}^{-\\mathrm{i} \\omega_{0} t} \\chi_{-\\frac{1}{2}}=\\mathrm{ie}^{-\\mathrm{i} \\omega_{0} t}[\\begin{array}{l}\n0 \\\\\n1\n\\end{array}] \\\\\ns_{\\tilde{z}}=-\\frac{\\hbar}{2}, \\quad\\langle\\boldsymbol{s}\\rangle=-\\frac{\\hbar}{2} e_{3} \\\\\nt=\\frac{\\pi}{\\omega_{1}}, \\quad \\chi=-\\mathrm{e}^{\\mathrm{i} \\omega_{0} t} \\chi_{\\frac{1}{2}}=-\\mathrm{e}^{\\mathrm{i} \\omega_{0} t}[\\begin{array}{l}\n1 \\\\\n0\n\\end{array}] \\\\\ns_{z}=\\frac{\\hbar}{2}, \\quad\\langle\\boldsymbol{s}\\rangle=\\frac{\\hbar}{2} e_{3}\n\\end{gathered}\n\nIn other words, the system's spin direction changes once every $\\Delta t=\\pi / 2 \\omega_{1}=\\pi \\hbar / 2 \\mu_{0} B_{1}$ . The spin state of the system undergoes periodic oscillation between $\\chi_{\\frac{1}{2}}$ and $\\chi_{-\\frac{1}{2}}$, with a period $T=2 \\Delta t=\\pi \\hbar / \\mu_{0} B_{1}$. \n\nThis problem illustrates the basic principle of magnetic resonance.", + "final_answer": [ + "\\pi \\hbar / 2 \\mu_{0} B_{1}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\pi$": "Mathematical constant, approximately 3.14159.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\mu_{0}$": "Constant relating magnetic moment to spin.", + "$B_{1}$": "Amplitude of the rotating magnetic field." + } + }, + { + "id": 29, + "context": "", + "question": "The magnetic moment (operator) of an electron is\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=\\boldsymbol{\\mu}_{l}+\\boldsymbol{\\mu}_{s}=-\\frac{e}{2 m_{\\mathrm{e}} c}(\\boldsymbol{l}+2 \\boldsymbol{s}) \\tag{1}\n\\end{equation*}\n\n\nTry to calculate the expectation value of $\\mu_{z}$ for the $|l j m_{j}\\rangle$ state.", + "answer": "If we use the Bohr magneton $\\mu_{\\mathrm{B}}=e \\hbar / 2 m_{\\mathrm{e}} c$ as the unit of magnetic moment, then the magnetic moment operator of an electron can be written as (here $\\hbar=1$ is taken)\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=-(\\boldsymbol{l}+2 \\boldsymbol{s})=-(\\boldsymbol{j}+\\boldsymbol{s})=-(\\boldsymbol{j}+\\frac{1}{2} \\boldsymbol{\\sigma}) \\tag{2}\n\\end{equation*}\n\n\nThus, we have\n\n\\begin{equation*}\n\\langle l j m_{j}| \\mu_{z}|l j m_{j}\\rangle=-g m_{j} \\tag{3}\n\\end{equation*}\n\n\nwhere\n\n\\begin{equation*}\ng=1+\\frac{j(j+1)-l(l+1)+3 / 4}{2 j(j+1)} \\quad \\text { (Landè } g \\text { factor) } \\tag{4}\n\\end{equation*}\n\n\nThe average value of $\\mu_{z}$ for $m_{j}=j$ (its maximum value) is usually taken as the definition of the magnetic moment observable, denoted as $\\mu$. For an electron,\n\n\\begin{equation*}\n\\mu=\\langle l j j| \\mu_{z}|l j j\\rangle=-g j \\tag{5}\n\\end{equation*}\n\n\nthat is\n\n\\mu= \\begin{cases}-(j+\\frac{1}{2}), & j=l+\\frac{1}{2} \\tag{$\\prime$}\\\\ -j(2 j+1) /(2 j+2), & j=l-\\frac{1}{2}\\end{cases}", + "final_answer": [ + "\\langle l j m_{j}| \\mu_{z}|l j m_{j}\\rangle=-(1+\\frac{j(j+1)-l(l+1)+3 / 4}{2 j(j+1)}) m_{j}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$l$": "Orbital angular momentum quantum number", + "$j$": "Total angular momentum quantum number", + "$m_{j}$": "Z-component of total angular momentum quantum number", + "$\\mu_{z}$": "Z-component of the magnetic moment operator" + } + }, + { + "id": 30, + "context": "", + "question": "A system composed of two spin-$1 / 2$ particles is placed in a uniform magnetic field, with the magnetic field direction as the $z$-axis. The Hamiltonian of the system related to spin is given by\n\n\\begin{equation*}\nH=a \\sigma_{1 z}+b \\sigma_{2 z}+c_{0} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2} \\tag{1}\n\\end{equation*}\n\n\nwhere $a, ~ b$ terms arise from the interaction between the magnetic field and the particles' intrinsic magnetic moments, and the $c_{0}$ term arises from the interaction between the two particles. $a, ~ b, ~ c_{0}$ are real constants. If the system is in the common eigenstate $\\chi_{11}=\\alpha(1) \\alpha(2)$ of the total spin operators $(\\boldsymbol{S}^{2}, S_z)$, find the energy level of the system in this case.", + "answer": "We will solve using matrix methods in spin state vector space. The basis vectors can be chosen as the common eigenstates of $(\\sigma_{1 z}, \\sigma_{2 z})$\n\n$$\\alpha(1) \\alpha(2), \\quad \\alpha(1) \\beta(2), \\quad \\beta(1) \\alpha(2), \\quad \\beta(1) \\beta(2) $$\n\n\nor as the common eigenstates $\\chi_{S M_{s}}$ of the total spin operators $(\\boldsymbol{S}^{2}, S_{z})$. For this problem, considering the diagonalization of $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$, it is more convenient to use $\\chi_{S M_{S}}$ as the basis vectors. For convenience, the basis vector order is as follows:\n\n\\begin{array}{l}\n\\chi_{1}=\\chi_{11}=\\alpha(1) \\alpha(2) \\tag{2}\\\\\n\\chi_{2}=\\chi_{1-1}=\\beta(1) \\beta(2) \\\\\n\\chi_{3}=\\chi_{10}=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)+\\beta(1) \\alpha(2)] \\\\\n\\chi_{4}=\\chi_{00}=\\frac{1}{\\sqrt{2}}[\\alpha(1) \\beta(2)-\\beta(1) \\alpha(2)]\n\\end{array}}\n\n\nThe Hamiltonian operator can be rewritten as\n\n\\begin{gather*}\nH=c_{0} \\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}+c_{1}(\\sigma_{1 z}+\\sigma_{2 z})+c_{2}(\\sigma_{1 z}-\\sigma_{2 z}) \\tag{1'}\\\\\nc_{1}=\\frac{1}{2}(a+b), \\quad c_{2}=\\frac{1}{2}(a-b) \\tag{3}\n\\end{gather*}\n\n\nAll four basis vectors are common eigenstates of $(\\boldsymbol{S}^{2}, S_{z})$, and also common eigenstates of $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$ and $(\\sigma_{1 z}+\\sigma_{2 z})$. It is easy to see that $\\chi_{1}$ and $\\chi_{2}$ are also eigenstates of $(\\sigma_{1 z}-\\sigma_{2 z})$, thus $\\chi_{1}$ and $\\chi_{2}$ are already eigenstates of $H$.\n\n\\begin{align*}\n& H \\chi_{1}=(c_{0}+2 c_{1}) \\chi_{1}=(c_{0}+a+b) \\chi_{1} \\tag{4}\\\\\n& H \\chi_{2}=(c_{0}-2 c_{1}) \\chi_{2}=(c_{0}-a-b) \\chi_{2} \\tag{5}\n\\end{align*}\n\n\nThus, we obtain two energy levels of the system: $E=c_{0} \\pm 2 c_{1}$.\nIt is easy to calculate the action of $(\\sigma_{1 z}-\\sigma_{2 z})$ on the basis vectors, which is\n\n\\begin{array}{cl}\n(\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{1}=0, & (\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{2}=0 \\\\\n(\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{3}=2 \\chi_{4}, & (\\sigma_{1 z}-\\sigma_{2 z}) \\chi_{4}=2 \\chi_{3} \\tag{7}\n\\end{array}\n\n\nTherefore, in the subspace ${\\chi_{3}, \\chi_{4}}$, the matrix elements of $(\\sigma_{1 z}-\\sigma_{2 z})$ are\n\n\\begin{array}{l}\n(\\sigma_{1 z}-\\sigma_{2 z})_{33}=(\\sigma_{1 z}-\\sigma_{2 z})_{44}=0 \\tag{7'}\\\\\n(\\sigma_{1 z}-\\sigma_{2 z})_{34}=(\\sigma_{1 z}-\\sigma_{2 z})_{43}=2\n\\end{array}\n\nAll matrix elements of $(\\sigma_{1 z}+\\sigma_{2 z})$ are zero. The matrix representation of $H$ is\n\\begin{equation}\n H=\\begin{pmatrix}\nc_{0} & 2 c_{2} \\tag{8}\\\\\n2 c_{2} & -3 c_{0}\n\\end{pmatrix}\n\\end{equation}\n\nAssume the energy eigenstate is\n\n\\begin{equation*}\n\\chi=f_{3} \\chi_{3}+f_{4} \\chi_{4} \\tag{9}\n\\end{equation*}\n\n\nSubstitute into the energy eigenvalue equation\n\n\\begin{equation*}\nH \\chi=E \\chi \\tag{10}\n\\end{equation*}\n\n\nresulting in\n\n[\\begin{array}{cc}\nc_{0}-E & 2 c_{2} \\tag{$\\prime$}\\\\\n2 c_{2} & -3 c_{0}-E\n\\end{array}][\\begin{array}{l}\nf_{1} \\\\\nf_{2}\n\\end{array}]=0\n\n\nThe energy level $E$ is determined by:\n\n\\begin{equation*}\n\\operatorname{det}(H-E)=0 \\tag{11}\n\\end{equation*}\n\n\nnamely\n\n|\\begin{array}{cc}\nc_{0}-E & 2 c_{2} \\tag{11'}\\\\\n2 c_{2} & -3 c_{0}-E\n\\end{array}|=(E-c_{0})(E+3 c_{0})-4 c_{2}^{2}=0\n\n\nSolving gives\n\nE=-c_{0} \\pm 2 \\sqrt{c_{0}^{2}+c_{2}^{2}}\n\n\nConclusion: This problem has four energy levels (excluding accidental degeneracy), which are\n\n\\begin{equation*}\nE=c_{0} \\pm 2 c_{1}, \\quad-c_{0} \\pm 2 \\sqrt{c_{0}^{2}+c_{2}^{2}} \\tag{12}\n\\end{equation*}\n\n\nThe energy eigenstates of the first two energy levels are $\\chi_{1}$ and $\\chi_{2}$, respectively, and the energy eigenstates of the last two energy levels are linear combinations of $\\chi_{3}$ and $\\chi_{4}$.", + "final_answer": [ + "c_{0} + a + b" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$c_{0}$": "Real constant arising from the interaction between the two particles", + "$a$": "Real constant arising from the interaction between the magnetic field and particle 1's intrinsic magnetic moment", + "$b$": "Real constant arising from the interaction between the magnetic field and particle 2's intrinsic magnetic moment" + } + }, + { + "id": 31, + "context": "", + "question": "Consider a system composed of three non-identical spin $1/2$ particles, with the Hamiltonian given by\n\n\\begin{equation*}\nH=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+B(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}) \\cdot \\boldsymbol{s}_{3}, \\tag{1}\n\\end{equation*}\nwhere $A, ~ B$ are real constants. Let $\\boldsymbol{S}_{12}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}$ and $\\boldsymbol{S}_{123}=\\boldsymbol{S}_{12}+\\boldsymbol{s}_{3}$. Try to express the Hamiltonian $H$ as a function of $\\boldsymbol{S}_{12}^{2}$ and $\\boldsymbol{S}_{123}^{2}$. (Take $\\hbar=1$ )", + "answer": "The sum of the spins of particles 1 and 2 is denoted as $\\boldsymbol{S}_{12}$, and the total spin is denoted as $\\boldsymbol{S}_{123}$, that is\n\\begin{equation*}\n\\boldsymbol{S}_{12}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}, \\quad \\boldsymbol{S}_{123}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}+\\boldsymbol{s}_{3}=\\boldsymbol{S}_{12}+\\boldsymbol{s}_{3} \\tag{2}\n\\end{equation*}\n\nEvidently, $\\boldsymbol{S}_{12}$ and $\\boldsymbol{S}_{123}$ both possess the properties of angular momentum, satisfying the commutation relation\n\n\\begin{array}{ll}\n\\boldsymbol{S}_{12} \\times \\boldsymbol{S}_{12}=\\mathrm{i} \\boldsymbol{S}_{12}, & {[\\boldsymbol{S}_{12}^{2}, \\boldsymbol{S}_{12}]=0} \\\\\n\\boldsymbol{S}_{123} \\times \\boldsymbol{S}_{123}=\\mathrm{i} \\boldsymbol{S}_{123}, & {[\\boldsymbol{S}_{123}^{2}, \\boldsymbol{S}_{123}]=0} \\tag{4}\n\\end{array}\n\n$\\boldsymbol{s}_{1}, ~ \\boldsymbol{s}_{2}, ~ \\boldsymbol{s}_{3}$ commute with each other, and\n\n\\begin{equation*}\ns_{1}^{2}=s_{2}^{2}=s_{3}^{2}=\\frac{3}{4} \\tag{5}\n\\end{equation*}\n\n\nTherefore\n\n\\begin{align*}\n& \\boldsymbol{S}_{12}^{2}=\\boldsymbol{s}_{1}^{2}+\\boldsymbol{s}_{2}^{2}+2 \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}=\\frac{3}{2}+2 \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} \\tag{6}\\\\\n& \\boldsymbol{S}_{123}^{2}=\\frac{9}{4}+2(\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+\\boldsymbol{s}_{2} \\cdot \\boldsymbol{s}_{3}+\\boldsymbol{s}_{3} \\cdot \\boldsymbol{s}_{1}) \\tag{7}\n\\end{align*}\n\n\nBased on this, $H$ can be written as\n\n\\begin{align*}\nH & =(A-B) \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+B(\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}+\\boldsymbol{s}_{2} \\cdot \\boldsymbol{s}_{3}+\\boldsymbol{s}_{3} \\cdot \\boldsymbol{s}_{1}) \\\\\n& =\\frac{1}{2}(A-B) \\boldsymbol{S}_{12}^{2}+\\frac{B}{2} \\boldsymbol{S}_{123}^{2}-\\frac{3}{8}(2 A+B) \\tag{$\\prime$}\n\\end{align*}", + "final_answer": [ + "H = \\frac{1}{2}(A-B) \\boldsymbol{S}_{12}^{2}+\\frac{B}{2} \\boldsymbol{S}_{123}^{2}-\\frac{3}{8}(2 A+B)" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$H$": "Hamiltonian of the system.", + "$A$": "Real constant in the Hamiltonian.", + "$B$": "Real constant in the Hamiltonian.", + "$\\boldsymbol{S}_{12}^{2}$": "Square of the magnitude of the sum of spins of particles 1 and 2.", + "$\\boldsymbol{S}_{123}^{2}$": "Square of the magnitude of the total spin of particles 1, 2, and 3." + } + }, + { + "id": 32, + "context": "", + "question": "Same as the previous question, for any value, find\n\n\\begin{equation*}\nd_{j m}^{j}(\\lambda)=\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j m\\rangle \\tag{1}\n\\end{equation*}", + "answer": "According to the general theory of angular momentum,\n\n\\begin{array}{rl}\nJ_{+}|j m\\rangle & =(J_{x}+i J_{y})|j m\\rangle \\tag{2}\\\\\nJ_{-}|j m\\rangle & =a_{j m}|j m+1\\rangle \\\\\nJ_{x}-i J_{y})|j m\\rangle & =a_{j,-m}|j m-1\\rangle\n\\end{array}}\n\n\nwhere\n\n\\begin{equation*}\na_{j m}=\\sqrt{(j-m)(j+m+1)} \\tag{3}\n\\end{equation*}\n\n\nWhen $m=j$\n\n\\begin{equation*}\nJ_{+}|j j\\rangle=0, \\quad\\langle j j| J_{-}=0 \\tag{4}\n\\end{equation*}\n\n(The second equation is the conjugate of the first equation.) Therefore\n\n\\begin{equation*}\n\\langle j j| J_{-} \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j m\\rangle=0. \\tag{5}\n\\end{equation*}\n\nWe also have:\n\\begin{align*}\nJ_{-} \\mathrm{e}^{-\\mathrm{i} \\mathrm{i} J_{y}} & =\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{z} \\sin \\lambda+J_{x} \\cos \\lambda-\\mathrm{i} J_{y}) \\\\\n& =\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}[J_{z} \\sin \\lambda+\\frac{1}{2}(\\cos \\lambda-1) J_{+}+\\frac{1}{2}(\\cos \\lambda+1) J_{-}] \\tag{6}\n\\end{align*}\n\nSubstitute into equation (5), and use equation (2) to obtain\n\n\\begin{gather*}\nm \\sin \\lambda d_{j m}^{j}(\\lambda)+\\frac{1}{2}(\\cos \\lambda-1) a_{j m} d_{j m+1}^{j}(\\lambda) \\\\\n\\quad+\\frac{1}{2}(\\cos \\lambda+1) a_{j,-m} d_{j m-1}^{j}(\\lambda)=0 \\tag{7}\n\\end{gather*}\n\nAdditionally, we can also obtain\n\n\\begin{equation*}\nJ_{z} \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}=\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{z} \\cos \\lambda-J_{x} \\sin \\lambda)=\\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}[J_{z} \\cos \\lambda-\\frac{1}{2} \\sin \\lambda(J_{+}+J_{-})] \\tag{8}\n\\end{equation*}\n\n\nMultiply the above equation from the left with $\\langle j|$ and from the right with $|j m\\rangle$, yielding\n\n\\begin{equation*}\n(j-m \\cos \\lambda) d_{j m}^{j}(\\lambda)+\\frac{1}{2} \\sin \\lambda[a_{j m} d_{j m+1}^{j}(\\lambda)+a_{j,-m} d_{j m-1}^{j}(\\lambda)]=0 \\tag{9}\n\\end{equation*}\n\n\nCombine equations (7) and (9), eliminate $d_{j m-1}^{j}(\\lambda)$, and obtain a simpler recursive relation:\n\n\\begin{equation*}\n\\sin \\lambda \\cdot a_{j m} d_{j m+1}^{j}(\\lambda)=-(j-m)(1+\\cos \\lambda) d_{j m}^{j}(\\lambda) \\tag{10}\n\\end{equation*}\n\nThat is\n\\begin{equation*}\nd_{j m}^{j}(\\lambda)=-\\frac{\\sin \\frac{\\lambda}{2}}{\\cos \\frac{\\lambda}{2}}(\\frac{j+m+1}{j-m})^{\\frac{1}{2}} d_{j m+1}^{j}(\\lambda) \\tag{$10^\\prime$}\n\\end{equation*}\n\n\nThus, once $d_{j j}^{j}$ is found, all $d_{j m}^{j}$ can be recursively derived. Let's first find $d_{j j}^{j}$.\n\n\\begin{equation*}\nd_{j j}^{j}(\\lambda)=\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j j\\rangle \\tag{11}\n\\end{equation*}\n\n\nNote that $d_{j j}^{j}(0)=1$, differentiating the above equation with respect to $\\lambda$, we get\n\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\lambda d_{j j}^{j}(\\lambda)} & =\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(-\\mathrm{i} J_{y})|j j\\rangle \\\\\n& =\\frac{1}{2}\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}(J_{-}-J_{+})|j j\\rangle \\\\\n& =\\frac{1}{2} a_{j-j} d_{j j-1}^{j}(\\lambda) \\tag{12}\n\\end{align*}\n\n\nWhile equation (7) takes $m=j$, noticing $a_{j j}=0$, we get\n\n\\begin{equation*}\n\\frac{1}{2}(\\cos \\lambda+1) a_{j,-j} d_{j j-1}^{j}(\\lambda)=-j \\sin \\lambda \\cdot d_{j j}^{j}(\\lambda) \\tag{13}\n\\end{equation*}\n\n\nSubstituting into equation (12), we get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} \\lambda} d_{j j}^{j}(\\lambda)=-j \\frac{\\sin \\lambda}{1+\\cos \\lambda} d_{j j}^{j}(\\lambda) \\tag{14}\n\\end{equation*}\n\n\nIntegrate, and using $d_{j j}^{j}(0)=1$, we obtain\n\n\\begin{equation*}\nd_{j j}^{j}(\\lambda)=(\\cos \\frac{\\lambda}{2})^{2 j} \\tag{15}\n\\end{equation*}\n\n\nThis formula can also be derived using the model in another format. Using equation (15) and the recursive relation ( $10^{\\prime}$), we can sequentially obtain\n\n\\begin{aligned}\n& d_{j j-1}^{j}(\\lambda)=-(2 j)^{1 / 2}(\\cos \\frac{\\lambda}{2})^{2 j-1} \\sin \\frac{\\lambda}{2} \\\\\n& d_{j j-2}^{j}(\\lambda)=[\\frac{2 j(2 j-1)}{2!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{2 j-2}(\\sin \\frac{\\lambda}{2})^{2} \\\\\n& \\vdots\n\\end{aligned}\n\n\n\\begin{align*}\n& d_{j m}^{j}(\\lambda)=(-1)^{j-m}[\\frac{(2 j)!}{(j+m)!(j-m)!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{j+m}(\\sin \\frac{\\lambda}{2})^{j-m} \\tag{16}\\\\\n& \\vdots \\\\\n& d_{j-j}^{j}(\\lambda)=(-1)^{2 j}(\\sin \\frac{\\lambda}{2})^{2 j}\n\\end{align*}\n\n\nDiscussion $1^{\\circ}$ When $\\lambda \\rightarrow-\\lambda$, clearly\n\nd_{j m}^{j_{m}}(-\\lambda)=(-1)^{\\jmath^{-m}} d_{j m}^{\\prime}(\\lambda)\n\n\nThis result matches the general property of $d_{m^{\\prime} m}^{\\prime}$.\n$\\mathbf{2}^{\\circ}$ When $\\lambda=\\pi$, the only non-zero matrix element is clearly\n\n\\begin{equation*}\nd_{j-\\jmath}^{j}(\\pi)=(-1)^{2 j} \\tag{18}\n\\end{equation*}\n\n\nThis is because\n\n\\begin{align*}\n& \\mathrm{e}^{-i \\pi J_{y}}|j m\\rangle=(-1)^{j-m}|j-m\\rangle \\tag{19}\\\\\n& \\mathrm{e}^{-i \\pi J_{y}}|j-j\\rangle=(-1)^{2 j}|j j\\rangle\n\\end{align*}\n\n[The meaning of operator $\\mathrm{e}^{-\\mathrm{i} \\pi_{y}}$ is to rotate the system by $180^{\\circ}$ about the y-axis, so the state $|j m\\rangle$ becomes the state $|j-m\\rangle$, and equation (19) specifies the relative phase factor for the relationship between $|j m\\rangle$ and $|j-m\\rangle$.]", + "final_answer": [ + "d_{j m}^{j}(\\lambda)=(-1)^{j-m}[\\frac{(2 j)!}{(j+m)!(j-m)!}]^{\\frac{1}{2}}(\\cos \\frac{\\lambda}{2})^{j+m}(\\sin \\frac{\\lambda}{2})^{j-m}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$d_{j m}^{j}(\\lambda)$": "Wigner d-matrix element, defined as $\\langle j j| \\mathrm{e}^{-\\mathrm{i} \\lambda J_{y}}|j m\\rangle$", + "$j$": "Total angular momentum quantum number", + "$m$": "Magnetic quantum number", + "$\\lambda$": "Angle of rotation about the y-axis" + } + }, + { + "id": 33, + "context": "", + "question": "Let $\\boldsymbol{J}_{1}$ and $\\boldsymbol{J}_{2}$ be angular momenta corresponding to different degrees of freedom, then their sum $\\boldsymbol{J}=\\boldsymbol{J}_{1}+\\boldsymbol{J}_{2}$ is also an angular momentum. Try to compute the expectation values of $J_{1 z}$ for the common eigenstate $|j_{1} j_{2} j m\\rangle$ of $(\\boldsymbol{J}_{1}^{2}, \\boldsymbol{J}_{2}^{2}, \\boldsymbol{J}^{2}, J_{z})$. (Take $\\hbar=1$)", + "answer": "$J_{1}, ~ J_{2}$ satisfy the fundamental commutation relations of angular momentum operators\n\n\\begin{equation*}\n\\boldsymbol{J}_{1} \\times \\boldsymbol{J}_{1}=\\mathrm{i} \\boldsymbol{J}_{1}, \\quad \\boldsymbol{J}_{2} \\times \\boldsymbol{J}_{2}=\\mathrm{i} \\boldsymbol{J}_{2} \\tag{1}\n\\end{equation*}\n\n$\\boldsymbol{J}_{1}, ~ J_{2}$ belong to different degrees of freedom and commute with each other, so\n\n\\begin{align*}\n& {[J_{x}, J_{1 x}]=[J_{1 x}+J_{2 x}, J_{1 x}]=0} \\tag{2}\\\\\n& {[J_{x}, J_{1 y}]=[J_{1 x}+J_{2 x}, J_{1 y}]=\\mathrm{i} J_{1 z}}\n\\end{align*}\n\n$\\boldsymbol{J}$ and $\\boldsymbol{J}_{2}$ have similar relationships. In summary, $\\boldsymbol{J}_{1}$ or $\\boldsymbol{J}_{2}$ and $\\boldsymbol{J}$ satisfy all the relations between the vector operator $\\boldsymbol{A}$ and $\\boldsymbol{J}$ from the previous problem. Moreover,\n\n\\begin{align*}\n& \\boldsymbol{J} \\cdot \\boldsymbol{J}_{1}=\\boldsymbol{J}_{1}^{2}+\\boldsymbol{J}_{2} \\cdot \\boldsymbol{J}_{1}=\\frac{1}{2}(\\boldsymbol{J}^{2}+\\boldsymbol{J}_{1}^{2}-\\boldsymbol{J}_{2}^{2}) \\tag{3}\\\\\n& \\boldsymbol{J} \\cdot \\boldsymbol{J}_{2}=\\boldsymbol{J}_{2}^{2}+\\boldsymbol{J}_{1} \\cdot \\boldsymbol{J}_{2}=\\frac{1}{2}(\\boldsymbol{J}^{2}+\\boldsymbol{J}_{2}^{2}-\\boldsymbol{J}_{1}^{2}) \\tag{4}\n\\end{align*}\n\nUsing equation (5) from the previous problem, we have\n\n\\begin{align*}\n& j(j+1)\\langle\\boldsymbol{J}_{1}\\rangle_{j_{1} j_{2} j m}=\\frac{1}{2}[j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)]\\langle\\boldsymbol{J}\\rangle_{j_{1} j_{2} j m} \\tag{5}\\\\\n& j(j+1)\\langle\\boldsymbol{J}_{2}\\rangle_{j_{1} j_{2} j m}=\\frac{1}{2}[j(j+1)+j_{2}(j_{2}+1)-j_{1}(j_{1}+1)]\\langle\\boldsymbol{J}\\rangle_{j_{1} j_{2} j m} \\tag{6}\n\\end{align*}\n\nSince in the state $|J_{z}=m\\rangle$\n\n$$ \\langle J_{x}\\rangle=\\langle J_{y}\\rangle=0, \\quad\\langle J_{z}\\rangle=m $$\n\nTherefore, equations (5) and (6) yield\n\n$$ \\langle J_{1 x}\\rangle_{j_{1} j_{2} j m}=\\langle J_{1 y}\\rangle_{j_{1} j_{2} j m}=\\langle J_{2 x}\\rangle_{j_{1} j_{2} j m}=\\langle J_{2 y}\\rangle_{j_{1} j_{2} j m}=0 $$\n\n\n\\begin{align*}\n\\langle J_{1 z}\\rangle_{j_{1} j_{2} j m} & =m \\frac{j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}{2 j(j+1)} \\tag{7}\\\\\n\\langle J_{2 z}\\rangle_{j_{1} j_{2} j m} & =m \\frac{j(j+1)+j_{2}(j_{2}+1)-j_{1}(j_{1}+1)}{2 j(j+1)} \\\\\n& =m-\\langle J_{1 z}\\rangle_{j_{1} j_{2}>m}\n\\end{align*}", + "final_answer": [ + "m \\frac{j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}{2 j(j+1)}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$m$": "Magnetic quantum number, representing the eigenvalue of $J_z$.", + "$j$": "Quantum number associated with the magnitude of the total angular momentum $\\boldsymbol{J}$, specifically for the eigenvalue of $\\boldsymbol{J}^{2}$.", + "$j_{1}$": "Quantum number associated with the magnitude of $\\boldsymbol{J}_{1}$, specifically for the eigenvalue of $\\boldsymbol{J}_{1}^{2}$.", + "$j_{2}$": "Quantum number associated with the magnitude of $\\boldsymbol{J}_{2}$, specifically for the eigenvalue of $\\boldsymbol{J}_{2}^{2}$." + } + }, + { + "id": 34, + "context": "", + "question": "Two angular momenta $\\boldsymbol{J}_{1}$ and $\\boldsymbol{J}_{2}$, of equal magnitude but belonging to different degrees of freedom, couple to form a total angular momentum $\\boldsymbol{J}=\\boldsymbol{J}_{1}+\\boldsymbol{J}_{2}$, with $\\hbar=1$, and assume $\\boldsymbol{J}_{1}^{2}=\\boldsymbol{J}_{2}^{2}=j(j+1)$. In the state where the total angular momentum quantum number $J=0$, what is the probability when $J_{1z}$ takes the value $m$ (with $J_{2z}$ simultaneously taking the value $-m$)?", + "answer": "The eigenstate of $(\\boldsymbol{J}_{1}^{2}, J_{1 z})$ is denoted by $|j m_{1}\\rangle_{1}$, and the eigenstate of $(\\boldsymbol{J}_{2}^{2}, J_{2 z})$ is denoted by $|j m_{2}\\rangle_{2}$. The common eigenstate of $(\\boldsymbol{J}_{1}^{2}, \\boldsymbol{J}_{2}^{2}$, $\\mathbf{J}^{2}, J_{z})$ is denoted by $|j j J M\\rangle$, where $M$ is the eigenvalue of $J_{z}$.\n$|j j J M\\rangle$ can also be abbreviated as $|J M\\rangle$. \nWhen $J=0$, $M=0, m_{1}=-m_{2}=m$, so the state under discussion can be expressed as \n\n\\begin{equation*}\n|j j 00\\rangle=\\sum_{m} C_{m}|j m\\rangle_{1}|j-m\\rangle_{2} \\tag{3}\n\\end{equation*}\n\n$C_{m}$ is the C.G. coefficient $\\langle j_{1} m_{1} j_{2} m_{2} \\mid J M\\rangle$ when $j_{1}=j_{2}=j, J=M=0, m_{1}=-m_{2}=m$. $|C_{m}|^{2}$ is the probability that $J_{1}$ takes the value $m$ (with $J_{2 z}$ taking the value $-m$ at the same time). Below, we solve for $C_{m}$.\n\nSince $\\boldsymbol{J}^{2}|j j 00\\rangle=0$, and $\\boldsymbol{J}^{2}$ is positive definite, it must be \n\n\\begin{equation*}\n\\boldsymbol{J}|j j 00\\rangle=0 \\tag{4}\n\\end{equation*}\n\n\nThus, \n\n\\begin{equation*}\n(J_{x}+\\mathrm{i} J_{y})|j j 00\\rangle=0 \\tag{5}\n\\end{equation*}\n\n\nWhere \n\n\\begin{equation*}\nJ_{x}+\\mathrm{i} J_{y}=(J_{1 x}+\\mathrm{i} J_{1 y})+(J_{2 x}+\\mathrm{i} J_{2 y})=J_{1+}+J_{2+} \\tag{6}\n\\end{equation*}\n\n\nAccording to the basic formula for angular momentum ladder operators, \n\n\\begin{align*}\nJ_{1+}|j m\\rangle_{1} & =(J_{1 x}+\\mathrm{i} J_{1 y})|j m\\rangle_{1}=a_{j m}|j m+1\\rangle_{1} \\\\\na_{j m} & =\\sqrt{(j-m)(j+m+1)} \\tag{7a}\\\\\nJ_{2+}|j,-m\\rangle_{2} & =(J_{2 x}+\\mathrm{i} J_{2 y})|j,-m\\rangle_{2}=a_{j,-m}|j, 1-m\\rangle_{2} \\\\\na_{j,-m} & =\\sqrt{(j+m)(j-m+1)}=a_{,, m-1} \\tag{7b}\n\\end{align*}\n\n\nSubstituting expressions (5) to (7) into (3), we obtain \n\n\\begin{equation*}\n\\sum_{m} C_{m}[a_{j m}|j, m+1\\rangle_{1}|j,-m\\rangle_{2}+a_{J, m-1}|j m\\rangle_{1}|j, 1-m\\rangle_{2}]=0 \\tag{8}\n\\end{equation*}\n\n\nSince \n\n\\sum_{m} C_{m} a_{j, m-1}|j m\\rangle_{1}|j, 1-m\\rangle_{2} \\xrightarrow{m \\rightarrow m+1} \\sum_{m} C_{m+1} a_{j m}|j m+1\\rangle_{1}|j,-m\\rangle_{2}\n\n\nTherefore, equation (8) becomes \n\n\\begin{equation*}\n\\sum_{m}(C_{m}+C_{m+1}) a_{j m}|j m+1\\rangle_{1}|j,-m\\rangle_{2}=0 \\tag{$\\prime$}\n\\end{equation*}\n\n\nSince each basis vector is linearly independent, it must be \n\n\\begin{equation*}\n(C_{m}+C_{m+1}) a_{j m}=0 \\tag{9}\n\\end{equation*}\n\n\nWhich implies \n\n\\begin{equation*}\nC_{m}=-C_{m+1}, \\quad m=j-1, j-2, \\cdots,(-j) \\tag{\\prime}\n\\end{equation*}\n\n\nIn equation (3), $|j j 00\\rangle, ~|j m\\rangle_{1}, ~|j,-m\\rangle_{2}$ etc. are all orthonormalized, and $m$ has a total of $(2 j+1)$ possible values. According to the normalization condition \n\n\\begin{equation*}\n\\sum_{m}|C_{m}|^{2}=1 \\tag{10}\n\\end{equation*}\n\n\nAnd equation ($9^{\\prime}$), we immediately have $|C_{m}|^{2}=1 /(2 j+1)$. If we take \n\nC_{j}=1 / \\sqrt{2 j+1}\n\n\nWe get \n\n\\begin{equation*}\nC_{m}=(-1)^{j-m} C_{j}=(-1)^{j-m} \\frac{1}{\\sqrt{2 j+1}} \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (3), we obtain \n\n\\begin{equation*}\n|j j 00\\rangle=\\frac{1}{\\sqrt{2 j+1}} \\sum_{m}(-1)^{1^{-m}}|j m\\rangle_{1}|j,-m\\rangle_{2} \\tag{12}\n\\end{equation*}\n\n\nClearly, under the premise $J_{1 z}=-J_{2 z}$, the probabilities that $J_{1 z}$ and $J_{2 z}$ take each eigenvalue $(j, j-1, \\cdots,-j)$ are equal, both being $1 /(2 j+1)$.", + "final_answer": [ + "1/(2j+1)" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$j$": "Angular momentum quantum number for $\\boldsymbol{J}_{1}$ and $\\boldsymbol{J}_{2}$." + } + }, + { + "id": 35, + "context": "", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a magnetic field $\\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}$, where the Hamiltonian is $H = \\frac{1}{2} \\mu \\boldsymbol{v}^{2}$, with $\\boldsymbol{v}$ as the velocity operator. Calculate $\\mathrm{d} \\boldsymbol{v} / \\mathrm{d} t$.", + "answer": "The Hamiltonian operator can be expressed as\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A})^{2}=\\frac{1}{2} \\mu \\boldsymbol{v}^{2} \\tag{2}\n\\end{equation*}\n\n\nUsing the commutation relations of $\\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{v}, H]=\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v}) \\tag{3}\n\\end{equation*}", + "final_answer": [ + "\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$q$": "Charge of the particle", + "$\\mu$": "Mass of the particle", + "$c$": "Speed of light", + "$\\boldsymbol{v}$": "Velocity operator", + "$\\boldsymbol{B}$": "Magnetic field, defined as $\\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}$" + } + }, + { + "id": 36, + "context": "", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a magnetic field $\\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}$, where the Hamiltonian is $H = \\frac{1}{2} \\mu \\boldsymbol{v}^{2}$, with $\\boldsymbol{v}$ as the velocity operator. Let $\\boldsymbol{L}$ be the the angular momentum operator. Calculate $\\mathrm{d} \\boldsymbol{L} / \\mathrm{d} t$.", + "answer": "The Hamiltonian operator can be expressed as\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A})^{2}=\\frac{1}{2} \\mu \\boldsymbol{v}^{2} \\tag{2}\n\\end{equation*}\n\n\nUsing the commutation relations of $\\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{v}, H]=\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v}) \\tag{3}\n\\end{equation*}\n\n\nIn classical electrodynamics, the Lorentz force is\n\n$$ \\boldsymbol{f}=\\frac{q}{c} \\boldsymbol{v} \\times \\boldsymbol{B}. $$\n\nThe equation of motion is\n\n\\begin{equation*}\n\\mu \\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{v}=\\boldsymbol{f}=\\frac{q}{c} \\boldsymbol{v} \\times \\boldsymbol{B} \\tag{$\\prime$}\n\\end{equation*}\n\n\nEquation (3) is the quantum mechanical extension of the classical equation of motion ($3^{\\prime}$). In equation (3),\n\n\\begin{equation*}\n\\boldsymbol{v} \\times \\boldsymbol{B}=\\frac{1}{\\mu}(\\boldsymbol{p}-\\frac{q}{c} \\boldsymbol{A}) \\times \\boldsymbol{B}=-\\boldsymbol{B} \\times \\boldsymbol{v}-\\frac{\\mathrm{i} \\hbar}{\\mu} \\nabla \\times \\boldsymbol{B} \\tag{4}\n\\end{equation*}\n\n\nFrom electrodynamics,\n\n$$\\nabla \\times \\boldsymbol{B}=\\frac{4 \\pi}{c} \\boldsymbol{j}$$\n\n$\\boldsymbol{j}$ is the source current that generates the magnetic field. Thus, if in the space where the particle moves, the source current $\\boldsymbol{j}=0$, then\n\n\\begin{equation*}\n\\frac{q^{v}}{c} \\boldsymbol{v} \\times \\boldsymbol{B}=-\\frac{q}{c} \\boldsymbol{B} \\times \\boldsymbol{v}=\\boldsymbol{f} \\tag{5}\n\\end{equation*}\n\n\nEquation (3) can still be rewritten in the form of equation ($3^{\\prime}$). For uniform fields, it is obvious that equations (3) and ($3^{\\prime}$) are equivalent.\nThe mechanical angular momentum can also be expressed as\n\n\\begin{equation*}\n\\boldsymbol{L}=\\mu \\boldsymbol{r} \\times \\boldsymbol{v}=\\frac{1}{2} \\mu(\\boldsymbol{r} \\times \\boldsymbol{v}-\\boldsymbol{v} \\times \\boldsymbol{r}) \\tag{$\\prime$}\n\\end{equation*}\n\n(Note that in $\\boldsymbol{r} \\times \\boldsymbol{v}$, the relevant components of $\\boldsymbol{r}$ and $\\boldsymbol{v}$ are commutative.) The time derivative of $\\boldsymbol{L}$ is\n\n$$\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{\\mu}{2}(\\frac{\\mathrm{~d} \\boldsymbol{r}}{\\mathrm{~d} t} \\times \\boldsymbol{v}+\\boldsymbol{r} \\times \\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{~d} t}-\\frac{\\mathrm{d} \\boldsymbol{v}}{\\mathrm{~d} t} \\times \\boldsymbol{r}-\\boldsymbol{v} \\times \\frac{\\mathrm{d} \\boldsymbol{r}}{\\mathrm{~d} t}).$$\n\nUsing equation (3) and $\\mathrm{d} \\boldsymbol{r} / \\mathrm{d} t=\\boldsymbol{v}$, we obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{q}{4 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}-\\boldsymbol{r} \\times(\\boldsymbol{B} \\times \\boldsymbol{v})-(\\boldsymbol{v} \\times \\boldsymbol{B}) \\times \\boldsymbol{r}] \\tag{6}\n\\end{equation*}\n\nUnder the conditions where equation (5) holds, equation (6) can be simplified to\n\n\\begin{equation*}\n\\frac{\\mathrm{d} \\boldsymbol{L}}{\\mathrm{~d} t}=\\frac{q}{2 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}]=\\frac{1}{2}(\\boldsymbol{r} \\times \\boldsymbol{f}-\\boldsymbol{f} \\times \\boldsymbol{r}) \\tag{7}\n\\end{equation*}\n\n\nNote that in $\\boldsymbol{r} \\times \\boldsymbol{f}$, the relevant components of $\\boldsymbol{r}$ and $\\boldsymbol{f}$ are non-commutative, and $\\boldsymbol{r} \\times \\boldsymbol{f} \\neq-\\boldsymbol{f} \\times \\boldsymbol{r}$, hence the right side of equation (7) is not equivalent to $\\boldsymbol{r} \\times \\boldsymbol{f}$.", + "final_answer": [ + "\\frac{q}{2 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}]" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$q$": "Charge of the particle", + "$c$": "Speed of light", + "$\\boldsymbol{r}$": "Position vector", + "$\\boldsymbol{v}$": "Velocity operator", + "$\\boldsymbol{B}$": "Magnetic field" + } + }, + { + "id": 37, + "context": "", + "question": "A three-dimensional isotropic oscillator with mass $\\mu$, charge $q$, and natural frequency $\\omega_{0}$ is placed in a uniform external magnetic field $\\boldsymbol{B}$. Find the formula for energy levels.", + "answer": "Compared to the previous two questions, an additional harmonic oscillator potential $\\frac{1}{2} \\mu \\omega_{0}^{2}(x^{2}+y^{2}+z^{2})$ should be added to the Hamiltonian in this question, thus\n\n\\begin{align*}\nH & =\\frac{1}{2 \\mu} \\boldsymbol{p}^{2}+\\frac{1}{2} \\mu \\omega_{0}^{2}(x^{2}+y^{2}+z^{2})+\\frac{q^{2} B^{2}}{8 \\mu c^{2}}(x^{2}+y^{2})-\\frac{q B}{2 \\mu c} l_{z} \\\\\n& =H_{1}+H_{2}-\\omega_{\\mathrm{L}} l_{z} \\tag{1}\n\\end{align*}\n\n\nwhere\n\n\\begin{align*}\n& H_{1}=\\frac{1}{2 \\mu} p_{z}^{2}+\\frac{1}{2} \\mu \\omega_{0}^{2} z^{2} \\\\\n& H_{2}=\\frac{1}{2 \\mu}(p_{x}^{2}+p_{y}^{2})+\\frac{1}{2} \\mu(\\omega_{0}^{2}+\\omega_{\\mathrm{L}}^{2})(x^{2}+y^{2}) \\tag{2}\\\\\n& \\omega_{\\mathrm{L}}=\\frac{q B}{2 \\mu c}\n\\end{align*}\n\n$H_{1}, ~ H_{2}, ~ l_{z}$ are mutually commuting conserved quantities and can be chosen as a complete set of commuting observables. $H_{1}$ corresponds to a one-dimensional harmonic oscillator energy operator, with eigenvalues given by\n\n\\begin{equation*}\nE_{n_{1}}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}, \\quad n_{1}=0,1,2, \\cdots \\tag{3}\n\\end{equation*}\n\n$\\mathrm{H}_{2}$ corresponds to a two-dimensional isotropic oscillator total energy operator, with eigenvalues given by\n\n\\begin{align*}\n& E_{n_{2} m}=(2 n_{2}+1+|m|) \\hbar \\omega \\tag{4}\\\\\n& n_{2}=0,1,2, \\cdots, \\quad m=0, \\pm 1, \\pm 2, \\cdots\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\n\\omega=\\sqrt{\\omega_{0}^{2}+\\omega_{L}^{2}} \\tag{5}\n\\end{equation*}\n\nThe eigenvalue of $l_{z}$ is $m \\hbar$. Therefore, the energy levels for this problem are\n\n\\begin{equation*}\nE_{n_{1} n_{2} m}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}+(2 n_{2}+1+|m|) \\hbar \\omega-m \\hbar \\omega_{\\mathrm{L}} \\tag{6}\n\\end{equation*}\n\n\nNote: $\\omega_{\\mathrm{L}}$ can be positive or negative depending on the sign of charge $q$; the magnetic quantum number $m$ can also be positive or negative. Therefore, the sign of $q$ does not affect the overall energy spectrum. For precision, only the case of $q>0(\\omega_{\\mathrm{L}}>0)$ is discussed below.\n\nThe relationship among the three frequencies $\\omega_{0}, ~ \\omega_{L}, ~ \\omega$ is given by:\n\n\\begin{equation*}\n0<(\\omega-\\omega_{\\mathrm{L}})<\\omega_{0} \\tag{7}\n\\end{equation*}\n\n\nThe dependence of the energy levels on the quantum numbers (listed in the order of their magnitude of change) is\n\n\\begin{array}{lll}\nm \\geqslant 0, & m \\text { increases by } 1, & E \\text { increases by } \\hbar(\\omega-\\omega_{\\mathrm{L}}) \\\\\n& n_{1} \\text { increases by } 1, & E \\text { increases by } \\hbar \\omega_{0} \\\\\nm \\leqslant 0, & |m| \\text { increases by } 1, & E \\text { increases by } \\hbar(\\omega+\\omega_{\\mathrm{L}}) \\\\\n& n_{2} \\text { increases by } 1, & E \\text { increases by } 2 \\hbar \\omega\n\\end{array}\n\n\nFor the ground state $E_{0}$, obviously, $n_{1}, ~ n_{2}, ~|m|$ all take their minimum values,\n\n\\begin{equation*}\nE_{0}=E_{000}=\\frac{1}{2} \\hbar \\omega_{0}+\\hbar \\omega \\tag{8}\n\\end{equation*}\n\n\nFor the first excited state $E_{1}, n_{1}=n_{2}=0, m=1$,\n\n\\begin{equation*}\nE_{1}=E_{001}=\\frac{1}{2} \\hbar \\omega_{0}+2 \\hbar \\omega-\\hbar \\omega_{\\mathrm{L}}=E_{0}+\\hbar(\\omega-\\omega_{\\mathrm{L}}) \\tag{9}\n\\end{equation*}", + "final_answer": [ + "E_{n_{1} n_{2} m}=(n_{1}+\\frac{1}{2}) \\hbar \\omega_{0}+(2 n_{2}+1+|m|) \\hbar \\omega-m \\hbar \\omega_{\\mathrm{L}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_{n_{1} n_{2} m}$": "Total energy level for the three-dimensional isotropic oscillator in a magnetic field.", + "$n_{1}$": "Quantum number for the one-dimensional harmonic oscillator along the z-axis.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega_{0}$": "Natural frequency of the three-dimensional isotropic oscillator.", + "$n_{2}$": "Radial quantum number for the two-dimensional isotropic oscillator in the xy-plane.", + "$m$": "Magnetic quantum number.", + "$\\omega$": "Effective frequency for the two-dimensional isotropic oscillator, defined as $\\omega=\\sqrt{\\omega_{0}^{2}+\\omega_{L}^{2}}$.", + "$\\omega_{\\mathrm{L}}$": "Larmor frequency, defined as $\\omega_{\\mathrm{L}}=\\frac{q B}{2 \\mu c}$." + } + }, + { + "id": 38, + "context": "", + "question": "A particle with mass $\\mu$ and charge $q$ moves in a uniform electric field $\\mathscr{E}$ (along the x-axis) and a uniform magnetic field $\\boldsymbol{B}$ (along the z-axis) that are perpendicular to each other. If the momentum of the particle in the y-direction is $p_y$ and in the z-direction is $p_z$, find the energy level expression of the system.", + "answer": "With the electric field direction as the $x$ axis and the magnetic field direction as the $z$ axis, then\n\n\\begin{equation*}\n\\mathscr{L}=(\\mathscr{E}, 0,0), \\quad \\boldsymbol{B}=(0,0, B) \\tag{1}\n\\end{equation*}\n\n\nTaking the scalar and vector potentials of the electromagnetic field as\n\n\\begin{equation*}\n\\phi=-\\mathscr{E} x, \\quad \\boldsymbol{A}=(0, B x, 0) \\tag{2}\n\\end{equation*}\n\n\nSatisfying the relation\n\n$$\\mathscr{E}=-\\nabla \\phi, \\quad \\boldsymbol{B}=\\nabla \\times \\boldsymbol{A}.$$\n\n\nThe Hamiltonian of the particle is\n\n\\begin{equation*}\nH=\\frac{1}{2 \\mu}[p_{x}^{2}+(p_{y}-\\frac{q B}{c} x)^{2}+p_{z}^{2}]-q \\mathscr{E} x \\tag{3}\n\\end{equation*}\n\n\nWith the constants of motion set as $(H, p_{y}, p_{z})$, their common eigenfunction can be written as\n\n\\begin{equation*}\n\\psi(x, y, z)=\\psi(x) \\mathrm{e}^{\\mathrm{i}(p_{y} y+p_{z} z) / \\hbar} \\tag{4}\n\\end{equation*}\n\n\nwhere $p_{y}$ and $p_{z}$ are eigenvalues and can be any real numbers.\n$\\psi(x, y, z)$ satisfies the energy eigen-equation\n\n$$ H \\psi(x, y, z)=E \\psi(x, y, z) $$\n\n\nThus, $\\psi(x)$ satisfies the equation\n\n\\begin{equation*}\n\\frac{1}{2 \\mu}[p_{x}^{2}+(p_{y}-\\frac{q B}{c} x)^{2}+p_{z}^{2}] \\psi-q \\mathscr{E} x \\psi=E \\psi \\tag{5}\n\\end{equation*}\n\n\nThat is, for $\\psi(x)$, $H$ is equivalent to the following:\n\n\\begin{align*}\nH & \\Rightarrow-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x^{2}-(q \\mathscr{E}+\\frac{q B}{\\mu c} p_{y}) x+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{x}^{2}) \\\\\n& =-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{q^{2} B^{2}}{2 \\mu c^{2}}(x-x_{0})^{2}-\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x_{0}^{2}+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{z}^{2}) \\tag{6}\n\\end{align*}\n\n\nwhere\n\n\\begin{equation*}\nx_{0}=\\frac{\\mu c^{2}}{q^{2} B^{2}}(q \\mathscr{E}+\\frac{q B}{\\mu c} p_{y})=\\frac{\\mu c}{q B}(\\frac{c \\mathscr{E}}{B}+\\frac{p_{y}}{\\mu}) \\tag{7}\n\\end{equation*}\n\n\nEquation (6) corresponds to a one-dimensional harmonic oscillator energy operator\n\n$$-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\partial^{2}}{\\partial x^{2}}+\\frac{1}{2} \\mu \\omega^{2}(x-x_{0})^{2}, \\quad \\omega=\\frac{|q| B}{\\mu c} $$\n\n\nPlus two constant terms. Therefore, the energy level of this problem is\n\n\\begin{align*}\nE & =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} B^{2}}{2 \\mu c^{2}} x_{0}^{2}+\\frac{1}{2 \\mu}(p_{y}^{2}+p_{z}^{2}) \\\\\n& =(n+\\frac{1}{2}) \\frac{\\hbar B|q|}{\\mu c}-\\frac{c^{2} \\mathscr{E}^{2} \\mu}{2 B^{2}}-\\frac{c \\mathscr{E}}{B} p_{y}+\\frac{1}{2 \\mu} p_{z}^{2} \\tag{8}\n\\end{align*}\n\n\nwhere $p_{y}, ~ p_{z}$ are any real numbers, $n=0,1,2, \\cdots$.\nIn equation (4), $\\psi(x)$ is the one-dimensional harmonic oscillator energy eigenfunction with $(x-x_{0})$ as the variable, i.e.,\n\n\\begin{equation*}\n\\psi(x)=\\psi_{n}(x-x_{0})=H_{n}(\\xi) \\mathrm{e}^{-\\frac{1}{2} \\xi^{2}} \\tag{9}\n\\end{equation*}\n\n$H_{n}(\\xi)$ is the Hermite polynomial, $\\xi=(\\frac{|q| B}{\\hbar c})^{\\frac{1}{2}}(x-x_{0})$.", + "final_answer": [ + "E = (n+\\frac{1}{2}) \\frac{\\hbar B|q|}{\\mu c}-\\frac{c^{2} \\mathscr{E}^{2} \\mu}{2 B^{2}}-\\frac{c \\mathscr{E}}{B} p_{y}+\\frac{1}{2 \\mu} p_{z}^{2}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E$": "Energy eigenvalue of the system.", + "$n$": "Quantum number for the energy levels, $n=0,1,2, \\cdots$.", + "$\\hbar$": "Reduced Planck's constant.", + "$B$": "Magnitude of the uniform magnetic field.", + "$|q|$": "Absolute value of the charge of the particle.", + "$\\mu$": "Mass of the particle.", + "$c$": "Speed of light.", + "$\\mathscr{E}$": "Magnitude of the uniform electric field, directed along the x-axis.", + "$p_y$": "Momentum of the particle in the y-direction.", + "$p_z$": "Momentum of the particle in the z-direction." + } + }, + { + "id": 39, + "context": "", + "question": "A particle moves in one dimension. When the total energy operator is\n\n\\begin{equation*}\nH_{0}=\\frac{p^{2}}{2 m}+V(x) \\tag{1}\n\\end{equation*}\n\n\nthe energy level is $E_{n}^{(0)}$. If the total energy operator becomes\n\n\\begin{equation*}\nH=H_{0}+\\frac{\\lambda p}{m} \\tag{2}\n\\end{equation*}\n\n\nfind the energy level $E_{n}$.", + "answer": "First, treat $\\lambda$ as a parameter, then\n\n\\begin{equation*}\n\\frac{\\partial H}{\\partial \\lambda}=\\frac{p}{m} \\tag{3}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, we have\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial \\lambda}=\\langle\\frac{\\partial H}{\\partial \\lambda}\\rangle_{n}=\\frac{1}{m}\\langle p\\rangle_{n} \\tag{4}\n\\end{equation*}\n\n\nHowever, since\n\n\\begin{equation*}\n\\frac{\\mathrm{d} x}{\\mathrm{~d} t}=\\frac{1}{\\mathrm{i} \\hbar}[x, H]=\\frac{1}{\\mathrm{i} \\hbar m}[x, \\frac{p^{2}}{2}+\\lambda p]=\\frac{1}{m}(p+\\lambda) \\tag{5}\n\\end{equation*}\n\n\nFor any bound state,\n\n$$ \\langle\\frac{\\mathrm{d} x}{\\mathrm{~d} t}\\rangle_{n}=\\frac{1}{\\mathrm{i} \\hbar}\\langle x H-H x\\rangle_{n}=0, $$\ntherefore\n\n\\begin{equation*}\n\\langle p\\rangle_{n}=-\\lambda \\tag{6}\n\\end{equation*}\n\n\nSubstitute into equation (4), obtaining\n\n\\begin{equation*}\n\\frac{\\partial E_{n}}{\\partial \\lambda}=-\\frac{\\lambda}{m} \\tag{7}\n\\end{equation*}\n\n\nIntegrate to get\n\n\\begin{equation*}\nE_{n}=-\\frac{\\lambda^{2}}{2 m}+C \\tag{8}\n\\end{equation*}\n\n$C$ is the integration constant. Since when $\\lambda=0$, $H=H_{0}, E_{n}=E_{n}^{(0)}$, so $C=E_{n}^{(0)}$. Substitute into equation (8), obtaining\n\n\\begin{equation*}\nE_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m} \\tag{9}\n\\end{equation*}\n\n\nSolution Two: Write $H$ as\n\n\\begin{equation*}\nH=\\frac{p^{2}}{2 m}+\\frac{\\lambda p}{m}+V(x)=\\frac{P^{2}}{2 m}+V(x)-\\frac{\\lambda^{2}}{2 m} \\tag{10}\n\\end{equation*}\n\n\nwhere\n\n\\begin{equation*}\nP=p+\\lambda \\tag{11}\n\\end{equation*}\n\n\nIn the momentum representation,\n\n\\begin{equation*}\nx=\\mathrm{i} \\hbar \\frac{\\partial}{\\partial p}=\\mathrm{i} \\hbar \\frac{\\partial}{\\partial P} \\tag{12}\n\\end{equation*}\n\n\nTherefore,\n\n\\begin{align*}\nH_{0} & =\\frac{p^{2}}{2 m}+V(i \\hbar \\frac{\\partial}{\\partial p}) \\tag{13}\\\\\nH & =\\frac{P^{2}}{2 m}+V(i \\hbar \\frac{\\partial}{\\partial P})-\\frac{\\lambda^{2}}{2 m} \\tag{14}\n\\end{align*}\n\nThe difference between $H$ and $H_{0}$, aside from the constant term, is just replacing $p$ with $P$, which does not affect the energy levels. Therefore,\n\n\\begin{equation*}\nE_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m} \\tag{15}\n\\end{equation*}", + "final_answer": [ + "E_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_n$": "Energy level corresponding to the modified total energy operator $H$.", + "$E_n^{(0)}$": "Initial energy level when the total energy operator is $H_0$.", + "$\\lambda$": "Parameter representing the strength of the perturbation.", + "$m$": "Mass of the particle." + } + }, + { + "id": 40, + "context": "", + "question": "A particle with mass $\\mu$ moves in a central force field,\n\n\\begin{equation*}\nV(r)=\\lambda r^{\\nu}, \\quad-2<\\nu, \\quad \\nu / \\lambda>0 \\tag{1}\n\\end{equation*}\n\n\nUse the Hellmann theorem and the virial theorem to analyze the dependence of the energy level structure on $\\hbar, ~ \\lambda, ~ \\mu$.", + "answer": "The energy operator is\n\n\\begin{equation*}\nH=T+V=-\\frac{\\hbar^{2}}{2 \\mu} \\nabla^{2}+\\lambda r^{\\nu} \\tag{2}\n\\end{equation*}\n\n\nLet $\\beta=\\hbar^{2} / 2 \\mu, \\lambda$ and $\\beta$ be independent parameters. It is evident that\n\n\\begin{equation*}\n\\beta \\frac{\\partial H}{\\partial \\beta}=T, \\quad \\lambda \\frac{\\partial H}{\\partial \\lambda}=V \\tag{3}\n\\end{equation*}\n\n\nAccording to the Hellmann theorem, for any bound state,\n\n\\begin{align*}\n& \\langle T\\rangle=\\frac{\\beta \\partial E}{\\partial \\beta} \\tag{4}\\\\\n& \\langle V\\rangle=\\frac{\\lambda \\partial E}{\\partial \\lambda} \\tag{5}\n\\end{align*}\n\n\nAdding the two equations yields\n\n\\begin{equation*}\n\\beta \\frac{\\partial E}{\\partial \\beta}+\\lambda \\frac{\\partial E}{\\partial \\lambda}=\\langle T+V\\rangle=E \\tag{6}\n\\end{equation*}\n\n\nAnd from the virial theorem, we have\n\n$$ \\langle T\\rangle=\\frac{\\nu}{2}\\langle V\\rangle$$\n\n\nThat is\n\n\\begin{equation*}\n\\beta \\frac{\\partial E}{\\partial \\beta}=\\frac{\\nu}{2} \\lambda \\frac{\\partial E}{\\partial \\lambda} \\tag{7}\n\\end{equation*}\n\n\nSubstituting equation (7) into equation (6), we get\n\n\\begin{align*}\n& (1+\\frac{\\nu}{2}) \\lambda \\frac{\\partial E}{\\partial \\lambda}=E \\tag{8}\\\\\n& (1+\\frac{2}{\\nu}) \\beta \\frac{\\partial E}{\\partial \\beta}=E \\tag{9}\n\\end{align*}\n\n\nIntegrating equation (8), we obtain the construction relationship between $E$ and $\\lambda$\n\n\\begin{equation*}\nE=C_{1} \\lambda^{2 /(2+\\iota)} \\tag{10}\n\\end{equation*}\n\n$C_{1}$ is the \"integration constant\" and is independent of $\\lambda$. Integrating equation (9), we obtain the construction relationship between $E$ and $\\beta$\n\n\\begin{equation*}\nE=C_{2} \\beta^{\\nu /(2 \\downarrow \\imath)} \\tag{11}\n\\end{equation*}\n\n$C_{2}$ is independent of $\\nu$. Comparing equations (10) and (11), it follows that\n\n\\begin{equation*}\nE=C \\lambda^{2 /(2+\\tau)} \\beta^{\\nu /(2+\\nu)}=C \\lambda^{2 /(2+\\tau)}(\\frac{\\hbar^{2}}{2 \\mu})^{\\nu /(2+\\nu)} \\tag{12}\n\\end{equation*}\n\n$C$ is independent of $\\lambda, ~ \\beta$, and is a dimensionless pure number (related to $\\nu$ and quantum numbers). It is easy to verify that the above expression is the only possible energy construction that is dimensionally correct.\n\nFrom equation (12), it is evident (note $\\nu>-2$) that as the interaction strength $|\\lambda|$ increases, $|E|$ increases, and the energy level spacing increases. If $\\lambda$ is independent of the particle's mass, then when $\\nu>0, ~ \\mu$ increases, $|E|$ decreases; when $\\nu<0, ~ \\mu$ increases, $|E|$ increases.\n\nIf $\\lambda$ is independent of $\\mu$, from equations (2) and (4) it can also be seen that\n\n\\begin{equation*}\n\\mu \\frac{\\partial E}{\\partial \\mu}=-\\langle T\\rangle<0 \\tag{13}\n\\end{equation*}\n\n\nThat is, an increase in the particle's mass always leads to a decrease in the algebraic value of the energy levels.\nFrom equation (12) it can also be seen that if $\\lambda \\propto \\mu^{\\nu / 2}$, then $E$ is independent of $\\mu$. A famous example of this situation is the harmonic oscillator, i.e.,\n\n\\begin{align*}\n& V(\\boldsymbol{r})=\\lambda r^{2}=\\frac{1}{2} \\mu \\omega^{2} r^{2} \\\\\n& (\\nu=2, \\quad \\lambda=\\frac{1}{2} \\mu \\omega^{2})\n\\end{align*}\n\n\nThe energy levels are\n\n$$ E_{N}=(N+\\frac{3}{2}) \\hbar \\omega, \\quad N=0,1,2, \\cdots $$\n\n\nIf $\\omega$ remains constant, $E_{N}$ is independent of $\\mu$.", + "final_answer": [ + "E=C \\lambda^{2 /(2+\\nu)}(\\frac{\\hbar^{2}}{2 \\mu})^{\\nu /(2+\\nu)}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E$": "Energy of a bound state, an eigenvalue of the Hamiltonian.", + "$C$": "Dimensionless pure number constant, independent of $\\lambda$ and $\\beta$, related to $\\nu$ and quantum numbers.", + "$\\lambda$": "Constant parameter defining the strength and sign of the central force potential.", + "$\\nu$": "Exponent parameter defining the radial dependence of the central force potential.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\mu$": "Mass of the particle.", + "$E_n$": "Energy level associated with quantum number $n$." + } + }, + { + "id": 41, + "context": "", + "question": "A particle moves in a potential field\n\\begin{equation*}\nV(x)=V_{0}|x / a|^{\\nu}, \\quad V_{0}, a>0 \\tag{1}.\n\\end{equation*}\n\nFind the dependence of energy levels on parameters as $\\nu \\rightarrow \\infty$.", + "answer": "The total energy operator is\n\n\\begin{equation*}\nH=T+V=-\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\mathrm{~d}^{2}}{\\mathrm{~d} x^{2}}+V_{0}|x / a|^{\\nu} \\tag{2}\n\\end{equation*}\n\n\nFrom dimensional analysis, if $x_{0}$ represents the characteristic length, we have\n\n\\begin{equation*}\nE \\sim \\frac{\\hbar^{2}}{\\mu x_{0}^{2}} \\sim \\frac{V_{0} x_{0}^{\\nu}}{a^{\\nu}} \\tag{3}\n\\end{equation*}\n\n\nIt is not difficult to solve for\n\n\\begin{equation*}\nx_{0} \\sim(\\frac{\\hbar^{2} a^{\\nu}}{\\mu V_{0}})^{\\frac{1}{\\nu+2}} \\xrightarrow{\\nu \\rightarrow \\infty} a \\tag{4}\n\\end{equation*}\n\n\n\\begin{equation*}\nE \\sim(\\frac{\\hbar^{2}}{\\mu})^{\\frac{1}{\\nu+2}} V_{0}^{2 /(\\nu+2)} a^{-2 \\nu /(\\nu+2)} \\xrightarrow{\\nu \\rightarrow \\infty} \\frac{\\hbar^{2}}{\\mu a^{2}} \\tag{5}\n\\end{equation*}\n\n\nThe construction of the characteristic length and energy levels is independent of $V_{0}$, \n\nIn fact, it is easy to see from Equation (1)\n\n$$ \\nu \\rightarrow \\infty, \\quad V(x) \\rightarrow \\begin{cases}0, & |x|a\\end{cases} $$\n\nThis is precisely an infinitely deep potential well of width $2 a$, with energy levels\n\n\\begin{equation*}\nE_{n}=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{8 \\mu a^{2}} . \\quad n=1,2,3, \\cdots \\tag{7}\n\\end{equation*}\n\nWhen $\\nu$ is finite, according to the virial theorem, for any bound state, there is a relationship between the average kinetic energy and the average potential energy\n\n\\begin{equation*}\n\\frac{2}{\\nu}\\langle T\\rangle=\\langle V\\rangle \\tag{8}\n\\end{equation*}\n\n\nThus, as $u \\rightarrow \\infty$, it follows\n\n\\begin{equation*}\n\\langle V\\rangle \\rightarrow 0, \\quad\\langle T\\rangle \\rightarrow E \\tag{9}\n\\end{equation*}\n\n\nThis conclusion is also consistent with the infinite potential well problem.", + "final_answer": [ + "E_{n}=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{8 \\mu a^{2}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_n$": "Energy levels for an infinite potential well.", + "$n$": "Principal quantum number, an integer ($n=1,2,3,...$).", + "$\\pi$": "Mathematical constant pi.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\mu$": "Mass of the particle.", + "$a$": "A positive constant parameter defining the characteristic length scale of the potential field." + } + }, + { + "id": 42, + "context": "", + "question": "A particle with mass $m$ moves in a uniform force field $V(x)=F x(F>0)$, with the motion constrained to the range $x \\geqslant 0$. Find its ground state energy level.", + "answer": "The total energy operator is\n\n\\begin{equation*}\nH=T+V=\\frac{p^{2}}{2 m}+F x \\tag{1}\n\\end{equation*}\n\n\nIn momentum representation, the operator for $x$ is given by\n\n\\begin{equation*}\n\\hat{x}=\\mathrm{i} \\hbar \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\tag{2}\n\\end{equation*}\n\nThe operator for $H$ is given by\n\n\\begin{equation*}\n\\hat{H}=\\frac{p^{2}}{2 m}+\\mathrm{i} \\hbar F \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\tag{3}\n\\end{equation*}\n\n\nThe time-independent Schrödinger equation is\n\n\\begin{equation*}\n\\frac{p^{2}}{2 m} \\varphi(p)+\\mathrm{i} \\hbar F \\frac{\\mathrm{~d}}{\\mathrm{~d} p} \\varphi(p)=E \\varphi(p) \\tag{4}\n\\end{equation*}\n\n\nHere $\\varphi(p)$ is the wave function in the momentum representation. Equation (4) is a very simple first-order differential equation, whose solution is\n\n\\begin{equation*}\n\\varphi(p)=A \\exp [\\frac{\\mathrm{i}}{\\hbar F}(\\frac{p^{3}}{6 m}-E p)] \\tag{5}\n\\end{equation*}\n\n$A$ is the normalization constant.\nTransforming to the $x$ representation, the wave function is\n\n\\begin{align*}\n\\psi(x) & =(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\mathrm{d} p \\\\\n& =A(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\exp \\frac{\\mathrm{i}}{\\hbar}[\\frac{p^{3}}{6 m F}+(x-\\frac{E}{F}) p] \\mathrm{d} p \\\\\n& =2 A(2 \\pi \\hbar)^{-1 / 2} \\int_{0}^{\\infty} \\cos [\\frac{p^{3}}{6 \\hbar m F}+\\frac{p}{\\hbar}(x-\\frac{E}{F})] \\mathrm{d} p \\\\\n& =\\frac{C}{\\sqrt{\\pi}} \\int_{0}^{\\infty} \\cos (\\frac{u^{3}}{3}+u \\xi) \\mathrm{d} u \\tag{6}\n\\end{align*}\n\n\nWhere\n\n\\begin{align*}\n& u=p(2 \\hbar m F)^{-1 / 3} \\tag{7}\\\\\n& \\xi=(x-\\frac{E}{F})(\\frac{2 m F}{\\hbar^{2}})^{1 / 3}=\\frac{x}{l}-\\lambda \\tag{8}\\\\\n& l=(\\frac{\\hbar^{2}}{2 m F})^{1 / 3} \\tag{9}\\\\\n& \\lambda=(\\frac{2 m}{\\hbar^{2} F^{2}})^{1 / 3} E=\\frac{2 m E}{\\hbar^{2}} l^{2} \\tag{10}\n\\end{align*}\n\n$l$ is the characteristic length of this problem.\nExcept for the normalization constant $C$, the right-hand side of equation (6) resembles the Airy function with $\\xi$ as the variable.\nWhen $\\xi>0(x>E / F$ , i.e., the classically forbidden region), the Airy function behaves like the modified Bessel function of the second kind, i.e.,\n\n\\begin{equation*}\n\\psi(x)=\\sqrt{\\xi} K \\frac{1}{3}(\\frac{2}{3} \\xi^{3 / 2}) \\xrightarrow{\\xi \\rightarrow \\infty} \\sqrt{\\xi}(\\frac{3 \\pi}{4 \\xi^{3 / 2}})^{1 / 2} \\mathrm{e}^{-\\frac{2}{3} \\xi^{3 / 2}} \\tag{11}\n\\end{equation*}\n\n\nWhen $\\xi<0$ ($x0$) through a $\\delta$ potential barrier $V(x)=V_{0} \\delta(x)$ in the momentum representation.", + "answer": "The stationary Schrödinger equation in the $x$ representation is\n\n\\begin{equation*}\n\\psi^{\\prime \\prime}+k^{2} \\psi-\\frac{2 m V_{0}}{\\hbar^{2}} \\delta(x) \\psi=0, \\quad k=\\sqrt{2 m E} / \\hbar \\tag{1}\n\\end{equation*}\n\nLet\n\n\\begin{equation*}\n\\psi(x)=(2 \\pi \\hbar)^{-1 / 2} \\int_{-\\infty}^{+\\infty} \\mathrm{d} p \\varphi(p) \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\tag{2}\n\\end{equation*}\n\nwhere $\\varphi(p)$ is the wave function in the momentum representation, which should satisfy the equation\n\n\\begin{equation*}\n\\frac{p^{2}}{2 m} \\varphi(p)+\\int_{-\\infty}^{+\\infty} V_{p p^{\\prime}} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=E \\varphi(p) \\tag{3}\n\\end{equation*}\n\nwhere\n\n\\begin{equation*}\nV_{p p^{\\prime}}=\\frac{1}{2 \\pi \\hbar} \\int_{-\\infty}^{+\\infty} \\mathrm{d} x V_{0} \\delta(x) \\mathrm{e}^{\\mathrm{i}(p^{\\prime}-p) x / \\hbar}=\\frac{V_{0}}{2 \\pi \\hbar} \\tag{4}\n\\end{equation*}\n\nThus, using equations (4) and (2), we can solve\n\n\\begin{equation*}\n\\int_{-\\infty}^{+\\infty} V_{p p^{\\prime}} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=\\frac{V_{0}}{2 \\pi \\hbar} \\int_{-\\infty}^{+\\infty} \\varphi(p^{\\prime}) \\mathrm{d} p^{\\prime}=\\frac{V_{0}}{\\sqrt{2 \\pi \\hbar}} \\psi(0) \\tag{5}\n\\end{equation*}\n\nSubstituting equations (4) and (5) into equation (3), we obtain\n\n\\begin{equation*}\n(p^{2}-\\hbar^{2} k^{2}) \\varphi(p)+\\frac{2 m V_{0}}{\\sqrt{2 \\pi \\hbar}} \\psi(0)=0 \\tag{6}\n\\end{equation*}\n\nAccording to the fundamental formula of the $\\delta$ function\n\n\\begin{equation*}\n(\\xi-\\xi_{0}) \\delta(\\xi-\\xi_{0})=0 \\tag{7}\n\\end{equation*}\n\nThe general solution of equation (6) is\n\n\\begin{equation*}\n\\varphi(p)=C_{1} \\delta(p-\\hbar k)+C_{2} \\delta(p+\\hbar k)-\\frac{2 m V_{0}}{\\sqrt{2 \\pi \\hbar}} \\frac{\\psi(0)}{(p^{2}-\\hbar^{2} k^{2})} \\tag{8}\n\\end{equation*}\n\nwhere $C_{1}, ~ C_{2}, ~ \\psi(0)$ are to be determined. Substituting equation (8) into equation (2), we get the wave function in the $x$ representation\n\n\\begin{equation*}\n\\psi(x)=\\frac{C_{1}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{\\mathrm{i} k x}+\\frac{C_{2}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{-\\mathrm{i} k x}-\\frac{2 m V_{0}}{2 \\pi \\hbar} \\psi(0) \\int_{-\\infty}^{+\\infty} \\frac{\\mathrm{d} p}{p^{2}-\\hbar^{2} k^{2}} \\mathrm{e}^{\\mathrm{i} p x / \\hbar} \\tag{9}\n\\end{equation*}\n\nThe last integral in equation (9) should be taken as the principal value, which can be calculated using the contour integral method in the complex $p$ plane, resulting in\n\n\\int_{-\\infty}^{+\\infty} \\frac{\\mathrm{d} p}{p^{2}-\\hbar^{2} k^{2}} \\mathrm{e}^{\\mathrm{i} k x / \\hbar}= \\begin{cases}\\frac{\\mathrm{i} \\pi}{2 \\hbar k}(\\mathrm{e}^{\\mathrm{i} k x}-\\mathrm{e}^{-\\mathrm{i} k x}), & x>0 \\tag{10}\\\\ \\frac{i \\pi}{2 \\hbar k}(\\mathrm{e}^{-\\mathrm{i} k x}-\\mathrm{e}^{\\mathrm{i} k x}), & x<0\\end{cases}\n\nWhen $x \\rightarrow 0$, the right side of equation (10) becomes 0, and from equation (9) we get\n\n\\begin{equation*}\n\\psi(0)=(C_{1}+C_{2}) / \\sqrt{2 \\pi \\hbar} \\tag{11}\n\\end{equation*}\n\nSubstituting equations (10) and (11) into equation (9), we get\n\n\\sqrt{2 \\pi \\hbar} \\psi(x)=C_{1} \\mathrm{e}^{\\mathrm{i} k x}+C_{2} \\mathrm{e}^{-\\mathrm{i} k x}-\\frac{\\mathrm{i} m V_{0}}{2 \\hbar^{2} k}(C_{1}+C_{2})(\\mathrm{e}^{\\mathrm{i} k x}-\\mathrm{e}^{-\\mathrm{i} k x}), \\quad x>0\n\nGiven that the incident wave is $\\mathrm{e}^{\\mathrm{i} k x}$ (i.e., the incident momentum $p=\\hbar k$), in the region $x>0$ there should only be the transmitted wave, i.e., the $\\mathrm{e}^{\\mathrm{i} k x}$ term, and no $\\mathrm{e}^{-\\mathrm{i} k}$ term. Therefore, $C_{1}, ~ C_{2}$ must satisfy the following relation:\n\n\\begin{equation*}\nC_{2}=-\\mathrm{i} \\frac{m V_{0}}{2 \\hbar^{2} k}(C_{1}+C_{2}) \\tag{12}\n\\end{equation*}\n\nThus,\n\n\\begin{equation*}\n\\psi(x)=\\frac{C_{1}+C_{2}}{\\sqrt{2 \\pi \\hbar}} \\mathrm{e}^{\\mathrm{i} k x}, \\quad x>0 \\tag{13}\n\\end{equation*}\n\nSimilarly, we can get\n\n\\begin{equation*}\n\\psi(x)=\\frac{1}{\\sqrt{2 \\pi \\hbar}}[(C_{1}-C_{2}) \\mathrm{e}^{i k x}+2 C_{2} \\mathrm{e}^{-\\mathrm{i} k x}], \\quad x<0 \\tag{14}\n\\end{equation*}\n\nwhere the $\\mathrm{e}^{\\mathrm{ik} x}$ term is the incident wave, and the $\\mathrm{e}^{-\\mathrm{i} k x}$ term is the reflected wave. If the incident wave amplitude is set to 1, then equation (14) can be written as\n\n\\begin{equation*}\n\\psi(x)=\\mathrm{e}^{\\mathrm{i} k \\tau}+R \\mathrm{e}^{-\\mathrm{i} k x} \\tag{14'}\n\\end{equation*}\n\nThen we should take\n\n\\begin{equation*}\nC_{1}-C_{2}=\\sqrt{2 \\pi \\hbar} \\tag{15}\n\\end{equation*}\n\nFrom equations (13) and (14), it is easy to see that\n\n$$\\text { Transmission coefficient }=|\\frac{C_{1}+C_{2}}{C_{1}-C_{2}}|^{2}=\\frac{1}{|1-\\frac{2 C_{2}}{C_{1}+C_{2}}|^{2}}$$\n\nUsing equation (12), we obtain\n\n\\begin{equation*}\n\\text { Transmission coefficient }=\\frac{1}{|1+\\mathrm{i} m V_{0} / \\hbar^{2} k|^{2}}=\\frac{1}{1+(m V_{0} / \\hbar^{2} k)^{2}} \\tag{16}\n\\end{equation*}", + "final_answer": [ + "\\frac{1}{1+(m V_{0} / \\hbar^{2} k)^{2}}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$m$": "Mass of the particle.", + "$V_0$": "Strength of the delta potential barrier.", + "$\\hbar$": "Reduced Planck's constant.", + "$k$": "Wave number, defined as $k=\\sqrt{2 m E} / \\hbar$." + } + }, + { + "id": 44, + "context": "", + "question": "The energy operator of a one-dimensional harmonic oscillator is\n\n\\begin{equation*}\nH=\\frac{p_{x}^{2}}{2 \\mu}+\\frac{1}{2} \\mu \\omega^{2} x^{2} \\tag{1}\n\\end{equation*}\n\n\nTry to derive its energy level expression using the Heisenberg equation of motion for operators and the fundamental commutation relation.", + "answer": "Using the Heisenberg equation of motion, we obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d} p}{\\mathrm{~d} t}=\\frac{1}{\\mathrm{i} \\hbar}[p, H]=\\frac{\\mu \\omega^{2}}{2 \\mathrm{i} \\hbar}[p, x^{2}]=-\\mu \\omega^{2} x \\tag{2}\n\\end{equation*}\n\n\nTake the matrix element in the energy representation, yielding\n\n\\begin{equation*}\n\\mathrm{i} \\omega_{k n} p_{k n}=-\\mu \\omega^{2} x_{k n} \\tag{3}\n\\end{equation*}\n\n\nFrom the previous result,\n\n\\begin{equation*}\np_{k n}=\\mathrm{i} \\omega_{k n} \\mu x_{k n} \\tag{4}\n\\end{equation*}\n\n\nCombining equations (3) and (4), we obtain\n\n\\begin{equation*}\n(\\omega^{2}-\\omega_{k n}^{2}) x_{k n}=0 \\tag{5}\n\\end{equation*}\n\n\nwhere $k, ~ n$ can be understood as quantum state indices. From equation (5) it is evident\n\n\\begin{array}{ll}\n\\text { If } \\omega_{k n} \\neq \\pm \\omega, & \\text { then } x_{k n}=0 \\tag{6}\\\\\n\\text { If } x_{k n} \\neq 0, \\quad \\text { then } \\omega_{k n}= \\pm \\omega\n\\end{array}\n\n\nSince\n\n\\begin{equation*}\n(x^{2})_{k k}=\\sum_{n} x_{k n} x_{n k}=\\sum_{n}|x_{n k}|^{2}>0 \\tag{7}\n\\end{equation*}\n\n\nFor any chosen energy level $E_{k}$, there must exist some $n$ such that $x_{n k} \\neq 0$, then from equation (6), the energy level difference $E_{n}-$ $E_{k}= \\pm \\hbar \\omega$, that is, given any energy level, there must exist another energy level differing by $\\hbar \\omega$. Therefore, all energy levels are\n\n$$ E=E_{0}, \\quad E_{0}+\\hbar \\omega, \\quad E_{0}+2 \\hbar \\omega, \\cdots $$\n\nThat is\n\n\\begin{equation*}\nE_{n}=E_{0}+n \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{8}\n\\end{equation*}\n\n\nTo find the ground state energy $E_{0}$, the equation proven in the previous problem can be used\n\n\\begin{equation*}\n\\sum_{n}(E_{n}-E_{k})|x_{n k}|^{2}=\\hbar^{2} / 2 \\mu \\tag{9}\n\\end{equation*}\n\n\nTaking $k$ as the ground state, from equations (6), (7), and (9), we get\n\n$$\\frac{\\hbar^{2}}{2 \\mu}=\\hbar \\omega \\sum_{n}|x_{n 0}|^{2}=\\hbar \\omega(x^{2})_{00}$$\n\n\nThus, the average potential energy of the ground state is\n\n\\begin{equation*}\n\\frac{1}{2} \\mu \\omega^{2}(x^{2})_{00}=\\frac{1}{4} \\hbar \\omega \\tag{10}\n\\end{equation*}\n\n\nAccording to the virial theorem, we have\n\n\\langle T\\rangle_{0}=\\langle V\\rangle_{0}=\\frac{1}{2} E_{0}\n\n\nComparing with equation (10), we obtain\n\n\\begin{equation*}\nE_{0}=\\frac{1}{2} \\hbar \\omega \\tag{11}\n\\end{equation*}\n\n\nSubstituting into equation (8), we get the energy level formula\n\n\\begin{equation*}\nE_{n}=(n+\\frac{1}{2}) \\hbar \\omega, \\quad n=0,1,2, \\cdots \\tag{$\\prime$}\n\\end{equation*}\n\n\nConsidering equation (6), equation (9) gives\n\n\\begin{equation*}\n|x_{k+1, k}|^{2}-|x_{k-1, k}|^{2}=\\frac{\\hbar}{2 \\mu \\omega} \\tag{12}\n\\end{equation*}\n\n\nBy appropriately choosing the phase factor $(\\mathrm{e}^{\\mathrm{i}})$ of each energy eigenfunction, all $x_{n k}$ can be made non-negative real numbers, and equation (12) can be rewritten as (changing $k$ to $n$)\n\n\\begin{equation*}\n(x_{n+1, n})^{2}-(x_{n, n-1})^{2}=\\frac{\\hbar}{2 \\mu \\omega} \\tag{$\\prime$}\n\\end{equation*}\n\n\nWhen $n=0$, the equation yields\n\n\\begin{equation*}\n(x_{10})^{2}=\\frac{\\hbar}{2 \\mu \\omega}, \\quad x_{10}=\\sqrt{\\frac{\\hbar}{2 \\mu \\omega}} \\tag{13}\n\\end{equation*}\n\n\nBy repeatedly using equation (12＇), we get\n\n\\begin{equation*}\nx_{n+1, n}=\\sqrt{\\frac{n+1}{2} \\frac{\\hbar}{\\mu \\omega}}, \\quad n=0,1,2, \\cdots \\tag{14}\n\\end{equation*}\n\n\nBy also using equation (4), we get\n\n\\begin{equation*}\np_{n+1, n}=\\mathrm{i} \\omega \\mu x_{n+1, n} \\tag{15}\n\\end{equation*}\n\n\nNote: The matrix elements of $x$ are real numbers, and the matrix elements of $p$ are purely imaginary numbers. Therefore\n\n\\begin{align*}\n& x_{n, n+1}=(x_{n+1, n})^{*}=x_{n+1, n} \\tag{16}\\\\\n& p_{n, n+1}=(p_{n+1, n})^{*}=-p_{n+1, n} \\tag{17}\n\\end{align*}", + "final_answer": [ + "E_{n}=(n+\\frac{1}{2}) \\hbar \\omega" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_n$": "Energy of quantum state $n$", + "$n$": "Quantum number for energy levels ($n=0,1,2,...$)", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega$": "Angular frequency of the harmonic oscillator" + } + }, + { + "id": 45, + "context": "", + "question": "The Hamiltonian of a one-dimensional harmonic oscillator is $H=\\frac{p^{2}}{2 m}+\\frac{1}{2} m \\omega^{2} x^{2}$. Compute $[x(t_1),p(t_2)]$ in the Heisenberg picture.", + "answer": "It is easy to obtain\n\n\\begin{align*}\n{[x(t_{1}), x(t_{2})] } & =[x, p] \\frac{1}{m \\omega}(\\cos \\omega t_{1} \\sin \\omega t_{2}-\\sin \\omega t_{1} \\cos \\omega t_{2}) \\\\\n& =\\frac{i \\hbar}{m \\omega} \\sin \\omega(t_{2}-t_{1}). \\tag{1}\n\\end{align*}\n\nSimilarly, it can be obtained\n\n\\begin{align*}\n& {[p(t_{1}), p(t_{2})]=\\mathrm{i} m \\omega \\hbar \\sin \\omega(t_{2}-t_{1})} \\tag{2}\\\\\n& {[x(t_{1}), p(t_{2})]=\\mathrm{i} \\hbar \\cos \\omega(t_{2}-t_{1})} \\tag{3}\n\\end{align*}\n\nIn equation (3), when $t_{1}=t_{2}=t$, we obtain\n\n\\begin{equation*}\n[x(t), p(t)]=\\mathrm{i} \\hbar \\tag{4}\n\\end{equation*}\n\n\nThis commutation relation corresponds to $[x, p]=\\mathrm{i} \\hbar$ in the Schrödinger picture, which applies not only to the harmonic oscillator problem but also to any other problem. As for equations (1), (2), and (3), they are specific to the harmonic oscillator.", + "final_answer": [ + "\\mathrm{i} \\hbar \\cos \\omega(t_{2}-t_{1})" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mathrm{i}$": "Imaginary unit", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega$": "Angular frequency of the harmonic oscillator", + "$t_2$": "Second time variable", + "$t_1$": "First time variable" + } + }, + { + "id": 46, + "context": "", + "question": "Two localized non-identical particles with spin $1 / 2$ (ignoring orbital motion) have an interaction energy given by\n\n$$H=A \\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2} $$\n\nwhere $\\boldsymbol{s}_{1}, \\boldsymbol{s}_{2}$ are spin operators (the eigenvalues of their $z$ components $s_{iz}$ being $\\pm 1/2$), and $A$ is a constant related to energy.\nAt time $t=0$, particle 1 has spin \"up\" (i.e., the measured value of $s_{1z}$ is $1/2$), and particle 2 has spin \"down\" (i.e., the measured value of $s_{2z}$ is $-1/2$). Determine the probability that at any time $t>0$, particle 1 has spin \"up\" (i.e., measuring $s_{1z}$ yields $1/2$) in the Heisenberg picture.", + "answer": "Solution 1: Starting from the Heisenberg equation of motion for the spin operators. It is easily observed from the construction of $H$ that the total spin $\\boldsymbol{S}$ commutes with $H$ and is a conserved quantity, hence the values and corresponding probabilities of the components of total $\\boldsymbol{S}$ remain constant, and the distribution probability of $\\boldsymbol{S}^{2}$ also does not change. At $t=0$, the spin state of the system $\\alpha(1) \\beta(2)$ is a common eigenstate of $s_{1 z}$ and $s_{2 z}$, and thus also an eigenstate of total $S_{z}$ with eigenvalue $S_{z}=0$. As $S_{z}$ is a conserved quantity, it can only have the eigenvalue 0 at any time point, and cannot take other eigenvalues. Hence, the probability that both particle 1 and 2 have spin \"up\" $(s_{1 z}=s_{2 z}=\\frac{1}{2}, S_{z}=1)$ is 0. At $t=0$, the total spin quantum numbers $S=1, ~ 0$, each with probabilities $1 / 2$. Since $\\boldsymbol{S}^{2}$ is conserved, this probability does not change over time. The above conclusions apply to parts (b) and (c).\n\nIn order to calculate $\\langle\\boldsymbol{s}_{1}\\rangle$ and $\\langle\\boldsymbol{s}_{2}\\rangle$, the Heisenberg equation of motion can be solved.\n\n\\begin{gather*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{s}_{1}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{s}_{1}, H]=\\mathrm{i} A[\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}, \\boldsymbol{s}_{1}]=-\\boldsymbol{A} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{1}\\\\\n\\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\boldsymbol{s}_{2}=\\frac{1}{\\mathrm{i} \\hbar}[\\boldsymbol{s}_{2}, H]=\\mathrm{i} A[\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}, \\boldsymbol{s}_{2}]=\\boldsymbol{A} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{2}\n\\end{gather*}\n\n\nAdding and subtracting the two equations gives\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2})=\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\boldsymbol{S}=0 \\tag{3}\\\\\n& \\frac{\\mathrm{~d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2})=-2 A \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\tag{4}\n\\end{align*}\n\nMultiplying equation (2) by $s_{1}$ and equation (1) by $s_{2}$ gives\n\n\\begin{align*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}) & =A[s_{1} \\times(s_{1} \\times s_{2})-(s_{1} \\times s_{2}) \\times s_{2}] \\\\\n& =\\frac{A}{2}(s_{1}-s_{2}) \\tag{5}\n\\end{align*}\n\nMultiply equation (5) by $2i$, and add it to equation (4) to get\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(s_{1}-s_{2}+2 \\mathrm{i} s_{1} \\times s_{2})=\\mathrm{i} A(s_{1}-s_{2}+2 \\mathrm{i} s_{1} \\times s_{2}) \\tag{6}\n\\end{equation*}\n\n\nTake the average value and integrate with respect to $t$, then\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}+2 \\mathrm{i} \\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}+2 \\mathrm{i} \\mathbf{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0} \\mathrm{e}^{\\mathrm{i} A t} \\tag{7}\n\\end{equation*}\n\n\nThe initial condition of the problem is\n\n\\begin{equation*}\n\\chi(t=0)=\\alpha(1) \\beta(2) \\tag{8}\n\\end{equation*}\n\n\nTherefore,\n\n\\begin{array}{l}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t=0}=(0,0, \\frac{1}{2}), \\quad\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=(0,0,-\\frac{1}{2}) \\tag{$\\prime$}\\\\\n\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0, \\quad\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t=0}=(0,0,1)=\\boldsymbol{e}_{3} \\\\\n\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0}=\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} \\times\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=0\n\\end{array}}\n\nSubstituting into equation (7), we get\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t}+2 \\mathrm{i}\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\mathrm{e}^{\\mathrm{i} A t} \\boldsymbol{e}_{3}=(\\cos A t+\\mathrm{i} \\sin A t) \\boldsymbol{e}_{3} \\tag{9}\n\\end{equation*}\n\n\nSince $s_{1}, s_{2}, s_{1} \\times s_{2}$ are all Hermitian operators, their averages are real numbers, so\n\n\\begin{gather*}\n\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t}=\\boldsymbol{e}_{3} \\cos A t \\tag{10}\\\\\n\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t}=\\frac{1}{2} \\boldsymbol{e}_{3} \\sin A t \\tag{11}\n\\end{gather*}\n\n\nEquation (3) shows that $\\boldsymbol{S}$ is conserved, and from the initial condition ($8^{\\prime}$):\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{12}\n\\end{equation*}\n\n\nAdding and subtracting equations (10) and (12), we get\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\langle\\boldsymbol{s}_{2}\\rangle_{t}=\\frac{1}{2} e_{3} \\cos A t \\tag{11}\n\\end{equation*}\n\n\nThat is:\n\n\\begin{align*}\n& \\langle s_{1 x}\\rangle_{t}=\\langle s_{1 y}\\rangle_{t}=\\langle s_{2 x}\\rangle_{t}=\\langle s_{2 y}\\rangle_{t}=0 \\\\\n& \\langle s_{1 z}\\rangle_{t}=-\\langle s_{2 z}\\rangle_{t}=\\frac{1}{2} \\cos A t \\tag{13'}\n\\end{align*}\n\n\nThis result is consistent with equation (5) obtained in Exercise 6.47. Equation (11) corresponds to\n\n\\begin{align*}\n& \\langle s_{1 x} s_{2 y}-s_{1 y} s_{2 x}\\rangle_{t}=\\frac{1}{2} \\sin A t \\\\\n& \\langle s_{1 y} s_{2 z}-s_{1 z} s_{2 y}\\rangle_{t}=0 \\tag{11'}\\\\\n& \\langle s_{1 z} s_{2 x}-s_{1 x} s_{2 z}\\rangle_{t}=0\n\\end{align*}\n\n\nReaders can easily use equation ($4^{\\prime}$) obtained in Exercise 6.47 to verify this conclusion, although it is not easy to see this result there.\n\nAssume that at $t>0$, the probability of particle 1 having spin \"up\" $(s_{1 z}=\\frac{1}{2})$ is $w(t)$, then\n\n\\langle s_{1 z}\\rangle_{t}=\\frac{1}{2} \\cos A t=\\frac{1}{2} w(t)-\\frac{1}{2}[1-w(t)]=w(t)-\\frac{1}{2}\n\n\nThus,\n\n\\begin{equation*}\nw(t)=\\frac{1}{2}(1+\\cos A t)=\\cos ^{2} \\frac{A t}{2} \\tag{1}\n\\end{equation*}\n\n\nSolution 2: Using the Heisenberg picture of the average value formula\n\n\\begin{equation*}\n\\langle s_{1}\\rangle_{t}=\\langle\\chi(0)| s_{1}(t)|\\chi(0)\\rangle=\\langle\\chi(0)| U^{+}(t) s_{1} U(t)|\\chi(0)\\rangle \\tag{15}\n\\end{equation*}\n\n\nwhere\n\n\\begin{array}{c}\\nu(t)=\\mathrm{e}^{-\\mathrm{i} H t}=\\mathrm{e}^{-\\mathrm{i} A s_{1} \\cdot s_{2}} \\tag{16}\\\\\nU^{+}(t)=\\mathrm{e}^{\\mathrm{i} H t}=\\mathrm{e}^{\\mathrm{i} t s_{1} \\cdot s_{2}}\n\\end{array}}\n\n$s_{1}(t)$ can be expressed as (let $\\lambda=A t)$\n\n\\begin{align*}\n\\boldsymbol{s}_{1}(t) & =U^{+}(t) \\boldsymbol{s}_{1} U(t) \\\\\n& =\\frac{1}{2}(\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2})+\\frac{1}{2}(\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}) \\cos A t-\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2} \\sin A t \\tag{17}\n\\end{align*}\n\n\nSubstituting into equation (15), and using the initial averages\n\n\\begin{align*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} & =\\langle\\chi(0)| \\boldsymbol{s}_{1}|\\chi(0)\\rangle=\\langle\\alpha(1)| \\boldsymbol{s}_{1}|\\alpha(1)\\rangle \\\\\n& =(0,0, \\frac{1}{2}) \\\\\n\\langle\\boldsymbol{s}_{2}\\rangle_{t=0} & =\\langle\\chi(0)| \\boldsymbol{s}_{2}|\\chi(0)\\rangle=\\langle\\beta(2)| \\boldsymbol{s}_{2}|\\beta(2)\\rangle \\\\\n& =(0,0,-\\frac{1}{2}) \\\\\n& \\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0}=\\langle\\boldsymbol{s}_{1}\\rangle_{t=0} \\times\\langle\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{18}\n\\end{align*}\n\n\nwe get\n\n\\begin{align*}\n\\langle\\boldsymbol{s}_{1}\\rangle_{t} & =\\frac{1}{2}\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}+\\frac{1}{2}\\langle\\boldsymbol{s}_{1}-\\boldsymbol{s}_{2}\\rangle_{t=0} \\cos A t-\\langle\\boldsymbol{s}_{1} \\times \\boldsymbol{s}_{2}\\rangle_{t=0} \\sin A t \\\\\n& =\\frac{1}{2} \\boldsymbol{e}_{3} \\cos A t \\tag{19}\n\\end{align*}\n\n\nThe total spin $\\boldsymbol{S}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}$ commutes with both $\\boldsymbol{s}_{1} \\cdot \\boldsymbol{s}_{2}$ and thus also with $U$ and $U^{+}$, so\n\n\\begin{gather*}\n\\boldsymbol{S}(t)=\\boldsymbol{S}=\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2} \\tag{20}\\\\\n\\langle\\boldsymbol{S}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{s}_{1}+\\boldsymbol{s}_{2}\\rangle_{t=0}=0 \\tag{21}\n\\end{gather*}\n\n\nSo\n\n\\begin{equation*}\n\\langle\\boldsymbol{s}_{2}\\rangle_{t}=\\langle\\boldsymbol{S}\\rangle_{t}-\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\langle\\boldsymbol{s}_{1}\\rangle_{t}=-\\frac{1}{2} \\boldsymbol{e}_{3} \\cos A t \\tag{22}\n\\end{equation*}\n\n\nAll these results match those of Solution 1. The method to calculate the probability is the same as Solution 1 and thus the result is also the same, omitted here.", + "final_answer": [ + "w(t)=\\cos ^{2} \\frac{A t}{2}" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$w(t)$": "Probability that particle 1 has spin 'up' at time $t$", + "$A$": "A constant related to energy", + "$t$": "Time" + } + }, + { + "id": 47, + "context": "", + "question": "For a spin $1/2$ particle, $\\langle\\boldsymbol{\\sigma}\\rangle$ is often called the polarization vector, denoted as $\\boldsymbol{P}$. It is also the spatial orientation of the spin angular momentum. Assume the particle is localized and subject to a magnetic field $\\boldsymbol{B}(t)$ along the $z$ direction with time-varying intensity, with the potential given by\n\n$$H=-\\mu_{0} \\boldsymbol{\\sigma} \\cdot \\boldsymbol{B}(t)=-\\mu_{0} \\sigma_{z} B(t),$$\n\nIn the Heisenberg picture, find the time evolution of the polarization vector, i.e., find $P_x(t)=\\langle{\\sigma}_x\\rangle_{t}$. Let $\\boldsymbol{P}(t=0)$ point in the direction $(\\theta_{0}, \\varphi_{0})$, where $\\theta_{0}=2 \\delta, \\varphi_{0}=2 \\alpha$.", + "answer": "According to the Heisenberg equation of motion.\n\n$$\\frac{\\mathrm{d}}{\\mathrm{~d} t} \\sigma_{z}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{z}, H]=0,$$\n\nTherefore,\n\n\\begin{equation*}\n\\langle\\sigma_{z}\\rangle_{t}=\\langle\\sigma_{z}\\rangle_{t=0}=\\cos \\theta_{0}=\\cos 2 \\delta \\tag{1}\n\\end{equation*}\n\nMoreover,\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} t} \\sigma_{x}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{x}, H]=-\\frac{\\mu_{0}}{\\mathrm{i} \\hbar} B(t)[\\sigma_{x}, \\sigma_{x}]=\\frac{2 \\mu_{0}}{\\hbar} B(t)_{\\sigma_{y}} \\tag{2}\\\\\n& \\frac{\\mathrm{~d}}{\\mathrm{~d} t} \\sigma_{y}=\\frac{1}{\\mathrm{i} \\hbar}[\\sigma_{y}, H]=-\\frac{2 \\mu_{0}}{\\hbar} B(t) \\sigma_{x}\n\\end{align*}\n\n\nCombining the two equations, it is easy to obtain\n\n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{~d} t}(\\sigma_{x}+\\mathrm{i} \\sigma_{y})=-\\mathrm{i} \\frac{2 \\mu_{0}}{\\hbar} B(t)(\\sigma_{x}+\\mathrm{i} \\sigma_{y}) \\tag{3}\n\\end{equation*}\n\n\nTaking the average value and integrating over $t$, we have\n\n\\begin{equation*}\n\\langle\\sigma_{x}+\\mathrm{i} \\sigma_{y}\\rangle_{t}=\\langle\\sigma_{x}+\\mathrm{i} \\sigma_{y}\\rangle_{t=0} \\exp [-\\mathrm{i} \\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau] \\tag{4}\n\\end{equation*}\n\nSince $\\langle\\sigma_{x}\\rangle$, $\\langle\\sigma_{y}\\rangle$ are real numbers, separate the real and imaginary parts in the above equation to get\n\n\\begin{align*}\n& \\langle\\sigma_{x}\\rangle_{t}=\\langle\\sigma_{x}\\rangle_{t=0} \\cos [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]+\\langle\\sigma_{y}\\rangle_{t=0} \\sin [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau] \\tag{5}\\\\\n& \\langle\\sigma_{y}\\rangle_{t}=\\langle\\sigma_{y}\\rangle_{t=0} \\cos [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]-\\langle\\sigma_{x}\\rangle_{t=0} \\sin [\\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau]\n\\end{align*}\n\n\nThe initial average values are\n\n\\begin{equation*}\n\\langle\\sigma_{x}\\rangle_{t=0}=\\sin 2 \\delta \\cos 2 \\alpha, \\quad\\langle\\sigma_{y}\\rangle_{t=0}=\\sin 2 \\delta \\sin 2 \\alpha \\tag{6}\n\\end{equation*}\n\n\nSubstituting into equation (5) and defining $\\varphi(t)$, we get\n\n\\begin{gather*}\n\\langle\\sigma_{x}\\rangle_{t}=\\sin 2 \\delta \\cos \\varphi(t), \\quad\\langle\\sigma_{y}\\rangle_{t}=\\sin 2 \\delta \\sin \\varphi(t) \\tag{7}\\\\\n\\boldsymbol{P}(t)=\\langle\\boldsymbol{\\sigma}\\rangle_{t}=(\\sin 2 \\delta \\cos \\varphi(t), \\sin 2 \\delta \\sin \\varphi(t), \\cos 2 \\delta) \\tag{8}\n\\end{gather*}", + "final_answer": [ + "\\langle\\sigma_{x}\\rangle_{t}=\\sin 2 \\delta \\cos \\varphi(t)" + ], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\langle\\sigma_{x}\\rangle_{t}$": "Expectation value of the x-component of the spin operator at time $t$.", + "$\\delta$": "Parameter related to the initial polar angle, where $\\theta_0 = 2\\delta$.", + "$\\varphi(t)$": "Time-dependent phase angle, defined as $\\varphi(t) = 2 \\alpha - \\frac{2 \\mu_{0}}{\\hbar} \\int_{0}^{t} B(\\tau) \\mathrm{d} \\tau$." + } + }, + { + "id": 48, + "context": "", + "question": "Evaluate the function\n\n$$ \\langle 0| \\phi(x) \\phi(y)|0\\rangle=D(x-y)=\\int \\frac{d^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{\\mathbf{p}}} e^{-i p \\cdot(x-y)} $$\n\nfor $(x-y)$ spacelike so that $(x-y)^{2}=-r^{2}$, explicitly in terms of Bessel functions.\n\n\\footnotetext{\n$\\ddagger$ With some additional work you can show that there are actually six conserved charges in the case of two complex fields, and $n(2 n-1)$ in the case of $n$ fields, corresponding to the generators of the rotation group in four and $2 n$ dimensions, respectively. The extra symmetries often do not survive when nonlinear interactions of the fields are included.\n}", + "answer": "We evaluate the correlation function of a scalar field at two points,\n\n\\begin{equation*}\nD(x-y)=\\langle 0| \\phi(x) \\phi(y)|0\\rangle \\tag{2.28}\n\\end{equation*}\n\nwith $x-y$ being spacelike. Since any spacelike interval $x-y$ can be transformed to a form such that $x^{0}-y^{0}=0$, thus we will simply take:\n\n\\begin{equation*}\nx^{0}-y^{0}=0, \\quad \\text { and } \\quad|\\mathbf{x}-\\mathbf{y}|^{2}=r^{2}>0 . \\tag{2.29}\n\\end{equation*}\n\n\nNow:\n\n\\begin{align*}\nD(x-y) & =\\int \\frac{\\mathrm{d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{p}} e^{-i p \\cdot(x-y)}=\\int \\frac{\\mathrm{d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 \\sqrt{m^{2}+p^{2}}} e^{i \\mathbf{p} \\cdot(\\mathbf{x}-\\mathbf{y})} \\\\\n& =\\frac{1}{(2 \\pi)^{3}} \\int_{0}^{2 \\pi} \\mathrm{~d} \\varphi \\int_{-1}^{1} \\mathrm{~d} \\cos \\theta \\int_{0}^{\\infty} \\mathrm{d} p \\frac{p^{2}}{2 \\sqrt{m^{2}+p^{2}}} e^{i p r \\cos \\theta} \\\\\n& =\\frac{-\\mathrm{i}}{2(2 \\pi)^{2} r} \\int_{-\\infty}^{\\infty} \\mathrm{d} p \\frac{p e^{\\mathrm{i} p r}}{\\sqrt{m^{2}+p^{2}}} \\tag{2.30}\n\\end{align*}\n\n\nNow we make the path deformation on $p$-complex plane, as is shown in Figure 2.3 of Peskin \\& Daniel V. Schroeder. Then the integral becomes,\n\n\\begin{equation*}\nD(x-y)=\\frac{1}{4 \\pi^{2} r} \\int_{m}^{\\infty} \\mathrm{d} \\rho \\frac{\\rho e^{-\\rho r}}{\\sqrt{\\rho^{2}-m^{2}}}=\\frac{m}{4 \\pi^{2} r} \\mathrm{~K}_{1}(m r) . \\tag{2.31}\n\\end{equation*}", + "final_answer": [ + "D(x-y) = \\frac{m}{4 \\pi^{2} r} K_{1}(m r)" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$D(x-y)$": "Two-point correlation function, representing the vacuum expectation value of the product of two scalar field operators, $D(x-y) = \\langle 0| \\phi(x) \\phi(y)|0\\rangle$.", + "$m$": "Mass of the scalar field particle.", + "$r$": "Magnitude of the spatial separation between points $x$ and $y$. For a spacelike interval, $(x-y)^2 = -r^2$, and when $x^0-y^0=0$, $r = |\\mathbf{x}-\\mathbf{y}|$.", + "$K_1$": "Notation for the modified Bessel function of the second kind, of order 1." + } + }, + { + "id": 49, + "context": "This problem concerns the discrete symmetries $P, C$, and $T$.", + "question": "Let $\\phi(x)$ be a complex-valued Klein-Gordon field. The current associated with this field is $J^{\\mu}=i(\\phi^{*} \\partial^{\\mu} \\phi-\\partial^{\\mu} \\phi^{*} \\phi)$. The Parity operator $P$ acts on the annihilation operator $a_{\\mathbf{p}}$ as $P a_{\\mathbf{p}} P=a_{-\\mathbf{p}}$, and on the field as $P \\phi(t, \\mathbf{x}) P =\\phi(t,-\\mathbf{x})$. Determine the transformation property of the current $J^{\\mu}(t, \\mathbf{x})$ under the Parity transformation $P$. Express your answer in terms of $J^{\\mu}(t,-\\mathbf{x})$ and the factor $(-1)^{s(\\mu)}$, where $s(\\mu)=0$ for $\\mu=0$ and $s(\\mu)=1$ for $\\mu=1,2,3$.", + "answer": "Now we work out the $C, P$ and $T$ transformation properties of a scalar field $\\phi$. Our starting point is\n\nP a_{\\mathbf{p}} P=a_{-\\mathbf{p}}, \\quad T a_{\\mathbf{p}} T=a_{-\\mathbf{p}}, \\quad C a_{\\mathbf{p}} C=b_{\\mathbf{p}}\n\n\nThen, for a complex scalar field\n\n\\begin{equation*}\n\\phi(x)=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{\\mathbf{k}} e^{-\\mathrm{i} k \\cdot x}+b_{\\mathbf{k}}^{\\dagger} e^{\\mathrm{i} k \\cdot x}], \\tag{3.63}\n\\end{equation*}\n\nwe have\n\n\\begin{align*}\n& P \\phi(t, \\mathbf{x}) P=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{-\\mathbf{k}} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+b_{-\\mathbf{k}}^{\\dagger} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi(t,-\\mathbf{x}) . \\tag{3.64a}\\\\\n& T \\phi(t, \\mathbf{x}) T=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[a_{-\\mathbf{k}} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+b_{-\\mathbf{k}}^{\\dagger} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi(-t, \\mathbf{x}) . \\tag{3.64b}\\\\\n& C \\phi(t, \\mathbf{x}) C=\\int \\frac{\\mathrm{d}^{3} k}{(2 \\pi)^{3}} \\frac{1}{\\sqrt{2 k^{0}}}[b_{\\mathbf{k}} e^{-\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}+a_{\\mathbf{k}}^{\\dagger} e^{\\mathrm{i}(k^{0} t-\\mathbf{k} \\cdot \\mathbf{x})}]=\\phi^{*}(t, \\mathbf{x}) . \\tag{3.64c}\n\\end{align*}\n\n\nAs a consequence, we can deduce the $C, P$, and $T$ transformation properties of the current $J^{\\mu}=\\mathrm{i}(\\phi^{*} \\partial^{\\mu} \\phi-(\\partial^{\\mu} \\phi^{*}) \\phi)$, as follows:\n\n\\begin{align*}\nP J^{\\mu}(t, \\mathbf{x}) P & =(-1)^{s(\\mu)} \\mathrm{i}[\\phi^{*}(t,-\\mathbf{x}) \\partial^{\\mu} \\phi(t,-\\mathbf{x})-(\\partial^{\\mu} \\phi^{*}(t,-\\mathbf{x})) \\phi(t,-\\mathbf{x})] \\\\\n& =(-1)^{s(\\mu)} J^{\\mu}(t,-\\mathbf{x}), \\tag{3.65a}\n\\end{align*}\n\nwhere $s(\\mu)$ is the label for space-time indices that equals to 0 when $\\mu=0$ and 1 when $\\mu=1,2,3$. In the similar way, we have\n\n\\begin{align*}\n& T J^{\\mu}(t, \\mathbf{x}) T=(-1)^{s(\\mu)} J^{\\mu}(-t, \\mathbf{x}) \\tag{3.65b}\\\\\n& C J^{\\mu}(t, \\mathbf{x}) C=-J^{\\mu}(t, \\mathbf{x}) \\tag{3.65c}\n\\end{align*}\n\n\nOne should be careful when playing with $T$ - it is antihermitian rather than hermitian, and anticommutes, rather than commutes, with $\\sqrt{-1}$.", + "final_answer": [ + "P J^{\\mu}(t, \\mathbf{x}) P =(-1)^{s(\\mu)} J^{\\mu}(t,-\\mathbf{x})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$P$": "Parity operator.", + "$J^{\\mu}$": "Current associated with the Klein-Gordon field.", + "$t$": "Time coordinate.", + "$\\mathbf{x}$": "Spatial coordinate vector.", + "$s(\\mu)$": "Function defining the parity factor, $s(\\mu)=0$ for $\\mu=0$ and $s(\\mu)=1$ for $\\mu=1,2,3$.", + "$\\mu$": "Spacetime index." + } + }, + { + "id": 50, + "context": "Let us return to the problem of the creation of Klein-Gordon particles by a classical source. This process can be described by the Hamiltonian\n\n$$ H=H_{0}+\\int d^{3} x(-j(t, \\mathbf{x}) \\phi(x))$$\n\nwhere $H_{0}$ is the free Klein-Gordon Hamiltonian, $\\phi(x)$ is the Klein-Gordon field, and $j(x)$ is a c-number scalar function. We found that, if the system is in the vacuum state before the source is turned on, the source will create a mean number of particles\n\n$$ \\langle N\\rangle=\\int \\frac{d^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{\\mathbf{p}}}|\\tilde{\\jmath}(p)|^{2} .$$\n\n\nIn this problem we will verify that statement, and extract more detailed information, by using a perturbation expansion in the strength of the source.", + "question": "Compute the probability that the source creates one particle of momentum $k$ to all orders in the source $j$ by summing the perturbation series.", + "answer": "The probability that the source creates one particle with momentum $\\mathbf{k}$ is given by,\n$$P(\\mathbf{k})=|\\langle\\mathbf{k}| T \\exp\\{i \\int \\mathrm{~d}^{4} x j(x) \\phi_{I}(x)\\}| 0\\rangle|^{2}. $$\nExpanding the amplitude to the first order in \\( j \\), we get:\n\\begin{equation*}\n\\begin{split}\nP(k) &= \\left| \\langle \\mathbf{k} | 0 \\rangle + i \\int \\mathrm{d}^4x \\, j(x) \\int \\frac{\\mathrm{d}^3p}{(2\\pi)^3} \\frac{e^{i p \\cdot x}}{\\sqrt{2E_p}} \\langle \\mathbf{k} | a_p^\\dagger | 0 \\rangle + O(j^2) \\right|^2 \\\\ \n&= \\left| i \\int \\frac{\\mathrm{d}^3p}{(2\\pi)^3} \\frac{\\tilde{\\jmath}(p)}{\\sqrt{2E_p}} (2\\pi)^3 \\delta(\\mathbf{p} - \\mathbf{k}) \\right|^2 = \\left| i \\frac{\\tilde{\\jmath}(k)}{\\sqrt{2E_k}} \\right|^2 = \\frac{1}{\\sqrt{2E_k}}|\\tilde{\\jmath}(k)|^2 + O(j^4).\n\\end{split}\n\\end{equation*}\n\nIf we go on to work out all the terms in the perturbation expansion, the transition amplitude factorizes into the first-order amplitude multiplied by the sum of all disconnected vacuum bubbles (vacuum-to-vacuum amplitude). The square of the vacuum-to-vacuum amplitude is $e^{-\\langle N \\rangle}$. Thus we obtain:\n\\begin{equation*}\nP(k)=\\frac{1}{2E_{\\mathbf{k}}}|\\tilde{\\jmath}(k)|^{2} e^{-\\langle N \\rangle}\n\\end{equation*}\nwhere $\\langle N \\rangle=\\int \\frac{d^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{\\mathbf{p}}}|\\tilde{\\jmath}(p)|^{2}$.", + "final_answer": [ + "P(k) = \\frac{1}{2 E_{\\mathbf{k}}} |\\tilde{\\jmath}(k)|^{2} e^{-\\langle N \\rangle}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$P(k)$": "Probability that the source creates one particle with momentum $k$", + "$\\tilde{j}(k)$": "Fourier transform of the source $j$ evaluated at momentum $k$" + } + }, + { + "id": 51, + "context": "", + "question": "Decay of a scalar particle. Consider the following Lagrangian, involving two real scalar fields $\\Phi$ and $\\phi$ :\n\n$$\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi .$$\n\n\nThe last term is an interaction that allows a $\\Phi$ particle to decay into two $\\phi$ 's, provided that $M>2 m$. Assuming that this condition is met, calculate the lifetime of the $\\Phi$ to lowest order in $\\mu$.", + "answer": "This problem is based on the following Lagrangian,\n\n\\begin{equation*}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi . \\tag{4.17}\n\\end{equation*}\n\n\nWhen $M>2 m$, a $\\Phi$ particle can decay into two $\\phi$ particles. We want to calculate the lifetime of the $\\Phi$ particle to lowest order in $\\mu$.\n\nThe two-body decay rate is given in (4.86) of $\\mathrm{P} \\& \\mathrm{~S}$,\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Gamma=\\frac{1}{2 M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}|\\mathcal{M}(\\Phi(0) \\rightarrow \\phi(p_{1}) \\phi(p_{2}))|^{2}(2 \\pi)^{4} \\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2}) . \\tag{4.18}\n\\end{equation*}\n\n\nTo lowest order in $\\mu$, the amplitude $\\mathcal{M}$ is given by,\n\n\\begin{equation*}\ni \\mathcal{M}=-2 i \\mu \\tag{4.19}\n\\end{equation*}\n\n\nThe delta function in our case reads,\n\n\\begin{equation*}\n\\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2})=\\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \\tag{4.20}\n\\end{equation*}\n\nthus,\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2} \\cdot \\frac{2 \\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}(2 \\pi)^{4} \\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \\tag{4.21}\n\\end{equation*}\n\nwhere an additional factor of $1 / 2$ takes account of two identical $\\phi$ 's in final state. Furthermore, there are two mass-shell constraints,\n\n\\begin{equation*}\nm^{2}+\\mathbf{p}_{i}^{2}=E_{\\mathbf{p}_{i}}^{2} . \\quad(i=1,2) \\tag{4.22}\n\\end{equation*}\n\n\nHence,\n\n\\begin{equation*}\n\\Gamma=\\frac{\\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1}}{(2 \\pi)^{3}} \\frac{1}{4 E_{\\mathbf{p}_{1}}^{2}}(2 \\pi) \\delta(M-2 E_{\\mathbf{p}_{1}})=\\frac{\\mu^{2}}{8 \\pi M}(1-\\frac{4 m^{2}}{M^{2}})^{1 / 2} \\tag{4.23}\n\\end{equation*}\n\n\nThen the lifetime $\\tau$ of $\\Phi$ is,\n\n\\begin{equation*}\n\\tau=\\Gamma^{-1}=\\frac{8 \\pi M}{\\mu^{2}}(1-\\frac{4 m^{2}}{M^{2}})^{-1 / 2} \\tag{4.24}\n\\end{equation*}", + "final_answer": [ + "\\tau = \\frac{8\\pi M}{\\mu^2} (1 - \\frac{4m^2}{M^2})^{-1/2}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\tau$": "Lifetime of the $\\Phi$ particle", + "$\\pi$": "Mathematical constant pi", + "$M$": "Mass of the $\\Phi$ particle", + "$\\mu$": "Coupling constant for the interaction term", + "$m$": "Mass of the $\\phi$ particle" + } + }, + { + "id": 52, + "context": "", + "question": "Consider a theory with two real scalar fields, $\\Phi$ with mass $M$ and $\\phi$ with mass $m$, described by the Lagrangian $\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi$. If $M > 2m$, the $\\Phi$ particle can decay into two $\\phi$ particles. To the lowest order in the coupling constant $\\mu$, the decay rate $\\Gamma$ for this process is given by $\\Gamma = \\frac{\\mu^2}{8\\pi M} \\sqrt{1 - \\frac{4m^2}{M^2}}$. Calculate the value of the product $\\Gamma M / \\mu^2$ if $M=3m$.", + "answer": "This problem is based on the following Lagrangian,\n\n\\begin{equation*}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\Phi)^{2}-\\frac{1}{2} M^{2} \\Phi^{2}+\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{1}{2} m^{2} \\phi^{2}-\\mu \\Phi \\phi \\phi . \n\\end{equation*}\n\n\nWhen $M>2 m$, a $\\Phi$ particle can decay into two $\\phi$ particles. We want to calculate the lifetime of the $\\Phi$ particle to lowest order in $\\mu$.\n\nThe two-body decay rate is given in (4.86) of $\\mathrm{P} \\& \\mathrm{~S}$,\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Gamma=\\frac{1}{2 M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}|\\mathcal{M}(\\Phi(0) \\rightarrow \\phi(p_{1}) \\phi(p_{2}))|^{2}(2 \\pi)^{4} \\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2}) . \n\\end{equation*}\n\n\nTo lowest order in $\\mu$, the amplitude $\\mathcal{M}$ is given by,\n\n\\begin{equation*}\ni \\mathcal{M}=-2 i \\mu \n\\end{equation*}\n\n\nThe delta function in our case reads,\n\n\\begin{equation*}\n\\delta^{(4)}(p_{\\Phi}-p_{1}-p_{2})=\\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \n\\end{equation*}\n\nthus,\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2} \\cdot \\frac{2 \\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1} \\mathrm{~d}^{3} p_{2}}{(2 \\pi)^{6}} \\frac{1}{4 E_{\\mathbf{p}_{1}} E_{\\mathbf{p}_{2}}}(2 \\pi)^{4} \\delta(M-E_{\\mathbf{p}_{1}}-E_{\\mathbf{p}_{2}}) \\delta^{(3)}(\\mathbf{p}_{1}+\\mathbf{p}_{2}), \n\\end{equation*}\n\nwhere an additional factor of $1 / 2$ takes account of two identical $\\phi$ 's in final state. Furthermore, there are two mass-shell constraints,\n\n\\begin{equation*}\nm^{2}+\\mathbf{p}_{i}^{2}=E_{\\mathbf{p}_{i}}^{2} . \\quad(i=1,2) \n\\end{equation*}\n\n\nHence,\n\n\\begin{equation*}\n\\Gamma=\\frac{\\mu^{2}}{M} \\int \\frac{\\mathrm{~d}^{3} p_{1}}{(2 \\pi)^{3}} \\frac{1}{4 E_{\\mathbf{p}_{1}}^{2}}(2 \\pi) \\delta(M-2 E_{\\mathbf{p}_{1}})=\\frac{\\mu^{2}}{8 \\pi M}(1-\\frac{4 m^{2}}{M^{2}})^{1 / 2}\n\\end{equation*}\n\n\nThen the lifetime $\\tau$ of $\\Phi$ is,\n\n\\begin{equation*}\n\\tau=\\Gamma^{-1}=\\frac{8 \\pi M}{\\mu^{2}}(1-\\frac{4 m^{2}}{M^{2}})^{-1 / 2}.\n\\end{equation*}\n\nWe thus have:\n\\begin{equation}\n \\Gamma M / \\mu^2 = \\frac{1}{8\\pi}\\sqrt{1-\\frac{4m^2}{M^2}} = \\frac{1}{8\\pi}\\sqrt{1-\\frac{4}{9}} = \\frac{\\sqrt{5}}{24\\pi}\n\\end{equation}", + "final_answer": [ + "\\frac{\\sqrt{5}}{24\\pi}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\pi$": "Mathematical constant pi." + } + }, + { + "id": 53, + "context": "Equivalent photon approximation. Consider the process in which electrons of very high energy scatter from a target. In leading order in $\\alpha$, the electron is connected to the target by one photon propagator. If the initial and final energies of the electron are $E$ and $E^{\\prime}$, the photon will carry momentum $q$ such that $q^{2} \\approx-2 E E^{\\prime}(1-\\cos \\theta)$. In the limit of forward scattering, whatever the energy loss, the photon momentum approaches $q^{2}=0$; thus the reaction is highly peaked in the forward direction. It is tempting to guess that, in this limit, the virtual photon becomes a real photon. Let us investigate in what sense that is true. \n\nYou should use $\\eta = (+,-,-,-)$, $\\gamma \\cdot \\epsilon = \\gamma_\\mu \\epsilon^\\mu$, assume $E > E'$, and choose $\\epsilon_1$ in the scattering plane with $q \\cdot \\epsilon_1 = 0$, $\\epsilon_1^0 = 0$, $\\epsilon_1^2 = -1$, and $\\epsilon_1 \\to (0,1,0,0)$ as $\\theta \\to 0^+$.", + "question": "Working in the frame where $p=(E, 0,0, E)$, compute explicitly the quantity $\\bar{u}_+(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_+(p)$ using massless electrons, where $u_+(p)$ and $u_+(p^{\\prime})$ are spinors of positive helicity, and $\\boldsymbol{\\epsilon}_{1}$ is a unit vector parallel to the plane of scattering. This quantity is needed only for scattering near the forward direction, and you need only provide the term of order $\\theta$. Note that for $\\boldsymbol{\\epsilon}_{1}$ (in the plane of scattering), the small $\\hat{3}$ component of $\\boldsymbol{\\epsilon}_{1}$ also gives a term of order $\\theta$ which must be taken into account.", + "answer": "It is easy to find that\n\n\\epsilon_{1}^{\\mu}=N(0, p^{\\prime} \\cos \\theta-p, 0,-p^{\\prime} \\sin \\theta), \\quad \\epsilon_{2}^{\\mu}=(0,0,1,0),\n\nwhere $N=(E^{2}+E^{\\prime 2}-2 E E^{\\prime} \\cos \\theta)^{-1 / 2}$ is the normalization constant. Then, for the righthanded electron with spinor $u_{+}(p)=\\sqrt{2 E}(0,0,1,0)^{T}$ and left-handed electron with $u_{-}(p)=$ $\\sqrt{2 E}(0,1,0,0)^{T}$, it is straightforward to show that\n\n\\begin{equation*}\\nu_{+}(p^{\\prime})=\\sqrt{2 E^{\\prime}}(0,0, \\cos \\frac{\\theta}{2}, \\sin \\frac{\\theta}{2})^{T}, \\quad u_{-}(p^{\\prime})=\\sqrt{2 E^{\\prime}}(-\\sin \\frac{\\theta}{2}, \\cos \\frac{\\theta}{2}, 0,0) \\tag{6.15}\n\\end{equation*}\n\nand,\n\n\\begin{align*}\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_{ \\pm}(p) \\simeq-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta, \\tag{6.16}\\\\\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{2} u_{ \\pm}(p) \\simeq \\pm \\mathrm{i} \\sqrt{E E^{\\prime}} \\theta \\tag{6.17}\\\\\n& \\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{1} u_{\\mp}(p)=\\bar{u}_{ \\pm}(p^{\\prime}) \\boldsymbol{\\gamma} \\cdot \\boldsymbol{\\epsilon}_{2} u_{\\mp}(p)=0 . \\tag{6.18}\n\\end{align*}\n\n\nThat is to say, we have,\n\n\\begin{equation*}\nC_{ \\pm}=-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta, \\quad \\quad D_{ \\pm}= \\pm \\mathrm{i} \\sqrt{E E^{\\prime}} \\theta \\tag{6.19}\n\\end{equation*}", + "final_answer": [ + "-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$E$": "Initial energy of the electron", + "$E^{\\prime}$": "Final energy of the electron", + "$\\theta$": "Scattering angle" + } + }, + { + "id": 54, + "context": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles.", + "question": "The unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to\n\n$$ H_{\\mathrm{int}}=\\int d^{3} x \\frac{\\lambda}{\\sqrt{2}} h \\bar{\\psi} \\psi $$\n\n\nCompute the contribution of a virtual Higgs boson to the electron $(g-2)$, in terms of $\\lambda$ and the mass $m_{h}$ of the Higgs boson.", + "answer": "The 1-loop vertex correction from Higgs boson is,\n\n\\begin{align*}\n\\bar{u}(p^{\\prime}) \\delta \\Gamma^{\\mu} u(p) & =(\\frac{\\mathrm{i} \\lambda}{\\sqrt{2}})^{2} \\int \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}} \\frac{\\mathrm{i}}{(k-p)^{2}-m_{h}^{2}} \\bar{u}(p^{\\prime}) \\frac{\\mathrm{i}}{\\not k+q-m} \\gamma^{\\mu} \\frac{\\mathrm{i}}{\\not k-m} u(p) \\\\\n& =\\frac{\\mathrm{i} \\lambda^{2}}{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{d} k^{\\prime}}{(2 \\pi)^{d}} \\frac{2 \\bar{u}(p^{\\prime}) N^{\\mu} u(p)}{(k^{\\prime 2}-\\Delta)^{3}}, \\tag{6.26}\n\\end{align*}\n\nwith\n\n\\begin{align*}\n& N^{\\mu}=(\\not k+q+m) \\gamma^{\\mu}(k+m), \\tag{6.27}\\\\\n& k^{\\prime}=k-x p+y q, \\tag{6.28}\\\\\n& \\Delta=(1-x) m^{2}+x m_{h}^{2}-x(1-x) p^{2}-y(1-y) q^{2}+2 x y p \\cdot q . \\tag{6.29}\n\\end{align*}\n\n\nTo put this correction into the following form,\n\n\\begin{equation*}\n\\Gamma^{\\mu}=\\gamma^{\\mu} F_{1}(q)+\\frac{i \\sigma^{\\mu \\nu} q_{\\nu}}{2 m} F_{2}(q) \\tag{6.30}\n\\end{equation*}\n\nwe first rewrite $N^{\\mu}$ as,\n\n\\begin{equation*}\nN^{\\mu}=A \\gamma^{\\mu}+B(p^{\\prime}+p)^{\\mu}+C(p^{\\prime}-p)^{\\mu} \\tag{6.31}\n\\end{equation*}\n\nwhere term proportional to $(p^{\\prime}-p)$ can be thrown away by Ward identity $q_{\\mu} \\Gamma^{\\mu}(q)=0$. This can be done by gamma matrix calculations, leading to the following result,\n\n\\begin{equation*}\nN^{\\mu}=[(\\frac{2}{d}-1) k^{\\prime 2}+(3+2 x-x^{2}) m^{2}+(y-x y-y^{2}) q^{2}] \\gamma^{\\mu}+(x^{2}-1) m(p^{\\prime}+p)^{\\mu} . \\tag{6.32}\n\\end{equation*}\n\n\nThen, using Gordon identity, we find,\n\n\\begin{equation*}\nN^{\\mu}=[(\\frac{2}{d}-1) k^{\\prime 2}+(x+1)^{2} m^{2}+(y-y^{2}-x y) q^{2}] \\gamma^{\\mu}+\\frac{\\mathrm{i} \\sigma^{\\mu \\nu} q_{\\nu}}{2 m} \\cdot 2 m^{2}(1-x^{2}) \\tag{6.33}\n\\end{equation*}\n\n\nComparing this with (6.30), we see that\n\n\\begin{align*}\n\\delta F_{2}(q=0) & =2 \\mathrm{i} \\lambda^{2} m^{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{4} k^{\\prime}}{(2 \\pi)^{4}} \\frac{1-x^{2}}{(k^{\\prime 2}-\\Delta)^{3}} \\\\\n& =\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{(1-x)^{2}(1+x)}{(1-x)^{2}+x(m_{h} / m)^{2}} . \\tag{6.34}\n\\end{align*}\n\n\nTo carry out the integration over $x$, we use the approximation that $m_{h} \\gg m$. Then,\n\n\\begin{align*}\n\\delta F_{2}(q=0) & \\simeq \\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x[\\frac{1}{1+x(m_{h} / m)^{2}}-\\frac{1+x-x^{2}}{(m_{h} / m)^{2}}] \\\\\n& \\simeq \\frac{\\lambda^{2}}{(4 \\pi)^{2}(m_{h} / m)^{2}}[\\log (m_{h}^{2} / m^{2})-\\frac{7}{6}] . \\tag{6.35}\n\\end{align*}", + "final_answer": [ + "\\frac{\\lambda^{2} m^2}{8 \\pi^{2} m_h^2}[\\log (\\frac{m_h^2}{m^2})-\\frac{7}{6}]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\lambda$": "Higgs-electron coupling constant.", + "$m$": "Mass of the electron.", + "$\\pi$": "Mathematical constant pi.", + "$m_h$": "Mass of the Higgs boson." + } + }, + { + "id": 55, + "context": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles.", + "question": "Some more complex versions of this theory contain a pseudoscalar particle called the axion, which couples to the electron according to\n\n$$ H_{\\mathrm{int}}=\\int d^{3} x \\frac{i \\lambda}{\\sqrt{2}} a \\bar{\\psi} \\gamma^{5} \\psi $$\n\n\nThe axion may be as light as the electron, or lighter, and may couple more strongly than the Higgs boson. Compute the contribution of a virtual axion to the $g-2$ of the electron.", + "answer": "The 1-loop correction from the axion is given by,\n\\begin{align*}\n\\bar{u}(p^{\\prime}) \\delta \\Gamma^{\\mu} u(p) & =(\\frac{-\\lambda}{\\sqrt{2}})^{2} \\int \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}} \\frac{\\mathrm{i}}{(k-p)^{2}-m_{h}^{2}} \\bar{u}(p^{\\prime}) \\gamma^{5} \\frac{\\mathrm{i}}{\\not k+\\not q-m} \\gamma^{\\mu} \\frac{\\mathrm{i}}{\\not k-m} \\gamma^{5} u(p) \\\\\n& =-\\frac{\\mathrm{i} \\lambda^{2}}{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{d} k^{\\prime}}{(2 \\pi)^{d}} \\frac{2 \\bar{u}(p^{\\prime}) N^{\\mu} u(p)}{(k^{\\prime 2}-\\Delta)^{3}}, \\tag{6.37}\n\\end{align*}\nin which $k^{\\prime}$ and $\\Delta$ are still defined as in (a) except the replacement $m_{h} \\rightarrow m_{a}$, while $N^{\\mu}$ is now given by,\n\\begin{equation*}\nN^{\\mu}=\\gamma^{5}(\\not k+q q+m) \\gamma^{\\mu}(\\not k+m) \\gamma^{5}=-(\\nless q+q-m) \\gamma^{\\mu}(\\not k-m) . \\tag{6.38}\n\\end{equation*}\n\nRepeating the same derivation as was done in (a), we get,\n\\begin{equation*}\nN^{\\mu}=[-(\\frac{2}{d}-1) k^{\\prime 2}-(1-x-y) y q^{2}+(1-x)^{2} m^{2}] \\gamma^{\\mu}-(1-x)^{2} m(p^{\\prime}+p)^{2} . \\tag{6.39}\n\\end{equation*}\n\nAgain, using Gordon identity, we get the contribution to the anomalous magnetic moment for arbitrary axion mass $m_a$:\n\\begin{align*}\n\\delta F_{2}(q=0) & =-2 \\mathrm{i} \\lambda^{2} m^{2} \\int_{0}^{1} \\mathrm{~d} x \\int_{0}^{1-x} \\mathrm{~d} y \\int \\frac{\\mathrm{~d}^{4} k^{\\prime}}{(2 \\pi)^{4}} \\frac{(1-x)^{2}}{(k^{\\prime 2}-\\Delta)^{3}} \\\\\n& =-\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{(1-x)^{3}}{(1-x)^{2}+x m_{a}^{2} / m^{2}} \\\\\n& =-\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{ m^{2} x^{3}}{ m^{2} x^{2} + m_{a}^{2} (1-x)} . \\tag{6.40}\n\\end{align*}", + "final_answer": [ + "-\\frac{\\lambda^{2}}{(4 \\pi)^{2}} \\int_{0}^{1} \\mathrm{~d} x \\frac{ m^{2} x^{3}}{ m^{2} x^{2} + m_{a}^{2} (1-x)}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\lambda$": "Coupling constant between the axion and the electron.", + "$m_{a}$": "Mass of the axion.", + "$m$": "Mass of the electron." + } + }, + { + "id": 56, + "context": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n$$\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu} $$\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n\\alpha_{g}=\\frac{g^{2}}{4 \\pi}\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n\n$$ \\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} . $$\n\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However (as we discussed in Problem 5.5), it is consistent to sum over polarization states of this massive boson by the replacement:\n\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n\nthis also implies that we may use the propagator\n\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon}", + "question": "Draw the Feynman diagrams for the process $e^{+} e^{-} \\rightarrow \\bar{q} q g$, to leading order in $\\alpha$ and $\\alpha_{g}$, and compute the differential cross section. You may throw away the information concerning the correlation between the initial beam axis and the directions of the final particles. This is conveniently done as follows: The usual trace tricks for evaluating the square of the matrix element give for this process a result of the structure\n\n\\int d \\Pi_{3} \\frac{1}{4} \\sum|\\mathcal{M}|^{2}=L_{\\mu \\nu} \\int d \\Pi_{3} H^{\\mu \\nu}\n\nwhere $L_{\\mu \\nu}$ represents the electron trace and $H^{\\mu \\nu}$ represents the quark trace. If we integrate over all parameters of the final state except $x_{1}$ and $x_{2}$, which are scalars, the only preferred 4 -vector characterizing the final state is $q^{\\mu}$. On the other hand, $H_{\\mu \\nu}$ satisfies\n\nq^{\\mu} H_{\\mu \\nu}=H_{\\mu \\nu} q^{\\nu}=0\n\n\nWhy is this true? (There is an argument based on general principles; however, you might find it a useful check on your calculation to verify this property explicitly.) Since, after integrating over final-state vectors, $\\int H^{\\mu \\nu}$ depends only on $q^{\\mu}$ and scalars, it can only have the form\n\n$$ \\int d \\Pi_{3} H^{\\mu \\nu}=(g^{\\mu \\nu}-\\frac{q^{\\mu} q^{\\nu}}{q^{2}}) \\cdot H $$\n\nwhere $H$ is a scalar. With this information, show that\n\n$$ L_{\\mu \\nu} \\int d \\Pi_{3} H^{\\mu \\nu}=\\frac{1}{3}(g^{\\mu \\nu} L_{\\mu \\nu}) \\cdot \\int d \\Pi_{3}(g^{\\rho \\sigma} H_{\\rho \\sigma})$$\n\n\nUsing this trick, derive the differential cross section\n\n$$ \\frac{d \\sigma}{d x_{1} d x_{2}}(e^{+} e^{-} \\rightarrow \\bar{q} q g)=\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}$$\n\nin the limit $\\mu \\rightarrow 0$. If we assume that each original final-state particle is realized physically as a jet of strongly interacting particles, this formula gives the probability for observing three-jet events in $e^{+} e^{-}$annihilation and the kinematic distribution of these events. The form of the distribution in the $x_{i}$ is an absolute prediction, and it agrees with experiment. The\nnormalization of this distribution is a measure of the strong-interaction coupling constant.", + "answer": "Now we calculate the differential cross section for the process $e^{+} e^{-} \\rightarrow q \\bar{g} g$ to lowest order in $\\alpha$ and $\\alpha_{g}$. First, the amplitude is\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}=Q_{f}(-\\mathrm{i} e)^{2}(-\\mathrm{i} g) \\epsilon_{\\nu}^{*}(k_{3}) \\bar{u}(k_{1})[\\gamma^{\\nu} \\frac{\\mathrm{i}}{\\not k_{1}+\\not k_{3}} \\gamma^{\\mu}-\\gamma^{\\mu} \\frac{\\mathrm{i}}{\\not k_{2}+\\not k_{3}} \\gamma^{\\nu}] v(k_{2}) \\frac{-\\mathrm{i}}{q^{2}} \\bar{v}(p_{2}) \\gamma_{\\mu} u(p_{1}) . \\tag{7.54}\n\\end{equation*}\n\n\nThen, the squared amplitude is\n\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}= & \\frac{Q_{f}^{2} g^{2} e^{4}}{4 s^{2}}(-g_{\\nu \\sigma}) \\operatorname{tr}(\\gamma_{\\mu} \\not p_{1} \\gamma_{\\rho} \\not \\not_{2}) \\\\\n& \\times \\operatorname{tr}[(\\gamma^{\\nu} \\frac{1}{\\not k_{1}+\\not k_{3}} \\gamma^{\\mu}-\\gamma^{\\mu} \\frac{1}{\\not k_{2}+\\not k_{3}} \\gamma^{\\nu}) \\not k_{2}(\\gamma^{\\rho} \\frac{1}{\\not k_{1}+\\not k_{3}} \\gamma^{\\sigma}-\\gamma^{\\sigma} \\frac{1}{\\not k_{2}+\\not k_{3}} \\gamma^{\\rho}) \\not k_{1}] \\\\\n= & \\frac{4 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{4(k_{1} \\cdot k_{2})(k_{1} \\cdot k_{2}+q \\cdot k_{3})}{(k_{1}+k_{3})^{2}(k_{2}+k_{3})^{2}} \\\\\n&+(\\frac{1}{(k_{1}+k_{3})^{4}}+\\frac{1}{(k_{2}+k_{3})^{4}})(2(k_{1} \\cdot k_{3})(k_{2} \\cdot k_{3})-\\mu^{2}(k_{1} \\cdot k_{2}))] . \\tag{7.55}\n\\end{align*}\n\n\nWe have used the trick described in Peskin's book (P261) when getting through the last equal sign. Now rewrite the quantities of final-state kinematics in terms of $x_{i}$, and set $\\mu \\rightarrow 0$, we obtain\n\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2} & =\\frac{2 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{2(1-x_{3})}{(1-x_{1})(1-x_{2})}+\\frac{1-x_{1}}{1-x_{2}}+\\frac{1-x_{2}}{1-x_{1}}] \\\\\n& =\\frac{8 Q_{f}^{2} g^{2} e^{4}}{3 s} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})} . \\tag{7.56}\n\\end{align*}\n\n\nThus the differential cross section, with 3 colors counted, reads\n\n\\begin{align*}\n\\frac{\\mathrm{d} \\sigma}{\\mathrm{~d} x_{1} \\mathrm{~d} x_{2}}|_{\\mathrm{COM}} & =\\frac{1}{2 E_{\\mathbf{p}_{1}} 2 E_{\\mathbf{p}_{2}}|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|} \\frac{s}{128 \\pi^{3}}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2}) \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})} \\tag{7.57}\n\\end{align*}\n\nwhere we have used the fact that the initial electron and positron are massless, which implies that $2 E_{\\mathbf{p}_{1}}=2 E_{\\mathbf{p}_{2}}=\\sqrt{s}$ and $|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|=2$ in COM frame.", + "final_answer": [ + "\\frac{d \\sigma}{d x_{1} d x_{2}}(e^{+} e^{-} \\rightarrow \\bar{q} q g)=\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\frac{x_{1}^{2}+x_{2}^{2}}{(1-x_{1})(1-x_{2})}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\sigma$": "Total cross section.", + "$x_{1}$": "Scalar kinematic variable, representing an energy fraction of a final-state particle.", + "$x_{2}$": "Scalar kinematic variable, representing an energy fraction of a final-state particle.", + "$\\alpha$": "Electromagnetic coupling constant (fine-structure constant).", + "$s$": "Mandelstam variable, square of the center-of-mass energy.", + "$Q_{f}$": "Electric charge of a quark of flavor $f$.", + "$\\alpha_{g}$": "Strong coupling constant, defined as $\\alpha_{g}=\\frac{g^{2}}{4 \\pi}$." + } + }, + { + "id": 57, + "context": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n$$\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu}$$\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n$$\\alpha_{g}=\\frac{g^{2}}{4 \\pi}$$\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n\n$$\\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} .$$\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However (as we discussed in Problem 5.5), it is consistent to sum over polarization states of this massive boson by the replacement:\n\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n\nthis also implies that we may use the propagator\n\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon}", + "question": "Now replace $\\mu \\neq 0$ in the formula of the differential cross section, and carefully integrate over the region. You may assume $\\mu^{2} \\ll q^{2}$. In this limit, you will find infrared-divergent terms of order $\\log (q^{2} / \\mu^{2})$ and also $\\log ^{2}(q^{2} / \\mu^{2})$, finite terms of order 1 , and terms explicitly suppressed by powers of $(\\mu^{2} / q^{2})$. You may drop terms of the last type throughout this calculation. For the moment, collect and evaluate only the infrared-divergent terms.", + "answer": "Now we reevaluate the averaged squared amplitude, with $\\mu$ kept nonzero in the formula\n\\begin{align*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}= & \\frac{4 Q_{f}^{2} g^{2} e^{4}}{3 s^{2}}(8 p_{1} \\cdot p_{2})[\\frac{4(k_{1} \\cdot k_{2})(k_{1} \\cdot k_{2}+q \\cdot k_{3})}{(k_{1}+k_{3})^{2}(k_{2}+k_{3})^{2}} \\\\\n&+(\\frac{1}{(k_{1}+k_{3})^{4}}+\\frac{1}{(k_{2}+k_{3})^{4}})(2(k_{1} \\cdot k_{3})(k_{2} \\cdot k_{3})-\\mu^{2}(k_{1} \\cdot k_{2}))].\n\\end{align*}\n\nThe result is\n\n\\begin{equation*}\n\\frac{1}{4} \\sum|\\mathrm{i} \\mathcal{M}|^{2}=\\frac{8 Q_{f}^{2} g^{2} e^{4}}{3 s} F(x_{1}, x_{2}, \\mu^{2} / s) \\tag{7.58}\n\\end{equation*}\n\nwhere\n\n\\begin{align*}\nF(x_{1}, x_{2}, \\frac{\\mu^{2}}{s})= & \\frac{2(x_{1}+x_{2}-1+\\frac{\\mu^{2}}{s})(1+\\frac{\\mu^{2}}{s})}{(1-x_{1})(1-x_{2})} \\\\\n& +[\\frac{1}{(1-x_{1})^{2}}+\\frac{1}{(1-x_{2})^{2}}]((1-x_{1})(1-x_{2})-\\frac{\\mu^{2}}{s}) \\tag{7.59}\n\\end{align*}\n\n\nThe cross section, then, can be got by integrating over $\\mathrm{d} x_{1} \\mathrm{~d} x_{2}$ :\n\n\\begin{align*}\n\\sigma(e^{+} e^{-} \\rightarrow q \\bar{q} g) & =\\frac{1}{2 E_{\\mathbf{p}_{1}} 2 E_{\\mathbf{p}_{2}}|v_{\\mathbf{p}_{1}}-v_{\\mathbf{p}_{2}}|} \\frac{s}{128 \\pi^{3}} \\int \\mathrm{~d} x_{1} \\mathrm{~d} x_{2}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2}) \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi} \\int_{0}^{1-\\frac{\\mu^{2}}{s}} \\frac{\\mathrm{~d} x_{1} \\int_{1-x_{1}-\\frac{\\mu^{2}}{s}}^{1-\\frac{t}{s(1-x_{1})}} \\mathrm{d} x_{2} F(x_{1}, x_{2}, \\frac{\\mu^{2}}{s})}{} \\\\\n& =\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi}[\\log ^{2} \\frac{\\mu^{2}}{s}+3 \\log \\frac{\\mu^{2}}{s}+5-\\frac{1}{3} \\pi^{2}+\\mathcal{O}(\\mu^{2})] \\tag{7.60}\n\\end{align*}", + "final_answer": [ + "\\frac{4 \\pi \\alpha^{2}}{3 s} \\cdot 3 Q_{f}^{2} \\cdot \\frac{\\alpha_{g}}{2 \\pi}[\\log ^{2} \\frac{\\mu^{2}}{s}+3 \\log \\frac{\\mu^{2}}{s}]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\pi$": "Mathematical constant pi.", + "$\\alpha$": "Electromagnetic fine-structure constant.", + "$s$": "Mandelstam variable, representing the square of the center-of-mass energy.", + "$Q_{f}$": "Electric charge of a quark of flavor $f$.", + "$\\alpha_{g}$": "Strong coupling constant in the given model, defined as $\\alpha_{g}=\\frac{g^{2}}{4 \\pi}$.", + "$\\mu$": "Small nonzero mass of gluons." + } + }, + { + "id": 58, + "context": "Scalar QED. This problem concerns the theory of a complex scalar field $\\phi$ interacting with the electromagnetic field $A^{\\mu}$. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4} F_{\\mu \\nu}^{2}+(D_{\\mu} \\phi)^{*}(D^{\\mu} \\phi)-m^{2} \\phi^{*} \\phi$$\n\nwhere $D_{\\mu}=\\partial_{\\mu}+i e A_{\\mu}$ is the usual gauge-covariant derivative.", + "question": "Compute, to lowest order, the differential cross section for $e^{+} e^{-} \\rightarrow \\phi \\phi^{*}$. Ignore the electron mass (but not the scalar particle's mass), and average over the electron and positron polarizations. Find the asymptotic angular dependence and total cross section. Compare your results to the corresponding formulae for $e^{+} e^{-} \\rightarrow \\mu^{+} \\mu^{-}$.\n\nCompute the contribution of the charged scalar to the photon vacuum polarization, using dimensional regularization. Note that there are two diagrams. To put the answer into the expected form,\n\n$$\\Pi^{\\mu \\nu}(q^{2})=(g^{\\mu \\nu} q^{2}-q^{\\mu} q^{\\nu}) \\Pi(q^{2})$$\n\nit is useful to add the two diagrams at the beginning, putting both terms over a common denominator before introducing a Feynman parameter. Show that, for $-q^{2} \\gg m^{2}$, the charged boson contribution to $\\Pi(q^{2})$ is exactly $1 / 4$ that of a virtual electron-positron pair.", + "answer": "Now we calculate the spin-averaged differential cross section for the process $e^{+} e^{-} \\rightarrow$ $\\phi^{*} \\phi$. The scattering amplitude is given by\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}=(-\\mathrm{i} e)^{2} \\bar{v}(k_{2}) \\gamma^{\\mu} u(k_{1}) \\frac{-\\mathrm{i}}{s}(p_{1}-p_{2})_{\\mu} . \\tag{9.3}\n\\end{equation*}\n\n\nThen the spin-averaged and squared amplitude is\n\n\\frac{1}{4} \\sum_{\\text {spins }}|\\mathcal{M}|^{2}=\\frac{e^{4}}{4 s^{2}} \\operatorname{tr}[(\\not p_{1}-\\not p_{2}) \\not k_{1}(\\not p_{1}-\\not p_{2}) \\not k_{2}]\n\n\n\\begin{equation*}\n=\\frac{e^{4}}{4 s^{2}}[8(k_{1} \\cdot p_{1}-k_{1} \\cdot p_{2})(k_{2} \\cdot p_{1}-k_{2} \\cdot p_{2})-4(k_{1} \\cdot k_{2})(p_{1}-p_{2})^{2}] . \\tag{9.4}\n\\end{equation*}\n\n\nWe may parameterize the momenta as\n\n\\begin{array}{ll}\nk_{1}=(E, 0,0, E), & p_{1}=(E, p \\sin \\theta, 0, p \\cos \\theta), \\\\\nk_{2}=(E, 0,0,-E), & p_{2}=(E,-p \\sin \\theta, 0,-p \\cos \\theta),\n\\end{array}\n\nwith $p=\\sqrt{E^{2}-m^{2}}$. Then we have\n\n\\begin{equation*}\n\\frac{1}{4} \\sum_{\\text {spins }}|\\mathcal{M}|^{2}=\\frac{e^{4} p^{2}}{2 E^{2}} \\sin ^{2} \\theta \\tag{9.5}\n\\end{equation*}\n\n\nThus the differential cross section is:\n\n\\begin{equation*}\n(\\frac{\\mathrm{d} \\sigma}{\\mathrm{~d} \\Omega})_{\\mathrm{CM}}=\\frac{1}{2(2 E)^{2}} \\frac{p}{8(2 \\pi)^{2} E}(\\frac{1}{4} \\sum|\\mathcal{M}|^{2})=\\frac{\\alpha^{2}}{8 s}(1-\\frac{m^{2}}{E^{2}})^{3 / 2} \\sin ^{2} \\theta \\tag{9.6}\n\\end{equation*}", + "final_answer": [ + "(\\frac{d\\sigma}{d\\Omega})_{CM} = \\frac{\\alpha^2}{8s} (1 - \\frac{m^2}{E^2})^{3/2} \\sin^2{\\theta}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$(\\frac{d\\sigma}{d\\Omega})_{\\mathrm{CM}}$": "Differential cross section in the center-of-mass frame.", + "$\\alpha$": "Fine-structure constant, $\\alpha = e^2 / (4\\pi)$.", + "$s$": "Mandelstam variable, square of the total energy in the center-of-mass frame.", + "$m$": "Mass of the scalar particle.", + "$E$": "Energy of each incoming/outgoing particle in the center-of-mass frame.", + "$\\theta$": "Scattering angle between the incoming electron and outgoing scalar particle in the center-of-mass frame." + } + }, + { + "id": 59, + "context": "We discussed the effective potential for an $O(N)$-symmetric $\\phi^{4}$ theory in four dimensions. We computed the perturbative corrections to this effective potential, and used the renormalization group to clarify the behavior of the potential for small values of the scalar field mass. After all this work, however, we found that the qualitative dependence of the theory on the mass parameter was unchanged by perturbative corrections. The theory still possessed a second-order phase transition as a function of the mass. The loop corrections affected this picture only in providing some logarithmic corrections to the scaling behavior near the phase transition.\n\nHowever, loop corrections are not always so innocuous. For some systems, they can change the structure of the phase transition qualitatively. This Final Project treats the simplest example of such a system, the Coleman-Weinberg model. The analysis of this model draws on a broad variety of topics discussed in Part II; it provides a quite nontrivial application of the effective potential formalism and the use of the renormalization group equation. The phenomenon displayed in this exercise reappears in many contexts, from displacive phase transitions in solids to the thermodynamics of the early universe.\n\nThis problem makes use of material in starred sections of the book Peskin \\& Schroeder, in particular, Sections 11.3, 11.4, and 13.2. Parts (a) and (e), however, depend only on the unstarred material of Part II. We recommend part (e) as excellent practice in the computation of renormalization group functions.\n\nThe Coleman-Weinberg model is the quantum electrodynamics of a scalar field in four dimensions, considered for small values of the scalar field mass. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4}(F_{\\mu \\nu})^{2}+(D_{\\mu} \\phi)^{\\dagger} D^{\\mu} \\phi-m^{2} \\phi^{\\dagger} \\phi-\\frac{\\lambda}{6}(\\phi^{\\dagger} \\phi)^{2},$$\n\nwhere $\\phi(x)$ is a complex-valued scalar field and $D_{\\mu} \\phi=(\\partial_{\\mu}+i e A_{\\mu}) \\phi$.", + "question": "Working in Landau gauge ( $\\partial^{\\mu} A_{\\mu}=0$ ), compute the one-loop correction to the effective potential $V(\\phi_{\\mathrm{cl}})$. Show that it is renormalized by counterterms for $m^{2}$ and $\\lambda$. Renormalize by minimal subtraction, introducing a renormalization scale $M$.", + "answer": "Now we calculate the 1-loop effective potential of the model. We know that 1-loop correction of the effective Lagrangian is given by,\n\\begin{equation*}\n\\Delta \\mathcal{L}=\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[-\\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0}+\\delta \\mathcal{L} \\tag{13.26}\n\\end{equation*}\nwhere $\\varphi$ is the fluctuating fields and $\\delta \\mathcal{L}$ denotes counterterms.\nLet the background value of the complex scalar be $\\phi_{\\mathrm{cl}}$. By the assumption of Poincaré symmetry, $\\phi_{\\mathrm{cl}}$ must be a constant. For the same reason, the background value of the vector field $A_{\\mu}$ must vanish. In addition, we can set $\\phi_{\\mathrm{cl}}$ to be real without loss of generality. Then we have,\n$$\\phi(x)=\\phi_{\\mathrm{cl}}+\\varphi_{1}(x)+\\mathrm{i} \\varphi_{2}(x),$$\nwhere $\\varphi_{1}(x), \\varphi_{2}(x)$, together with $A_{\\mu}(x)$, now serve as fluctuating fields. Expanding the Lagrangian around the background fields and keeping terms quadratic in fluctuating fields only, we get,\n\\begin{align*}\n\\mathcal{L}= & -\\frac{1}{2} F_{\\mu \\nu} F^{\\mu \\nu}+|(\\partial_{\\mu}+\\mathrm{i} e A_{\\mu})(\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2})|^{2} \\\\\n& -m^{2}|\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2}|^{2}-\\frac{\\lambda}{6}|\\phi_{\\mathrm{cl}}+\\varphi_{1}+\\mathrm{i} \\varphi_{2}|^{4} \\\\\n= & \\frac{1}{2} A_{\\mu}[g^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})-\\partial^{\\mu} \\partial^{\\nu}] A_{\\nu}+\\frac{1}{2} \\varphi_{1}(-\\partial^{2}-m^{2}-\\lambda \\phi_{\\mathrm{cl}}^{2}) \\varphi_{1} \\\\\n& +\\frac{1}{2} \\varphi_{2}(-\\partial^{2}-m^{2}-\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}) \\varphi_{2}-2 e \\phi_{\\mathrm{cl}} A_{\\mu} \\partial^{\\mu} \\varphi_{2}+\\cdots, \\tag{13.27}\n\\end{align*}\nwhere \". . .\" denotes terms other than being quadratic in fluctuating fields. Now we impose the Landau gauge condition $\\partial_{\\mu} A^{\\mu}=0$ to the Lagrangian, which removes the off-diagonal term $-2 e \\phi_{\\mathrm{cl}} A_{\\mu} \\partial^{\\mu} \\varphi_{2}$. Then, according to (13.26), the 1-loop effective Lagrangian can be evaluated as,\n\\begin{align*}\n\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[-\\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0}= & \\frac{\\mathrm{i}}{2}[\\log \\operatorname{det}(-\\eta^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})+\\partial^{\\mu} \\partial^{\\nu}) \\\\\n& +\\log \\operatorname{det}(\\partial^{2}+m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})+\\log \\operatorname{det}(\\partial^{2}+m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})] \\\\\n= & \\frac{\\mathrm{i}}{2} \\frac{\\mathrm{~d}^{d} k}{(2 \\pi)^{d}}[\\operatorname{tr} \\log (-k^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{3} .\n\\end{align*} \n\nIn the second equality we use the following identity,\n\\begin{equation*}\n\\operatorname{det}(\\lambda I+A B)=\\lambda^{n-1}(\\lambda+B A) \\tag{13.29}\n\\end{equation*}\nwhere $A$ and $B$ are matrices of $n \\times 1$ and $1 \\times n$, respectively, $\\lambda$ is an arbitrary complex number and $I$ is the $n \\times n$ identity matrix. In our case, this gives,\n\\begin{equation*}\n\\operatorname{det}(-\\eta^{\\mu \\nu}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})+\\partial^{\\mu} \\partial^{\\nu})=-2 e^{2} \\phi_{\\mathrm{cl}}^{2}(\\partial^{2}+2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{3} \\tag{13.30}\n\\end{equation*}\n\nThen the second equality follows up to an irrelevant constant term. The third equality makes use of the trick in (11.72) of P\\&S. Note that for the vector field in $d$ dimensions, the trace corresponds to $d-1$ degrees of freedom, leading to a modification of the finite part constant relative to the scalar case. Then, for $d=4-\\epsilon$ and $\\epsilon \\rightarrow 0$, we have,\n\\begin{align*}\n\\frac{\\mathrm{i}}{2} \\log \\operatorname{det}[-\\frac{\\delta^{2} \\mathcal{L}}{\\delta \\varphi \\delta \\varphi}]_{\\varphi=0} & =\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (2 e^{2} \\phi_{\\mathrm{cl}}^{2})) \\\\\n& +(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})) \\\\\n& +(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\Delta-\\log (m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}))], \\tag{13.31}\n\\end{align*}\nwhere we define $\\Delta \\equiv \\frac{2}{\\epsilon}-\\gamma+\\log 4 \\pi+\\frac{3}{2}$ for brevity.\nNow, with $\\overline{M S}$ scheme, we can determine the counterterms in (13.26) to be\n\\begin{equation*}\n\\delta \\mathcal{L}=\\frac{-1}{4(4 \\pi)^{2}}[\\frac{2}{\\epsilon}-\\gamma+\\log 4 \\pi-\\log M^{2}](3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}+(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}) . \\tag{13.32}\n\\end{equation*}\nwhere $M$ is the renormalization scale. Now the effective potential follows directly from (13.26), (13.31) and (13.32),\n\\begin{align*}\n& V_{\\mathrm{eff}}[\\phi_{\\mathrm{cl}}]=m^{2} \\phi_{\\mathrm{cl}}^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}-\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{5}{6}) \\\\\n& \\quad+(m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})] . \\tag{13.33}\n\\end{align*}", + "final_answer": [ + "V_{\\mathrm{eff}}[\\phi_{\\mathrm{cl}}]= m^{2} \\phi_{\\mathrm{cl}}^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}-\\frac{1}{4(4 \\pi)^{2}}[3(2 e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{5}{6}) + (m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\lambda \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{3} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})] \\quad \\text{or} \\quad V_{\\mathrm{eff}}[\\phi_{\\mathrm{cl}}]= \\frac{1}{2} m^{2} \\phi_{\\mathrm{cl}}^{2}+\\frac{\\lambda}{24} \\phi_{\\mathrm{cl}}^{4}-\\frac{1}{4(4 \\pi)^{2}}[3(e^{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{e^{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{5}{6}) + (m^{2}+\\frac{\\lambda}{2} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{2} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})+(m^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{2})^{2}(\\log \\frac{M^{2}}{m^{2}+\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{2}}+\\frac{3}{2})]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$V_{\\mathrm{eff}}$": "One-loop effective potential.", + "$\\phi_{\\mathrm{cl}}$": "Classical background value of the scalar field.", + "$m$": "Scalar field mass parameter.", + "$\\lambda$": "Quartic coupling constant for the scalar field self-interaction.", + "$\\pi$": "Mathematical constant pi.", + "$e$": "Electric charge, the coupling constant for quantum electrodynamics.", + "$M$": "Renormalization scale introduced in the minimal subtraction scheme." + } + }, + { + "id": 60, + "context": "We discussed the effective potential for an $O(N)$-symmetric $\\phi^{4}$ theory in four dimensions. We computed the perturbative corrections to this effective potential, and used the renormalization group to clarify the behavior of the potential for small values of the scalar field mass. After all this work, however, we found that the qualitative dependence of the theory on the mass parameter was unchanged by perturbative corrections. The theory still possessed a second-order phase transition as a function of the mass. The loop corrections affected this picture only in providing some logarithmic corrections to the scaling behavior near the phase transition.\n\nHowever, loop corrections are not always so innocuous. For some systems, they can change the structure of the phase transition qualitatively. This Final Project treats the simplest example of such a system, the ColemanWeinberg model. The analysis of this model draws on a broad variety of topics discussed in Part II; it provides a quite nontrivial application of the effective potential formalism and the use of the renormalization group equation. The phenomenon displayed in this exercise reappears in many contexts, from displacive phase transitions in solids to the thermodynamics of the early universe.\n\nThis problem makes use of material in starred sections of the book Peskin \\% Schroeder, in particular, Sections 11.3, 11.4, and 13.2. Parts (a) and (e), however, depend only on the unstarred material of Part II. We recommend part (e) as excellent practice in the computation of renormalization group functions.\n\nThe Coleman-Weinberg model is the quantum electrodynamics of a scalar field in four dimensions, considered for small values of the scalar field mass. The Lagrangian is\n\n$$\\mathcal{L}=-\\frac{1}{4}(F_{\\mu \\nu})^{2}+(D_{\\mu} \\phi)^{\\dagger} D^{\\mu} \\phi-m^{2} \\phi^{\\dagger} \\phi-\\frac{\\lambda}{6}(\\phi^{\\dagger} \\phi)^{2},\n$$\nwhere $\\phi(x)$ is a complex-valued scalar field and $D_{\\mu} \\phi=(\\partial_{\\mu}+i e A_{\\mu}) \\phi$.", + "question": "Construct the renormalization-group-improved effective potential at $\\mu^{2}=$ 0 by applying the results of part (e) to the calculation of part (c). Compute $\\langle\\phi\\rangle$ and the mass of the $\\sigma$ particle as a function of $\\lambda, e^{2}, M$. Compute the ratio $m_{\\sigma} / m_{A}$ to leading order in $e^{2}$, for $\\lambda \\ll e^{2}$.", + "answer": "The effective potential obtained in (c) is not a solution to the renormalization group equation, since it is only a first order result in perturbation expansion. However, it is possible to find an effective potential as a solution to the RG equation, with the result in (c) serving as a sort of \"initial condition\". The effective potential obtained in this way is said to be RG improved.\n\nThe Callan-Symansik equation for the effective potential reads\n\n\\begin{equation*}\n(M \\frac{\\partial}{\\partial M}+\\beta_{\\lambda} \\frac{\\partial}{\\partial \\lambda}+\\beta_{e} \\frac{\\partial}{\\partial e}-\\gamma_{\\phi} \\phi_{\\mathrm{cl}} \\frac{\\partial}{\\partial \\phi_{\\mathrm{cl}}}) V_{\\mathrm{eff}}(\\phi_{\\mathrm{cl}}, \\lambda, e ; M)=0 . \\tag{13.53}\n\\end{equation*}\n\n\nThe solution to this equation is well known, that is, the dependence of the sliding energy scale $M$ is described totally by running parameters,\n\n\\begin{equation*}\nV_{\\mathrm{eff}}(\\phi_{\\mathrm{cl}}, \\lambda, e ; M)=V_{\\mathrm{eff}}(\\bar{\\phi}_{\\mathrm{cl}}(M^{\\prime}), \\bar{\\lambda}(M^{\\prime}), \\bar{e}(M^{\\prime}) ; M^{\\prime}), \\tag{13.54}\n\\end{equation*}\n\nwhere barred quantities satisfy\n\n\\begin{equation*}\nM \\frac{\\partial \\bar{\\lambda}}{\\partial M}=\\beta_{\\lambda}(\\bar{\\lambda}, \\bar{e}), \\quad M \\frac{\\partial \\bar{e}}{\\partial M}=\\beta_{e}(\\bar{\\lambda}, \\bar{e}), \\quad M \\frac{\\partial \\bar{\\phi}_{\\mathrm{cl}}}{\\partial M}=-\\gamma_{\\phi}(\\bar{\\lambda}, \\bar{e}) \\bar{\\phi}_{\\mathrm{cl}} \\tag{13.55}\n\\end{equation*}\n\n\nThe RG-improved effective potential should be such that when expanded in terms of coupling constants $\\lambda$ and $e$, it will recover the result in (c) at the given order. For simplicity here we work under the assumption that $\\lambda \\sim e^{4}$, so that all terms of higher orders of coupling constants than $\\lambda$ and $e^{4}$ can be ignored. In this case, the perturbative calculation in (c) gives\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{\\lambda}{6} \\phi_{\\mathrm{cl}}^{4}+\\frac{3 e^{4} \\phi_{\\mathrm{cl}}^{4}}{(4 \\pi)^{2}}(\\log \\frac{2 e^{2} \\phi_{\\mathrm{cl}}^{2}}{M^{2}}-\\frac{3}{2}) . \\tag{13.56}\n\\end{equation*}\n\n\nNow we claim that the RG-improved edition of this result reads\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{\\bar{\\lambda}}{6} \\bar{\\phi}_{\\mathrm{cl}}^{4}+\\frac{3 \\bar{e}^{4} \\bar{\\phi}_{\\mathrm{cl}}^{4}}{(4 \\pi)^{2}}(\\log 2 \\bar{e}^{2}-\\frac{3}{2}) . \\tag{13.57}\n\\end{equation*}\n\n\nTo see this, we firstly solve the renormalization group equations (13.55),\n\n\\begin{align*}\n\\bar{\\lambda}(M^{\\prime}) & =\\bar{e}^{4}(\\frac{\\lambda}{e^{4}}+\\frac{9}{4 \\pi^{2}} \\log \\frac{M^{\\prime}}{M}) \\tag{13.58}\\\\\n\\bar{e}^{2}(M^{\\prime}) & =\\frac{e^{2}}{1-(e^{2} / 24 \\pi^{2}) \\log (M^{\\prime} / M)} \\tag{13.59}\\\\\n\\bar{\\phi}_{\\mathrm{cl}}(M^{\\prime}) & =\\phi_{\\mathrm{cl}}(\\frac{M^{\\prime}}{M})^{2 e^{2} /(4 \\pi)^{2}} \\tag{13.60}\n\\end{align*}\n\nwhere the unbarred quantities $\\lambda, e$ and $\\phi_{\\mathrm{cl}}$ are evaluated at scale $M$. Now we substitute these results back into the RG-improved effective potential (13.57) and expand in terms of coupling constants. Then it is straightforward to see that the result recovers (13.56). To see the spontaneous symmetry breaking still occurs, we note that the running coupling $\\bar{\\lambda}(M^{\\prime})$ flows to negative value rapidly for small $M^{\\prime}=\\phi_{\\mathrm{cl}}$, while $\\bar{e}(M^{\\prime})$ changes mildly along the $\\phi_{\\mathrm{cl}}$ scale, as can be seen directly from Figure 13.2. Therefore the the coefficient before $\\phi_{\\mathrm{cl}}^{4}$ is negative for small $\\phi_{\\mathrm{cl}}$ and positive for large $\\phi_{\\mathrm{cl}}$. As a consequence, the minimum of this effective potential should be away from $\\phi_{\\mathrm{cl}}=0$, namely the $U(1)$ symmetry is spontaneously broken.\n\nTo find the scalar mass $m_{\\sigma}$ in this case (with $\\mu=0$ ), we calculate the second derivative of the effective potential $V_{\\text {eff }}$ with respect to $\\phi_{\\mathrm{cl}}$. Since the renormalization scale $M$ can be arbitrarily chosen, we set it to be $M^{2}=2 e^{2}\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle$ to simplify the calculation. Then the vanishing of the first derivative of $V_{\\mathrm{eff}}$ at $\\phi_{\\mathrm{cl}}=\\langle\\phi_{\\mathrm{cl}}\\rangle$ implies that $\\lambda=9 e^{4} / 8 \\pi^{2}$. Insert this back to $V_{\\text {eff }}$ in (13.56), we find that\n\n\\begin{equation*}\nV_{\\mathrm{eff}}=\\frac{3 e^{4} \\phi_{\\mathrm{cl}}^{4}}{16 \\pi^{2}}(\\log \\frac{\\phi_{\\mathrm{cl}}^{2}}{\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle}-\\frac{1}{2}) . \\tag{13.61}\n\\end{equation*}\n\n\nThen, taking the second derivative of this expression with respect to $\\phi_{\\mathrm{cl}}$, we get the scalar mass $m_{\\sigma}^{2}=3 e^{4}\\langle\\phi_{\\mathrm{cl}}^{2}\\rangle / 4 \\pi^{2}=3 e^{4} v^{2} / 8 \\pi^{2}$. Recall that the gauge boson's mass $m_{A}$ is given by $m_{A}=e^{2} v^{2}$ at the leading order, thus we conclude that $m_{\\sigma}^{2} / m_{A}^{2}=3 e^{2} / 8 \\pi^{2}$ at the leading order in $e^{2}$.", + "final_answer": [ + "m_{\\sigma}/m_A = \\frac{\\sqrt{6}e}{4\\pi}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$m_{\\sigma}$": "Mass of the scalar $\\sigma$ particle.", + "$m_A$": "Mass of the gauge boson.", + "$e$": "Electric charge coupling constant.", + "$\\pi$": "Mathematical constant pi." + } + }, + { + "id": 61, + "context": "Deep inelastic scattering from a photon. Consider the problem of deepinelastic scattering of an electron from a photon. This process can actually be measured by analyzing the reaction $e^{+} e^{-} \\rightarrow e^{+} e^{-}+X$ in the regime where the positron goes forward, with emission of a collinear photon, which then has a hard reaction with the electron. Let us analyze this process to leading order in QED and to leading-log order in QCD. To predict the photon structure functions, it is reasonable to integrate the renormalization group equations with the initial condition that the parton distribution for photons in the photon is $\\delta(x-1)$ at $Q^{2}=(\\frac{1}{2} \\mathrm{GeV})^{2}$. Take $\\Lambda=150 \\mathrm{MeV}$. Assume for simplicity that there are four flavors of quarks, $u, d, c$, and $s$, with charges $2 / 3$, $-1 / 3,2 / 3,-1 / 3$, respectively, and that it is always possible to ignore the masses of these quarks.", + "question": "Use the Altarelli-Parisi equations to compute the parton distributions for quarks and antiquarks in the photon, to leading order in QED and to zeroth order in QCD.", + "answer": "The A-P equation for parton distributions in the photon can be easily written down by using the QED splitting functions listed in (17.121) of Peskin \\& Schroeder. Taking account of quarks' electric charge properly, we have,\n\n\\begin{align*}\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{q}(x, Q)= \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{e \\leftarrow e}(z) f_{q}(\\frac{x}{z}, Q)+P_{e \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.68}\\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{\\bar{q}}(x, Q)= \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{e \\leftarrow e}(z) f_{\\bar{q}}(\\frac{x}{z}, Q)+P_{e \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.69}\\\\\n& \\frac{\\mathrm{d}}{\\mathrm{~d} \\log Q} f_{\\gamma}(x, Q)= \\sum_{q} \\frac{3 Q_{q}^{2} \\alpha}{\\pi} \\int_{x}^{1} \\frac{\\mathrm{d} z}{z}{P_{\\gamma \\leftarrow e}(z)[f_{q}(\\frac{x}{z}, Q)+f_{\\bar{q}}(\\frac{x}{z}, Q)] \\\\\n&\\quad+P_{\\gamma \\leftarrow \\gamma}(z) f_{\\gamma}(\\frac{x}{z}, Q)}, \\tag{18.70}\n\\end{align*}\n\nwhere the splitting functions are\n\n\\begin{align*}\n& P_{e \\leftarrow e}(z)=\\frac{1+z^{2}}{(1-z)_{+}}+\\frac{3}{2} \\delta(1-z), \\tag{18.71}\\\\\n& P_{\\gamma \\leftarrow e}(z)=\\frac{1+(1-z)^{2}}{z}, \\tag{18.72}\\\\\n& P_{e \\leftarrow \\gamma}(z)=z^{2}+(1-z)^{2} \\tag{18.73}\\\\\n& P_{\\gamma \\leftarrow \\gamma}(z)=-\\frac{2}{3} \\delta(1-z) . \\tag{18.74}\n\\end{align*}\n\n\nWe take $q=u, d, c, s$, and $Q_{u, c}=+2 / 3, Q_{d, s}=-1 / 3$. The factor 3 in the A-P equations above takes account of 3 colors. Since no more leptons appear in final states other than original $e^{+} e^{-}$, they are not included in the photon structure. With the initial condition\n$f_{\\gamma}(x, Q_{0})=\\delta(1-x)$ and $f_{q, \\bar{q}}(x, Q_{0})=0$ where $Q_{0}=0.5 \\mathrm{GeV}$, these distribution functions can be solved from the equations above to the first order in $\\alpha$, to be\n\n\\begin{align*}\n& f_{q}(x, Q)=f_{\\bar{q}}(x, Q)=\\frac{3 Q_{q}^{2} \\alpha}{2 \\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}[x^{2}+(1-x)^{2}] \\tag{18.75}\\\\\n& f_{\\gamma}(x, Q)=(1-\\sum_{q} \\frac{Q_{q}^{2} \\alpha}{\\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}) \\delta(1-x) \\tag{18.76}\n\\end{align*}", + "final_answer": [ + "f_{q}(x, Q)=f_{\\bar{q}}(x, Q)=\\frac{3 Q_{q}^{2} \\alpha}{2 \\pi} \\log \\frac{Q^{2}}{Q_{0}^{2}}[x^{2}+(1-x)^{2}]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$f_q(x, Q)$": "Parton distribution function for a quark of flavor $q$ in the photon.", + "$f_{\\\\bar{q}}(x, Q)$": "Parton distribution function for an antiquark of flavor $\\\\bar{q}$ in the photon.", + "$Q_q$": "Electric charge of a quark of flavor $q$.", + "$\\\\alpha$": "Fine-structure constant (QED coupling constant).", + "$Q$": "Momentum transfer scale.", + "$Q_0$": "Initial momentum transfer scale, $Q_0=0.5 \\\\mathrm{GeV}$.", + "$x$": "Bjorken scaling variable, representing the momentum fraction of the parton." + } + }, + { + "id": 62, + "context": "Neutral-current deep inelastic scattering.", + "question": "In Eq. (17.35) of Peskin \\& Schroeder, we wrote formulae for neutrino and antineutrino deep inelastic scattering with $W^{\\pm}$ exchange. Neutrinos and antineutrinos can also scatter by exchanging a $Z^{0}$. This process, which leads to a hadronic jet but no observable outgoing lepton, is called the neutral current reaction. Compute $\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\nu p \\rightarrow \\nu X)$ for neutral current deep inelastic scattering of neutrinos from protons, accounting for scattering from $u$ and $d$ quarks and antiquarks.", + "answer": "In this problem we study the neutral-current deep inelastic scattering. The process is mediated by $Z^{0}$ boson. Assuming $m_{Z}$ is much larger than the energy scale of the scattering process, we can write down the corresponding effective operators, from the neutral-current Feynman rules in electroweak theory,\n\\begin{align*}\n\\Delta \\mathcal{L}=\\frac{g^{2}}{4 m_{W}^{2}}(\\bar{\\nu} \\gamma^{\\mu}) P_{L} \\nu & {[\\bar{u} \\gamma_{\\mu}((1-\\frac{4}{3} s_{w}^{2}) P_{L}-\\frac{4}{3} s_{w}^{2} P_{R}) u} \\\\\n& +\\bar{d} \\gamma_{\\mu}((1-\\frac{2}{3} s_{w}^{2}) P_{L}-\\frac{2}{3} s_{w}^{2} P_{R}) d]+ \\text { h.c. } \\tag{20.30}\n\\end{align*}\nwhere $P_{L}=(1-\\gamma^{5}) / 2$ and $P_{R}=(1+\\gamma^{5}) / 2$ are left- and right-handed projectors, respectively. Compare the effective operator with the charged-operator in (17.31) of Peskin \\& Schroeder, we can write down directly the differential cross section for neutrino scattering by modifying (17.35) in Peskin \\& Schroeder properly, as\n\\begin{align*}\n\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{~d} x \\mathrm{~d} y}(\\nu p \\rightarrow \\nu X)=\\frac{G_{F}^{2} s x}{4 \\pi} & \\{[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y)^{2}] f_{u}(x) \\\\ + & {[(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y)^{2}] f_{d}(x) } \\\\ + & [\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{\\bar{u}}(x) \\\\ + & [\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{\\bar{d}}(x)\\}. \\tag{20.31}\n\\end{align*}", + "final_answer": [ + "\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\nu p \\rightarrow \\nu X)=\\frac{G_{F}^{2} s x}{4 \\pi} {[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y)^{2}] f_{u}(x) + [(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y)^{2}] f_{d}(x) + [\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{\\bar{u}}(x) + [\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{\\bar{d}}(x)}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\sigma$": "Cross-section for deep inelastic scattering.", + "$x$": "Bjorken scaling variable, representing the fraction of the proton's momentum carried by the struck quark.", + "$y$": "Inelasticity variable, representing the fraction of the lepton's energy transferred to the hadron system.", + "$\\nu$": "Neutrino.", + "$p$": "Proton.", + "$X$": "Hadronic jet, representing the unspecified final hadronic state.", + "$G_{F}$": "Fermi coupling constant.", + "$s$": "Mandelstam variable, representing the square of the center-of-mass energy.", + "$\\pi$": "Mathematical constant pi.", + "$s_{w}$": "Sine of the Weinberg angle (or weak mixing angle).", + "$f_{u}(x)$": "Parton distribution function for up quarks, dependent on Bjorken $x$.", + "$f_{d}(x)$": "Parton distribution function for down quarks, dependent on Bjorken $x$.", + "$f_{\\bar{u}}(x)$": "Parton distribution function for anti-up quarks, dependent on Bjorken $x$.", + "$f_{\\bar{d}}(x)$": "Parton distribution function for anti-down quarks, dependent on Bjorken $x$." + } + }, + { + "id": 63, + "context": "Neutral-current deep inelastic scattering.", + "question": "In Eq. (17.35) of Peskin \\& Schroeder, we wrote formulae for neutrino and antineutrino deep inelastic scattering with $W^{ \\pm}$exchange. Neutrinos and antineutrinos can also scatter by exchanging a $Z^{0}$. This process, which leads to a hadronic jet but no observable outgoing lepton, is called the neutral current reaction. Compute $\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X)$ for neutral current deep inelastic scattering of antineutrinos from protons, accounting for scattering from $u$ and $d$ quarks and antiquarks.", + "answer": "In this problem we study the neutral-current deep inelastic scattering. The process is mediated by $Z^{0}$ boson. Assuming $m_{Z}$ is much larger than the energy scale of the scattering process, we can write down the corresponding effective operators, from the neutral-current Feynman rules in electroweak theory,\n\\begin{align*}\n\\Delta \\mathcal{L}=\\frac{g^{2}}{4 m_{W}^{2}}(\\bar{\\nu} \\gamma^{\\mu}) P_{L} \\nu & {[\\bar{u} \\gamma_{\\mu}((1-\\frac{4}{3} s_{w}^{2}) P_{L}-\\frac{4}{3} s_{w}^{2} P_{R}) u} \\\\\n& +\\bar{d} \\gamma_{\\mu}((1-\\frac{2}{3} s_{w}^{2}) P_{L}-\\frac{2}{3} s_{w}^{2} P_{R}) d]+ \\text { h.c. } \\tag{20.30}\n\\end{align*}\nwhere $P_{L}=(1-\\gamma^{5}) / 2$ and $P_{R}=(1+\\gamma^{5}) / 2$ are left- and right-handed projectors, respectively. Compare the effective operator with the charged-operator in (17.31) of Peskin \\& Schroeder, we can write down directly the differential cross section for antineutrino scattering by modifying (17.35) in Peskin \\& Schroeder properly, as\n\\begin{align*}\n\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{~d} x \\mathrm{~d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X)=\\frac{G_{F}^{2} s x}{4 \\pi} & \\{[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{u}(x) \\\\\n& +[\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{d}(x) \\\\\n& +[(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y)^{2}] f_{\\bar{u}}(x) \\\\\n& +[(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y)^{2}] f_{\\bar{d}}(x)\\} . \\tag{20.32}\n\\end{align*}", + "final_answer": [ + "\\frac{\\mathrm{d}^{2} \\sigma}{\\mathrm{d} x \\mathrm{d} y}(\\bar{\\nu} p \\rightarrow \\bar{\\nu} X) = \\frac{G_{F}^{2} s x}{4 \\pi} \\{[\\frac{16}{9} s_{w}^{4}+(1-\\frac{4}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{u}(x) + [\\frac{4}{9} s_{w}^{4}+(1-\\frac{2}{3} s_{w}^{2})^{2}(1-y)^{2}] f_{d}(x) + [(1-\\frac{4}{3} s_{w}^{2})^{2}+\\frac{16}{9} s_{w}^{4}(1-y)^{2}] f_{\\bar{u}}(x) + [(1-\\frac{2}{3} s_{w}^{2})^{2}+\\frac{4}{9} s_{w}^{4}(1-y)^{2}] f_{\\bar{d}}(x) \\}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\sigma$": "Cross section for deep inelastic scattering.", + "$x$": "Bjorken scaling variable, representing the fraction of the proton's momentum carried by the struck quark.", + "$y$": "Bjorken scaling variable, representing the inelasticity of the scattering.", + "$\\bar{\\nu}$": "Antineutrino.", + "$p$": "Proton.", + "$X$": "Unobserved final state, typically a hadronic jet.", + "$G_F$": "Fermi coupling constant.", + "$s$": "Mandelstam variable, representing the square of the center-of-mass energy.", + "$\\pi$": "Mathematical constant pi.", + "$s_w$": "Sine of the Weinberg angle (or weak mixing angle).", + "$f_u$": "Parton distribution function for up quarks, dependent on $x$.", + "$f_d$": "Parton distribution function for down quarks, dependent on $x$.", + "$f_{\\bar{u}}$": "Parton distribution function for anti-up quarks, dependent on $x$.", + "$f_{\\bar{d}}$": "Parton distribution function for anti-down quarks, dependent on $x$." + } + }, + { + "id": 64, + "context": "A model with two Higgs fields.", + "question": "Find the couplings of the physical charged Higgs boson $\\phi^{+}$ to the quark mass eigenstates, given the Lagrangian $$\\mathcal{L}_{m} = -\\lambda_{d}^{ij} \\bar{Q}_{L}^{i} \\cdot \\phi_{1} d_{R}^{j} - \\lambda_{u}^{ij} \\epsilon^{ab} \\bar{Q}_{La}^{i} \\phi_{2b}^{\\dagger} u_{R}^{j} + \\text{h.c.}$$", + "answer": "Assuming that the Yukawa interactions between quarks and scalars take the following form:\n\n\\begin{align}\n\\mathcal{L}_m &= -\\left(\\bar{u}_L \\quad \\bar{d}_L\\right)\\left[\\lambda_d\\left(\\frac{\\pi_1^+}{\\sqrt{2}v_1}\\right)d_R + \\lambda_u\\left(\\frac{\\frac{1}{\\sqrt{2}}v_2}{\\pi^-}\\right)u_R\\right] + \\text{h.c.}, \\tag{20.44}\n\\end{align}\nwhere we have suppressed flavor indices and neglected neutral scalar components. We focus on charged components only. Using Peskin \\& Schroeder's notation, we perform the replacements \\( u_L \\to U_u u_L \\), \\( d_L \\to U_d d_L \\), \\( u_R \\to W_u u_R \\), and \\( d_R \\to W_d d_R \\). With the diagonalization \\(\\lambda_d = U_d D_d W_d^\\dagger\\) and \\(\\lambda_u = U_u D_u W_u^\\dagger\\) (where \\(D_d\\) and \\(D_u\\) are diagonal matrices), we derive:\n\n\\begin{align}\n\\mathcal{L}_m &= -\\frac{1}{\\sqrt{2}}\\left(v_1\\bar{d}_L D_d d_R + v_2\\bar{u}_L D_u u_R\\right) \\nonumber \\\\\n&\\quad - \\bar{u}_L V_{\\text{CKM}} D_d d_R \\pi_1^+ + \\bar{d}_L V_{\\text{CKM}}^\\dagger D_u u_R \\pi_2^- + \\text{h.c.}. \\tag{20.45}\n\\end{align}\n\nFrom the first term, the diagonal quark mass matrices are given by \\( m_u = \\frac{v_1}{\\sqrt{2}}D_u \\) and \\( m_d = \\frac{v_2}{\\sqrt{2}}D_d \\). Defining \\( v = \\sqrt{v_1^2 + v_2^2} \\), and using the relations \\( \\pi_1^+ = -\\phi^+ \\sin\\beta + \\cdots \\), \\( \\pi_2^+ = \\phi^+ \\cos\\beta + \\cdots \\), the Yukawa interactions with charged bosons become:\n\n\\begin{align}\n\\mathcal{L}_m &\\Rightarrow -\\frac{\\sqrt{2}}{v_1} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\pi_1^+ + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\pi_2^- \\right) + \\text{h.c.} \\nonumber \\\\\n&\\Rightarrow \\frac{\\sqrt{2}}{v} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\phi^+ \\tan\\beta + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\phi^- \\cot\\beta \\right) + \\text{h.c.}. \\tag{20.46}\n\\end{align}", + "final_answer": [ + "\\frac{\\sqrt{2}}{v} \\left( \\bar{u}_L V_{\\text{CKM}} m_d d_R \\phi^+ \\tan\\beta + \\bar{d}_L V_{\\text{CKM}}^\\dagger m_u u_R \\phi^- \\cot\\beta \\right) + \\text{h.c.}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$v$": "Total vacuum expectation value, $v = \\sqrt{v_1^2 + v_2^2}$", + "$\\bar{u}_L$": "Left-handed up-type quark (anti-quark)", + "$V_{\\text{CKM}}$": "Cabibbo-Kobayashi-Maskawa (CKM) matrix", + "$m_d$": "Mass matrix for down-type quarks, $m_d = \\frac{v_2}{\\sqrt{2}}D_d$", + "$d_R$": "Right-handed down-type quark", + "$\\phi^+$": "Physical charged Higgs boson (positive charge)", + "$\\tan\\beta$": "Tangent of the mixing angle $\\beta$", + "$\\bar{d}_L$": "Left-handed down-type quark (anti-quark)", + "$V_{\\text{CKM}}^\\dagger$": "Hermitian conjugate of the Cabibbo-Kobayashi-Maskawa (CKM) matrix", + "$m_u$": "Mass matrix for up-type quarks, $m_u = \\frac{v_1}{\\sqrt{2}}D_u$", + "$u_R$": "Right-handed up-type quark", + "$\\phi^-$": "Physical charged Higgs boson (negative charge)", + "$\\cot\\beta$": "Cotangent of the mixing angle $\\beta$", + "h.c.": "Hermitian conjugate" + } + }, + { + "id": 65, + "context": "We discussed the mystery of the origin of spontaneous symmetry breaking in the weak interactions. The simplest hypothesis is that the $S U(2) \\times U(1)$ gauge symmetry of the weak interactions is broken by the expectation value of a two-component scalar field $\\phi$. However, since we have almost no experimental information about the mechanism of this symmetry breaking, many other possibilities can be suggested.\n\nEventually, this problem should be resolved by experimental observation of the particles associated with the symmetry breaking. To form incisive experimental tests, we should compute the properties expected for these particles. We saw in Section 20.2 of Peskin \\& Schroeder that, if the symmetry is indeed broken by a single scalar field $\\phi$, the symmetry-breaking sector contributes only one new particle, a scalar $h^{0}$ called the Higgs boson. The mass $m_{h}$ of this particle is unknown. However, the couplings of the $h^{0}$ to known fermions and bosons are completely determined by the masses of those particles and the weak interaction coupling constants. Thus, it is possible to compute the amplitudes for production and decay of the $h^{0}$ in some detail. More complicated models of $S U(2) \\times U(1)$ symmetry breaking typically contain one or more particles that share some properties with the $h^{0}$. Thus, this study is a useful starting point for the more general analysis of experimental tests of these models.\n\nIn this Final Project you will compute the amplitudes for the Higgs boson $h^{0}$ to decay to pairs of quarks, leptons, and gauge bosons. The computations beautifully illustrate the working of perturbation theory for non-Abelian gauge fields. Those decays of the Higgs boson that involve quarks and gluons bring in aspects of QCD. Thus, this exercise reviews all of the important technical methods of Part III. Except for a question raised at the end of part (a), the problem relies only on material from unstarred sections of Part III. The material in Chapter 20 plays an essential role. Material from Chapter 21 enters the analysis only in parts (b) and (f), and the other parts of the problem (except for the final summary in part (h)) do not rely on these. If you have studied Section 19.5, you will have some additional insight into the results of parts (c) and (f), but this insight is not necessary to work the problem.\n\nConsider, then, the minimal form of the Glashow-Weinberg-Salam electroweak theory with one Higgs scalar field $\\phi$. The physical Higgs boson $h^{0}$ of\nthis theory was discussed in Section 20.2, and we listed there the couplings of this particle to quarks, leptons, and gauge bosons. You can now use that information to compute the amplitudes for the various possible decays of the $h^{0}$ as a function of its mass $m_{h}$. You will discover that the decay pattern has a complicated dependence on the mass of the Higgs boson, with different decay modes dominating in different mass ranges. The dependences of the various decay rates on $m_{h}$ illustrate many aspects of the physics of gauge theories that we have discussed in Part III.\n\nIn working this exercise, you should consider $m_{h}$ as a free parameter. For the other parameters of weak-interaction theory, you might use the following values: $m_{b}=5 \\mathrm{GeV}, m_{t}=175 \\mathrm{GeV}, m_{W}=80 \\mathrm{GeV}, m_{Z}=91 \\mathrm{GeV}, \\sin ^{2} \\theta_{w}$ $=0.23, \\alpha_{s}(m_{Z})=0.12$.", + "question": "Provide the tree-level decay width $\\Gamma(h^0 \\rightarrow f\\bar{f})$ for a Higgs boson $h^0$ decaying into a fermion-antifermion pair $f\\bar{f}$ (where $f$ is a quark or lepton of the Standard Model), expressed in terms of the fine-structure constant $\\alpha$, the Higgs mass $m_h$, the fermion mass $m_f$, the W boson mass $m_W$, the weak mixing angle $\\theta_w$, and the color factor $N_c(f)$ (1 for leptons, 3 for quarks).", + "answer": "In this final project, we calculate partial widths of various decay channels of the standard model Higgs boson. Although a standard-model-Higgs-like boson has been found at the LHC with mass around 125 GeV , it is still instructive to treat the mass of the Higgs boson as a free parameter in the following calculation.\n\nThe main decay modes of Higgs boson include $h^{0} \\rightarrow f \\bar{f}$ with $f$ the standard model fermions, $h^{0} \\rightarrow W^{+} W^{-}, h^{0} \\rightarrow Z^{0} Z^{0}, h^{0} \\rightarrow g g$ and $h^{0} \\rightarrow \\gamma \\gamma$. The former three processes appear at the tree level, while the leading order contributions to the latter two processes are at one-loop level. We will work out the decay widths of these processes in the following.\n\nIn this problem we only consider the two-body final states. The calculation of decay width needs the integral over the phase space of the two-body final states. By momentum conservation and rotational symmetry, we can always parameterize the momenta of two final particles in CM frame to be $p_{1}=(E, 0,0, p)$ and $p_{2}=(E, 0,0,-p)$, where $E=\\frac{1}{2} m_{h}$ by energy conservation. Then the amplitude $\\mathcal{M}$ will have no angular dependence. Then the phase space integral reads\n\n\\begin{equation*}\n\\int \\mathrm{d} \\Pi_{2}|\\mathcal{M}|^{2}=\\frac{1}{4 \\pi} \\frac{p}{m_{h}}|\\mathcal{M}|^{2} . \\tag{21.50}\n\\end{equation*}\n\n\nThen the decay width is given by\n\n\\begin{equation*}\n\\Gamma=\\frac{1}{2 m_{h}} \\int \\mathrm{~d} \\Pi_{2}|\\mathcal{M}|^{2}=\\frac{1}{8 \\pi} \\frac{p}{m_{h}^{2}}|\\mathcal{M}|^{2} . \\tag{21.51}\n\\end{equation*}\n\n\nIn part (d) of this problem, we will also be dealing with the production of the Higgs boson from two-gluon initial state, thus we also write down the formula here for the cross section of the one-body final state from two identical initial particle. This time, the two ingoing particles have momenta $k_{1}=(E, 0,0, k)$ and $k_{2}=(E, 0,0,-k)$, with $E^{2}=k^{2}+m_{i}^{2}$ and $2 E=m_{f}$ where $m_{i}$ and $m_{f}$ are masses of initial particles and final particle, respectively. The final particle has momentum $p=(m_{f}, 0,0,0)$. Then, the cross section is given by\n\n\\begin{align*}\n\\sigma & =\\frac{1}{2 \\beta s} \\int \\frac{\\mathrm{~d}^{3} p}{(2 \\pi)^{3}} \\frac{1}{2 E_{p}}|\\mathcal{M}|^{2}(2 \\pi)^{4} \\delta^{(4)}(p-k_{1}-k_{2}) \\\\\n& =\\frac{1}{4 m_{f} \\beta s}|\\mathcal{M}|^{2}(2 \\pi) \\delta(2 k-m_{f})=\\frac{\\pi}{\\beta m_{f}^{2}}|\\mathcal{M}|^{2} \\delta(s-m_{f}^{2}), \\tag{21.52}\n\\end{align*}\n\nwhere $\\beta=\\sqrt{1-(4 m_{i} / m_{f})^{2}}$ is the magnitude of the velocity of the initial particle in the center-of-mass frame.\n(a) The easiest calculation of above processes is $h^{0} \\rightarrow f \\bar{f}$, where $f$ represents all quarks and charged leptons. The tree level contribution to this process involves a single Yukawa vertex only. The corresponding amplitude is given by\n\n\\begin{equation*}\n\\mathrm{i} \\mathcal{M}(h^{0} \\rightarrow f \\bar{f})=-\\frac{\\mathrm{i} m_{f}}{v} \\bar{u}^{*}(p_{1}) v(p_{2}) . \\tag{21.53}\n\\end{equation*}\n\n\nThen it is straightforward to get the squared amplitude with final spins summed to be\n\n\\begin{equation*}\n\\sum|\\mathcal{M}(h^{0} \\rightarrow f \\bar{f})|^{2}=\\frac{m_{f}^{2}}{v^{2}} \\operatorname{tr}[(\\not p_{1}+m_{f})(\\not p_{2}-m_{f})]=\\frac{2 m_{f}^{2}}{v^{2}}(m_{h}^{2}-4 m_{f}^{2}) \\tag{21.54}\n\\end{equation*}\n\n\nIn CM frame, the final states momenta can be taken to be $p_{1}=(E, 0,0, p)$ and $p_{2}=$ $(E, 0,0,-p)$, with $E=\\frac{1}{2} m_{h}$ and $p^{2}=E^{2}-m_{f}^{2}$. Then the decay width is given by\n\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow f \\bar{f})=\\frac{1}{8 \\pi} \\frac{p}{m_{h}^{2}}|\\mathcal{M}|^{2}=\\frac{m_{h} m_{f}^{2}}{8 v^{2}}(1-\\frac{4 m_{f}^{2}}{m_{h}^{2}})^{3 / 2} \\tag{21.55}\n\\end{equation*}\n\n\nThis expression can be expressed in terms of the fine structure constant $\\alpha$, the mass of $W$ boson $m_{w}$ and Weinberg angle $\\sin \\theta_{w}$, as\n\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow f \\bar{f})=\\frac{\\alpha m_{h}}{8 \\sin ^{2} \\theta_{w}} \\frac{m_{f}^{2}}{m_{W}^{2}}(1-\\frac{4 m_{f}^{2}}{m_{h}^{2}})^{3 / 2} \\tag{21.56}\n\\end{equation*}", + "final_answer": [ + "\\Gamma(h^0 \\rightarrow f\\bar{f}) = (\\frac{\\alpha m_h}{8\\sin^2\\theta_w}) \\cdot \\frac{m_f^2}{m_W^2} (1 - \\frac{4m_f^2}{m_h^2})^{3/2}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\Gamma$": "Decay width", + "$h^0$": "Higgs boson", + "$f$": "Standard Model fermion (quark or lepton)", + "$\\bar{f}$": "Antifermion", + "$\\alpha$": "Fine-structure constant", + "$m_h$": "Mass of the Higgs boson", + "$\\theta_w$": "Weak mixing angle (Weinberg angle)", + "$m_f$": "Mass of the fermion", + "$m_W$": "Mass of the W boson" + } + }, + { + "id": 66, + "context": "We discussed the mystery of the origin of spontaneous symmetry breaking in the weak interactions. The simplest hypothesis is that the $S U(2) \\times U(1)$ gauge symmetry of the weak interactions is broken by the expectation value of a two-component scalar field $\\phi$. However, since we have almost no experimental information about the mechanism of this symmetry breaking, many other possibilities can be suggested.\n\nEventually, this problem should be resolved by experimental observation of the particles associated with the symmetry breaking. To form incisive experimental tests, we should compute the properties expected for these particles. We saw in Section 20.2 that, if the symmetry is indeed broken by a single scalar field $\\phi$, the symmetry-breaking sector contributes only one new particle, a scalar $h^{0}$ called the Higgs boson. The mass $m_{h}$ of this particle is unknown. However, the couplings of the $h^{0}$ to known fermions and bosons are completely determined by the masses of those particles and the weak interaction coupling constants. Thus, it is possible to compute the amplitudes for production and decay of the $h^{0}$ in some detail. More complicated models of $S U(2) \\times U(1)$ symmetry breaking typically contain one or more particles that share some properties with the $h^{0}$. Thus, this study is a useful starting point for the more general analysis of experimental tests of these models.\n\nIn this Final Project you will compute the amplitudes for the Higgs boson $h^{0}$ to decay to pairs of quarks, leptons, and gauge bosons. The computations beautifully illustrate the working of perturbation theory for non-Abelian gauge fields. Those decays of the Higgs boson that involve quarks and gluons bring in aspects of QCD. Thus, this exercise reviews all of the important technical methods of Part III. Except for a question raised at the end of part (a), the problem relies only on material from unstarred sections of Part III. The material in Chapter 20 plays an essential role. Material from Chapter 21 enters the analysis only in parts (b) and (f), and the other parts of the problem (except for the final summary in part (h)) do not rely on these. If you have studied Section 19.5, you will have some additional insight into the results of parts (c) and (f), but this insight is not necessary to work the problem.\n\nConsider, then, the minimal form of the Glashow-Weinberg-Salam electroweak theory with one Higgs scalar field $\\phi$. The physical Higgs boson $h^{0}$ of\nthis theory was discussed in Section 20.2, and we listed there the couplings of this particle to quarks, leptons, and gauge bosons. You can now use that information to compute the amplitudes for the various possible decays of the $h^{0}$ as a function of its mass $m_{h}$. You will discover that the decay pattern has a complicated dependence on the mass of the Higgs boson, with different decay modes dominating in different mass ranges. The dependences of the various decay rates on $m_{h}$ illustrate many aspects of the physics of gauge theories that we have discussed in Part III.\n\nIn working this exercise, you should consider $m_{h}$ as a free parameter. For the other parameters of weak-interaction theory, you might use the following values: $m_{b}=5 \\mathrm{GeV}, m_{t}=175 \\mathrm{GeV}, m_{W}=80 \\mathrm{GeV}, m_{Z}=91 \\mathrm{GeV}, \\sin ^{2} \\theta_{w}$ $=0.23, \\alpha_{s}(m_{Z})=0.12$.", + "question": "If the Higgs boson mass $m_h$ is sufficiently large (specifically, if $m_h > 2m_Z$), it can also decay to $Z^{0} Z^{0}$. Compute the decay width $\\Gamma(h^0 \\rightarrow Z^0Z^0)$. As context from the original problem, if $m_h \\gg m_Z$, this decay width can be approximated by $\\Gamma(h^0 \\rightarrow Z^0Z^0) \\approx \\Gamma(h^0 \\rightarrow \\phi^3\\phi^3)$, where $\\phi^3$ is a Goldstone boson, and an explanation and verification of this approximation was requested.", + "answer": "For $h^{0} \\rightarrow Z^{0} Z^{0}$ process, the calculation is similar to the $h^0 \\rightarrow W^+W^-$ decay. However, there are two key differences: the coupling of the Higgs to the $Z$ boson involves an additional factor of $1/\\cos\\theta_w$ compared to the $W$ boson, and an additional factor $1 / 2$ is needed to account for the identical particles in the final state. Additionally, the factor of $\\pi$ cancels out when expressing the result in terms of $\\alpha$. Therefore, replacing $m_W$ with $m_Z$ and including these factors, we have\n\\begin{equation*}\n\\Gamma(h^{0} \\rightarrow Z^{0} Z^{0})=\\frac{\\alpha m_{h}^{3}}{32 m_{Z}^{2} \\sin ^{2} \\theta_{w}(1-\\sin^2\\theta_w)}(1-4 \\tau_{Z}^{-1}+12 \\tau_{Z}^{-2})(1-4 \\tau_{Z}^{-1})^{1 / 2} \\tag{21.60}\n\\end{equation*}\nwhere $\\tau_{Z} \\equiv(m_{h} / m_{Z})^{2}$.", + "final_answer": [ + "\\Gamma(h^{0} \\rightarrow Z^{0} Z^{0})=\\frac{\\alpha m_{h}^{3}}{32 m_{Z}^{2} \\sin ^{2} \\theta_{w}(1-\\sin^2\\theta_w)}(1-4 \\tau_{Z}^{-1}+12 \\tau_{Z}^{-2})(1-4 \\tau_{Z}^{-1})^{1 / 2}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\Gamma$": "Decay width, representing the probability per unit time for a particle to decay.", + "$h^0$": "Higgs boson, the single new scalar particle predicted by the minimal Glashow-Weinberg-Salam electroweak theory.", + "$Z^0$": "Z boson, a neutral gauge boson of the weak interaction.", + "$\\alpha$": "Fine-structure constant, the electromagnetic coupling constant.", + "$m_h$": "Mass of the Higgs boson.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$m_Z$": "Mass of the Z boson.", + "$\\sin^2 \\theta_w$": "Square of the sine of the Weinberg angle (or weak mixing angle), a parameter of the electroweak theory.", + "$\\tau_Z$": "Dimensionless parameter defined as $\\tau_Z = (m_h / m_Z)^2$." + } + }, + { + "id": 67, + "context": "", + "question": "Derive the commutator $[M^{\\mu \\nu}, M^{\\rho \\sigma}]$. Hints :\n- Denote $\\Lambda^{\\prime} \\mu_{\\nu} \\approx \\delta^{\\mu}{ }_{v}+\\chi^{\\mu}{ }_{v}$. Check $(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)_{\\rho \\sigma} \\approx \\delta_{\\rho \\sigma}+\\chi_{\\mu \\nu} \\Lambda^{\\mu}{ }_{\\rho} \\Lambda^{v}{ }_{\\sigma}$.\n- Denote $\\Lambda^{\\mu}{ }_{v} \\approx \\delta^{\\mu}{ }_{v}+\\omega^{\\mu}{ }_{v}$. Check $U^{-1}(\\Lambda) M^{\\mu \\nu} U(\\Lambda) \\approx M^{\\mu v}+\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]$.\n- Identify this expression with $\\wedge^{\\mu}{ }_{\\rho} \\Lambda^{\\nu}{ }_{\\sigma} M^{\\rho \\sigma}$ (when simplifying by $\\omega_{\\rho \\sigma}$, one needs to enforce the antisymmetry of the residual factors).", + "answer": "We can write\n\n\\begin{aligned}\n(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)_{\\rho \\sigma} & =\\Lambda_{\\rho}^{-1 \\mu} \\Lambda_{\\mu}^{\\prime}{ }^{\\nu} \\Lambda_{v \\sigma}=\\Lambda_{\\rho}^{\\mu}(\\delta_{\\mu}{ }^{\\nu}+\\chi_{\\mu}{ }^{\\nu}+\\mathcal{O}(\\chi^{2})) \\Lambda_{v \\sigma} \\\\\n& =\\delta_{\\rho \\sigma}+\\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{v} \\chi_{\\mu \\nu}+\\mathcal{O}(\\chi^{2})\n\\end{aligned}\n\n\nTherefore, we have\n\n\\begin{equation*}\n\\mathrm{U}(\\Lambda^{-1} \\Lambda^{\\prime} \\Lambda)=1+\\frac{i}{2} \\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{\\nu} \\chi_{\\mu \\nu} M^{\\rho \\sigma}+\\mathcal{O}(\\chi^{2}) \\tag{*}\n\\end{equation*}\n\n\nOn the other hand, we also have\n\n\\begin{aligned}\n\\mathrm{U}(\\Lambda)^{-1} \\mathcal{M}^{\\mu \\nu} \\mathrm{U}(\\Lambda) & =(1-\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\mathcal{M}^{\\mu \\nu}(1+\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\\\\n& =M^{\\mu v}+\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]+\\mathcal{O}(\\omega^{2})\n\\end{aligned}\n\n\nThe second term of the right hand side must coincide at order $\\omega^{1}$ with the coefficient of $\\frac{i}{2} \\chi_{\\mu \\nu}$ in $(*)$ :\n\n\\begin{aligned}\n\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu v}, M^{\\rho \\sigma}] & =\\Lambda_{\\rho}^{\\mu} \\Lambda_{\\sigma}^{v} M^{\\rho \\sigma}|_{\\text {order } \\omega^{1}} \\\\\n& =(\\delta^{\\mu}{ }_{\\rho} \\omega^{v}{ }_{\\sigma}+\\omega^{\\mu}{ }_{\\rho} \\delta^{v}{ }_{\\sigma}) M^{\\rho \\sigma}=\\omega^{v}{ }_{\\sigma} M^{\\mu \\sigma}+\\omega_{\\rho}^{\\mu} M^{\\rho v} \\\\\n& =\\omega_{\\rho \\sigma}(g^{v \\rho} M^{\\mu \\sigma}+g^{\\mu \\sigma} M^{v \\rho})\n\\end{aligned}\n\n\nAt this stage, we cannot simply \"divide\" by $\\omega_{\\rho \\sigma}$, because $\\omega_{\\rho \\sigma}$ is not linearly independent of $\\omega_{\\sigma \\rho}$. Before we can perform this operation, we should explicitly antisymmetrize the coefficient of $\\omega_{\\rho \\sigma}$ in the right hand side by writing\n\n$$\\frac{i}{2} \\omega_{\\rho \\sigma}[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\frac{1}{2} \\omega_{\\rho \\sigma}(g^{\\nu \\rho} M^{\\mu \\sigma}+g^{\\mu \\sigma} M^{v \\rho}-g^{\\nu \\sigma} M^{\\mu \\rho}-g^{\\mu \\rho} M^{v \\sigma})$$\n\nThen, we obtain the following expression for the commutator of two generators of the Lorentz algebra\n\n$$[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\nu \\sigma} M^{\\mu \\rho}+g^{\\mu \\rho} M^{v \\sigma}-g^{\\nu \\rho} M^{\\mu \\sigma}-g^{\\mu \\sigma} M^{v \\rho}) $$", + "final_answer": [ + "[M^{\\mu \\nu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\nu \\sigma} M^{\\mu \\rho}+g^{\\mu \\rho} M^{v \\sigma}-g^{\\nu \\rho} M^{\\mu \\sigma}-g^{\\mu \\sigma} M^{v \\rho})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$M^{\\mu \\nu}$": "Generators of the Lorentz algebra.", + "$\\mathfrak{i}$": "Imaginary unit, as used in the final expression for the commutator.", + "$g^{\\nu \\sigma}$": "Metric tensor." + } + }, + { + "id": 68, + "context": "", + "question": "What are the commutation relations for $[\\mathrm{M}^{\\mu \\nu}, \\mathrm{P}^{\\rho}]$ and $[\\mathrm{P}^{\\mu}, \\mathrm{P}^{\\nu}]$ in the Poincaré algebra?\n\nYou should use $U(\\Lambda)^{-1} P^\\rho U(\\Lambda) = \\Lambda^\\rho{}_\\sigma P^\\sigma$, with $U(\\Lambda) = \\exp( +\\frac{i}{2} \\omega_{\\mu\\nu} M^{\\mu\\nu})$. (and $[A, B] = AB - BA$ if you want zero ambiguity)", + "answer": "The action of $\\Lambda^{-1} \\mathrm{a} \\Lambda$ on a point $x$ can be written explicitly as follows,\n\n\\begin{aligned}\n{[(\\Lambda^{-1} \\mathrm{a} \\Lambda) x]_{\\rho} } & =\\Lambda_{\\rho}^{-1 \\mu}(a_{\\mu}+\\Lambda_{\\mu}{ }^{\\nu} \\chi_{v})=\\Lambda_{\\rho}^{-1 \\mu} a_{\\mu}+\\delta_{\\rho}{ }^{\\nu} \\chi_{v} \\\\\n& =x_{\\rho}+a_{\\mu} \\Lambda_{\\rho}^{\\mu}\n\\end{aligned}\n\n\nTherefore, $\\Lambda^{-1} \\mathrm{a} \\Lambda$ acts as a translation of $x_{\\rho}$ by an amount $a_{\\mu} \\Lambda^{\\mu}{ }_{\\rho}$. Consider now the representation $\\mathrm{U}(\\Lambda^{-1} \\mathrm{a} \\Lambda)$. On the one hand, the previous results tells us that\n\n\\begin{equation*}\n\\mathrm{U}(\\Lambda^{-1} \\mathrm{a} \\Lambda)=1+i \\mathrm{a}_{\\mu} \\Lambda_{\\rho}^{\\mu}{ }_{\\rho}^{\\rho} \\mathrm{P}^{\\rho}+\\mathcal{O}(\\mathrm{a}^{2}) \\tag{*}\n\\end{equation*}\n\n\nOn the other hand, we have\n\n\\begin{aligned}\n\\mathrm{U}(\\Lambda^{-1}) \\mathrm{P}^{\\mu} \\mathrm{U}(\\Lambda) & =(1-\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\mathrm{P}^{\\mu}(1+\\frac{i}{2} \\omega_{\\rho \\sigma} M^{\\rho \\sigma}+\\mathcal{O}(\\omega^{2})) \\\\\n& =\\mathrm{P}^{\\mu}+\\frac{i}{2} \\omega_{\\rho \\sigma}[\\mathrm{P}^{\\mu}, M^{\\rho \\sigma}]+\\mathcal{O}(\\omega^{2})\n\\end{aligned}\n\n\nThe second term of the right hand side must coincide at order $\\omega^{1}$ with the coefficient of $i a_{\\mu}$ in $(*)$ :\n\n\\begin{aligned}\n\\frac{i}{2} \\omega_{\\rho \\sigma}[\\mathrm{P}^{\\mu}, M^{\\rho \\sigma}] & =\\Lambda_{\\rho}^{\\mu} \\mathrm{P}^{\\rho}|_{\\text {order } \\omega^{1}} \\\\\n& =\\omega_{\\rho}^{\\mu}{ }_{\\rho} \\mathrm{P}^{\\rho}=-\\omega_{\\rho \\sigma} g^{\\mu \\sigma} \\mathrm{P}^{\\rho}=\\frac{1}{2} \\omega_{\\rho \\sigma}(g^{\\mu \\rho} \\mathrm{P}^{\\sigma}-g^{\\mu \\sigma} \\mathrm{P}^{\\rho})\n\\end{aligned}\n\nwhere in the last equality we have performed the antisymmetrization on the indices $\\rho, \\sigma$. Therefore, we conclude that\n\n[P^{\\mu}, M^{\\rho \\sigma}]=\\mathfrak{i}(g^{\\mu \\sigma} P^{\\rho}-g^{\\mu \\rho} P^{\\sigma})\n\n\nThe fact that $P^{\\mu}$ and $P^{v}$ commute simply follows from the fact that the order in which two successive translations are performed does not matter.", + "final_answer": [ + "[\\mathrm{M}^{\\mu \\nu}, \\mathrm{P}^{\\rho}] = \\mathfrak{i}(g^{\\rho \\mu} \\mathrm{P}^{\\nu} - g^{\\rho \\nu} \\mathrm{P}^{\\mu})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{M}^{\\mu \\nu}$": "Lorentz generator, representing rotations and boosts in spacetime.", + "$\\mathrm{P}^{\\rho}$": "Momentum operator, representing translations in spacetime.", + "$\\mathfrak{i}$": "Imaginary unit.", + "$g^{\\rho \\mu}$": "Component of the metric tensor.", + "$\\mathrm{P}^{\\nu}$": "Momentum operator, representing translations in spacetime.", + "$g^{\\rho \\nu}$": "Component of the metric tensor.", + "$\\mathrm{P}^{\\mu}$": "Momentum operator, representing translations in spacetime." + } + }, + { + "id": 69, + "context": "", + "question": "Calculate the expression in coordinate space of the retarded propagator given in eq. \\begin{align}\n\\tilde{G}_R^0(\\kappa) = \\frac{i}{(\\kappa_0 + i0^+)^2 - (\\kappa^2 + m^2)}.\n\\end{align}\n\nHint : perform the $\\mathrm{k}_{0}$ integral in the complex plane with the theorem of residues. The remaining integrals are elementary.", + "answer": "The free retarded propagator in coordinate space is given by the following Fourier integral:\n$$ G_{R}^{0}(x, y)=i \\int \\frac{d^{4} k}{(2 \\pi)^{4}} \\frac{e^{-i k \\cdot(x-y)}}{(k^{0}+i 0^{+})^{2}-k^{2}}. $$\n\nThe integrand has two poles in the complex plane of the variable $k^{0}$, located at $k^{0}= \\pm|\\mathbf{k}|-\\mathfrak{i} 0^{+}$. The integration over $k^{0}$ can be performed with the theorem of residues by completing the real axis with a semi-circle at infinity. Whether this semi-circle should be in the upper or lower half plane is determined by the request that the exponential factor in the numerator does not diverge when going to infinity in the imaginary direction. To see this, write $k^{0}=k_{r}^{0}+i k_{i}^{0}$. Then, we have\n\ne^{-i k^{0}(x^{0}-y^{0})}=e^{-i k_{r}^{0}(x^{0}-y^{0})} e^{k_{i}^{0}(x^{0}-y^{0})} .\n\n\nThe dangerous factor is the second one, since the argument of the exponential is real. If $x^{0}-y^{0}<0$, this exponential remains bounded if we close the contour in the upper half plane. Since the integrand has no poles on this side, the integral is zero. If on the contrary $x^{0}-y^{0}>0$, we must close the contour in the lower half plane, and the theorem of residues gives non-zero contributions from the two poles. By writing\n\n$$\\frac{\\mathfrak{i}}{(k^{0}+i 0^{+})^{2}-\\mathbf{k}^{2}}=\\frac{\\mathfrak{i}}{2|\\mathbf{k}|}[\\frac{1}{k^{0}+\\mathfrak{i} 0^{+}-|\\mathbf{k}|}-\\frac{1}{k^{0}+\\mathfrak{i} 0^{+}+|\\mathbf{k}|}],$$\n\nwe obtain easily the corresponding residues, and the propagator now reads\n\n$$ G_{R}^{0}(x, y)=\\theta(x^{0}-y^{0}) \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3}} \\frac{e^{i \\mathbf{k} \\cdot(x-y)}}{2|\\mathbf{k}|}[e^{-i|\\mathbf{k}|(x^{0}-y^{0})}-e^{i|\\mathbf{k}|(x^{0}-y^{0})}] .$$\n\n\nIn order to make the notations more compact, let us denote $r^{0} \\equiv x^{0}-y^{0}$ and $\\mathbf{r} \\equiv \\boldsymbol{x}-\\mathbf{y}$. It is convenient to perform the integration over $k$ in spherical coordinates with a polar axis in the direction of $r$. This leads to\n\n\\begin{aligned}\nG_{R}^{0}(x, y) & =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{0}^{\\infty} d k[e^{i k r}-e^{-i k r}][e^{i k r^{0}}-e^{-i k r^{0}}] \\\\\n& =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{0}^{\\infty} d k[e^{i k(r+r^{0})}+e^{-i k(r+r^{0})}-e^{i k(r-r^{0})}-e^{-i k(r-r^{0})}] \\\\\n& =\\frac{i}{8 \\pi^{2} r} \\theta(r^{0}) \\int_{-\\infty}^{+\\infty} d k[e^{i k(r+r^{0})}-e^{i k(r-r^{0})}] \\\\\n& =\\frac{i}{4 \\pi r} \\theta(r^{0})[\\delta(r+r^{0})-\\delta(r-r^{0})]=-\\frac{i}{2 \\pi} \\theta(r^{0}) \\delta(r_{0}^{2}-r^{2})\n\\end{aligned}\n\n\nThe proportionality to $\\theta(r^{0})$ makes this propagator retarded, while the delta function with support on the light-cone makes it causal. With a mass, the integral over $k$ would be much more complicated (the result is expressible in terms of Bessel functions), with a support restricted to $r_{0}^{2}-r^{2} \\geq 0$ (therefore, it is still causal).", + "final_answer": [ + "G_{R}^{0}(x, y) = -\\frac{i}{2 \\pi} \\theta(r^{0}) \\delta(r_{0}^{2}-r^{2})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$G_{R}^{0}$": "Free retarded propagator in coordinate space.", + "$x$": "Four-position vector.", + "$y$": "Four-position vector.", + "$i$": "Imaginary unit.", + "$\\pi$": "Pi, mathematical constant.", + "$\\theta$": "Heaviside step function.", + "$r^{0}$": "Time difference, defined as $r^0 \\equiv x^0 - y^0$. (Appears as $r_0$ in $\\delta(r_{0}^{2}-r^{2})$ due to a likely typo).", + "$\\delta$": "Dirac delta function.", + "$r$": "Magnitude of the spatial difference vector $\\mathbf{r}$." + } + }, + { + "id": 70, + "context": "", + "question": "Consider a hypothetical quantum field theory with a kinetic term $$\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi,$$ where $\\mu$ is a constant with the dimension of mass. What is the expression for Källen-Lehman spectral function for this theory in this theory? \n\nConvention: Use metric $\\eta_{\\mu\\nu} = \\mathrm{diag}(+,-,-,-)$, $\\Box = \\partial_\\mu \\partial^\\mu$, and define the retarded propagator and spectral density by $G_R(p) = \\int_0^\\infty \\frac{dM^2}{2\\pi}\\,\\rho(M^2)\\,\\frac{i}{(p^0 + i0^+)^2 - \\mathbf{p}^2 - M^2}$.", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n$$\\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}} $$\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}$$\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n$$\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)$$\n\n\nThe Källen-Lehman representation of the standard retarded propagator, $G_{R}^{0}(p)$, reads\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi} \\rho(M^{2}) \\frac{\\mathfrak{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{M}^{2}}$$\n\n(Of course, in the case of the free propagator, the spectral function is trivial, namely $\\rho(M^{2})=2 \\pi \\delta(M^{2}-$ $\\mathrm{m}^{2}$ ). In the interacting case, it would be more complicated, but still positive definite). Note that in this representation, the dependence on the mass $m^{2}$ is carried only by the spectral function $\\rho(M^{2})$. Taking a derivative with respect to $\\mathrm{m}^{2}$ of this equation, we obtain\n\n$$\\mathcal{G}_{\\mathrm{R}}^{0}(p)=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi}[-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})] \\frac{\\mathfrak{i}}{(p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-M^{2}}$$\n\nand the (free) spectral function in the theory with the higher order kinetic term is the factor between the square brackets, i.e.\n\n$$-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})=2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})$$\n\nThis distribution is not positive definite. Indeed, when applied to a positive definite test function $f(M^{2})$, we obtain by integration by parts\n\n$$ \\int_{0}^{\\infty} \\frac{d M^{2}}{2 \\pi} f(M^{2}) 2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})=-\\mu^{2} f^{\\prime}(m^{2}) $$\n\nthat can have either sign. The non-positivity of this spectral function means that this theory must contain states that contribute negatively to the sum in $$\\begin{align}\n\\rho(M^2) \\equiv 2\\pi \\sum_{\\text{classes } \\alpha} \\delta(M^2 - m_\\alpha^2) \n\\left\\langle 0_{\\text{out}} \\left| \\phi(0) \\right| \\alpha_0 \\right\\rangle\n\\left\\langle \\alpha_0 \\left| \\phi(0) \\right| 0_{\\text{in}} \\right\\rangle.\n\\end{align}$$, and therefore is not unitary in the usual sense.", + "final_answer": [ + "2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mu$": "Constant with the dimension of mass.", + "M": "Mass, integration variable in the Källen-Lehman representation.", + "$\\delta(M^2-m^2)$": "Dirac delta function.", + "m": "Mass parameter." + } + }, + { + "id": 71, + "context": "", + "question": "For the theory with the kinetic term $\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi$, what is the relationship between its retarded propagator $\\mathcal{G}_{R}^{0}(p)$ and the free retarded propagator $G_{R}^{0}(p)$ of a standard scalar field? (Express $\\mathcal{G}_{R}^{0}(p)$ in terms of $G_{R}^{0}(p)$)", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n$$ \\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}} $$\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n$$\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}$$\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n$$ \\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p) $$", + "final_answer": [ + "\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathcal{G}_{R}^{0}(p)$": "Retarded propagator for the theory with the kinetic term $\\mathcal{L}_{0}$", + "$\\mu$": "Parameter in the kinetic term", + "$m$": "Mass of the scalar field", + "$G_{R}^{0}(p)$": "Free retarded propagator of a standard scalar field", + "$p$": "Four-momentum" + } + }, + { + "id": 72, + "context": "", + "question": "For the theory with the kinetic term $\\mathcal{L}_{0} \\equiv-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m^{2})^{2} \\phi$, what is its the free retarded propagator?", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. \nThe free retarded propagator reads\n\n\\begin{aligned}\n \\mathcal{G}_{\\mathrm{R}}^{0}(\\mathrm{p})&=-\\frac{\\mathfrak{i} \\mu^{2}}{((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{1}^{2})((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{2}^{2})} \\\\\n& =\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}] .\n\\end{aligned}", + "final_answer": [ + "\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mu$": "A parameter appearing in the kinetic term and the propagator.", + "$m_{1}$": "First mass parameter introduced in the propagator expression.", + "$m_{2}$": "Second mass parameter introduced in the propagator expression.", + "$i$": "Imaginary unit.", + "$p^{0}$": "Time component of the four-momentum.", + "$0^{+}$": "Infinitesimal positive quantity, used for defining the retarded propagator pole prescription.", + "$p^{2}$": "Magnitude squared of the spatial momentum vector, $p^2 = |\\vec{p}|^2$." + } + }, + { + "id": 73, + "context": "", + "question": "For the theory with the kinetic term $-\\frac{1}{2 \\mu^{2}} \\phi(\\square+m_{1}^{2})(\\square+m_{2}^{2}) \\phi$, what is its Källen-Lehman spectral function? \n\nDefine the (free) Feynman propagator $G_F(p)$ by $G_F(p) = i\\,K^{-1}(p)$ with metric $\\eta = (+,-,-,-)$ and Fourier convention $e^{-i p \\cdot x}$. Define the spectral density $\\rho$ via\n$$G_F(p) = \\int_0^\\infty \\frac{d M^2}{2 \\pi}\\,\\rho(M^2)\\,\\frac{i}{p^2 - M^2 + i0^+}. $$\nFind $\\rho(M^2)$.", + "answer": "The free propagator is obtained from the inverse of the operator between the two fields. In the case of the first example, its momentum space expression reads\n\n\\mathcal{G}_{\\mathrm{R}}^{0}(p)=-\\frac{\\mathfrak{i} \\mu^{2}}{((p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-m^{2})^{2}}\n\nwhere the $\\mathrm{i}^{+}$prescription we have chosen ensures that all the poles are located below the real axis in the complex plane of the variable $p^{0}$. Therefore, this is indeed the retarded propagator. Recall now that the\nfree retarded propagator for the usual scalar kinetic term is\n\n\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\frac{\\mathrm{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{m}^{2}}\n\n\nThe two propagators are thus related to one another by a derivative with respect to the squared mass. More precisely, one has\n\n\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)\n\n\nThe Källen-Lehman representation of the standard retarded propagator, $G_{R}^{0}(p)$, reads\n\n\\mathrm{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi} \\rho(M^{2}) \\frac{\\mathfrak{i}}{(\\mathrm{p}^{0}+i 0^{+})^{2}-\\mathrm{p}^{2}-\\mathrm{M}^{2}}\n\n(Of course, in the case of the free propagator, the spectral function is trivial, namely $\\rho(M^{2})=2 \\pi \\delta(M^{2}-$ $\\mathrm{m}^{2}$ ). In the interacting case, it would be more complicated, but still positive definite). Note that in this representation, the dependence on the mass $m^{2}$ is carried only by the spectral function $\\rho(M^{2})$. Taking a derivative with respect to $\\mathrm{m}^{2}$ of this equation, we obtain\n\n\\mathcal{G}_{\\mathrm{R}}^{0}(p)=\\int_{0}^{\\infty} \\frac{\\mathrm{d} M^{2}}{2 \\pi}[-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})] \\frac{\\mathfrak{i}}{(p^{0}+\\mathfrak{i} 0^{+})^{2}-\\mathbf{p}^{2}-M^{2}}\n\nand the (free) spectral function in the theory with the higher order kinetic term is the factor between the square brackets, i.e.\n\n-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} \\rho(M^{2})=2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})\n\n\nThis distribution is not positive definite. Indeed, when applied to a positive definite test function $f(M^{2})$, we obtain by integration by parts\n\n\\int_{0}^{\\infty} \\frac{d M^{2}}{2 \\pi} f(M^{2}) 2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})=-\\mu^{2} f^{\\prime}(m^{2})\n\nthat can have either sign. The non-positivity of this spectral function means that this theory must contain states that contribute negatively to the sum in eq. (1.113), and therefore is not unitary in the usual sense.\n\nConsider now the second example, more general, of theory with higher derivatives in the kinetic term. This time, the free retarded propagator reads\n\n\\begin{aligned}\n& \\mathcal{G}_{\\mathrm{R}}^{0}(\\mathrm{p})=-\\frac{\\mathfrak{i} \\mu^{2}}{((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{1}^{2})((\\mathrm{p}^{0}+\\mathfrak{i 0 ^ { + }})^{2}-\\mathrm{p}^{2}-\\mathrm{m}_{2}^{2})} \\\\\n& =\\frac{\\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{2}^{2}}-\\frac{i}{(p^{0}+i 0^{+})^{2}-p^{2}-m_{1}^{2}}] .\n\\end{aligned}\n\n\nBy the same reasoning as before, we see that this corresponds to the following spectral function,\n\n$$\\frac{2 \\pi \\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\delta(M^{2}-m_{2}^{2})-\\delta(M^{2}-m_{1}^{2})]$$\n\nwhich is also not positive definite (one delta function contributes positively and the other one negatively). The bottom line of this exercise is that kinetic terms with higher derivatives in general lead to serious problems with unitarity. In the case of the second example, the theory could nevertheless have a practical use if $m_{1} \\gg m_{2}$, provided one stays in an energy range much lower than $m_{1}$ in order not to excite the mode that has a negative norm (such a kinetic term is in fact the basis of the Pauli-Villars regularization).", + "final_answer": [ + "\\frac{2 \\pi \\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\delta(M^{2}-m_{2}^{2})-\\delta(M^{2}-m_{1}^{2})]" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\pi$": "Mathematical constant pi.", + "$\\mu$": "Parameter in the kinetic term of the theory.", + "$m_1$": "First mass parameter in the kinetic term.", + "$m_2$": "Second mass parameter in the kinetic term.", + "$\\delta$": "Dirac delta function.", + "$M$": "Integration variable for mass squared in the Källen-Lehman representation." + } + }, + { + "id": 74, + "context": "", + "question": "Consider the momentum-space six-point function in $\\lambda\\phi^4$ theory with $\\mathcal L_{\\text{int}}=-\\frac{\\lambda}{4!}\\phi^4$. All external momenta are incoming and paired as \n$$p=k_1+k_2,\\qquad q=k_3+k_4,\\qquad r=k_5+k_6,\\qquad p+q+r=0.$$\n\nUse Minkowski Feynman rules: propagator $\\frac{i}{p^2-m^2+i0^+}$, vertex factor $-i\\lambda$.\n\nOmit $(2\\pi)^D\\delta^{(D)}(p+q+r)$, omit any sum over external-leg permutations, and do not shift loop momenta.\nWrite the integrands (no integration measures). The two 2-loop integrands carry an overall factor 1/2.\n\n(1) 1-loop triangle: internal momenta $\\ell,\\ \\ell+p,\\ \\ell-q$.\n\n(2) 2-loop coupled topology: internal momenta $\\ell,\\ \\ell+p,\\ \\ell-q,\\ \\ell',\\ \\ell+\\ell'+p+r$, with an overall factor 1/2.\n\n(3) 2-loop factorized topology: internal momenta $\\ell,\\ \\ell+p,\\ \\ell-q,\\ \\ell',\\ \\ell'-p-q$, with an overall factor 1/2.", + "answer": "(1) The 1-loop triangle topology integral is given by:\n\\begin{equation*}\n \\int \\frac{\\mathrm{d}^{D} \\ell}{(2 \\pi)^{D}} \\frac{(-i \\lambda)^{3} i^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell + p)^{2} - m^{2} + i 0^{+})((\\ell - q)^{2} - m^{2} + i 0^{+})}\n\\end{equation*}\n\n(2) The 2-loop coupled topology integral is given by:\n\\begin{equation*}\n\\begin{aligned}\n\\frac{1}{2} \\int \\frac{\\mathrm{d}^D \\ell \\mathrm{d}^D \\ell'}{(2 \\pi)^{2D}} & \\frac{(-i\\lambda)^4 i^2}{(\\ell'^2-m^2+i0^+)((\\ell+\\ell'+p+r)^2-m^2+i0^+)} \\\\\n& \\times \\frac{i^3}{(\\ell^2-m^2+i0^+)((\\ell+p)^2-m^2+i0^+)((\\ell - q)^2-m^2+i0^+)}\n\\end{aligned}\n\\end{equation*}\n\n(3) The 2-loop factorized topology integral is given by:\n\\begin{equation*}\n\\begin{aligned}\n\\frac{1}{2} \\int \\frac{\\mathrm{d}^D \\ell \\mathrm{d}^D \\ell'}{(2 \\pi)^{2D}} & \\frac{(-i\\lambda)^4 i^2}{(\\ell'^2-m^2+i0^+)((\\ell'-p-q)^2-m^2+i0^+)} \\\\\n& \\times \\frac{i^3}{(\\ell^2-m^2+i0^+)((\\ell+p)^2-m^2+i0^+)((\\ell - q)^2-m^2+i0^+)}\n\\end{aligned}\n\\end{equation*}", + "final_answer": [ + "(1) The 1-loop triangle topology integral is given by:\n\\begin{equation*}\n \\int \\frac{\\mathrm{d}^{D} \\ell}{(2 \\pi)^{D}} \\frac{(-i \\lambda)^{3} i^{3}}{(\\ell^{2}-m^{2}+i 0^{+})((\\ell + p)^{2} - m^{2} + i 0^{+})((\\ell - q)^{2} - m^{2} + i 0^{+})}\n\\end{equation*}\n\n(2) The 2-loop coupled topology integral is given by:\n\\begin{equation*}\n\\begin{aligned}\n\\frac{1}{2} \\int \\frac{\\mathrm{d}^D \\ell \\mathrm{d}^D \\ell'}{(2 \\pi)^{2D}} & \\frac{(-i\\lambda)^4 i^2}{(\\ell'^2-m^2+i0^+)((\\ell+\\ell'+p+r)^2-m^2+i0^+)} \\\\\n& \\times \\frac{i^3}{(\\ell^2-m^2+i0^+)((\\ell+p)^2-m^2+i0^+)((\\ell - q)^2-m^2+i0^+)}\n\\end{aligned}\n\\end{equation*}\n\n(3) The 2-loop factorized topology integral is given by:\n\\begin{equation*}\n\\begin{aligned}\n\\frac{1}{2} \\int \\frac{\\mathrm{d}^D \\ell \\mathrm{d}^D \\ell'}{(2 \\pi)^{2D}} & \\frac{(-i\\lambda)^4 i^2}{(\\ell'^2-m^2+i0^+)((\\ell'-p-q)^2-m^2+i0^+)} \\\\\n& \\times \\frac{i^3}{(\\ell^2-m^2+i0^+)((\\ell+p)^2-m^2+i0^+)((\\ell - q)^2-m^2+i0^+)}\n\\end{aligned}\n\\end{equation*}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$i$": "Imaginary unit", + "$\\lambda$": "Coupling constant for the $\\phi^4$ interaction", + "$\\ell$": "Loop momentum, a variable of integration", + "$m$": "Mass of the scalar particle", + "$0^{+}$": "Infinitesimal positive quantity, part of the Feynman prescription for propagators", + "$p$": "External momentum", + "$q$": "External momentum", + "$\\mathfrak{i}$": "Imaginary unit (stylistic variant of $i$)", + "$\\ell^{\\prime}$": "Second loop momentum, a variable of integration", + "$r$": "External momentum" + } + }, + { + "id": 75, + "context": "", + "question": "Calculate $\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\rho} \\gamma^{\\sigma})$.", + "answer": "With four Dirac matrices, we can write\n\n\\begin{align}\n\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\rho} \\gamma^{\\sigma})= & -\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{v} \\gamma^{\\sigma} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu}) \\\\\n= & +\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\sigma} \\gamma^{v} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-2 g^{v \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho}) \\\\\n= & -\\operatorname{tr}(\\gamma^{\\sigma} \\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\rho})+2 g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-2 g^{v \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho}) \\\\\n& +2 g^{\\mu \\sigma} \\operatorname{tr}(\\gamma^{\\nu} \\gamma^{\\rho}) .\n\\end{align}\n\nThis gives:\n\n\\begin{align}\n\\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\rho} \\gamma^{\\sigma}) & =g^{\\rho \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\nu})-g^{\\nu \\sigma} \\operatorname{tr}(\\gamma^{\\mu} \\gamma^{\\rho})+g^{\\mu \\sigma} \\operatorname{tr}(\\gamma^{\\nu} \\gamma^{\\rho}) \\\\\n& =4(g^{\\rho \\sigma} g^{\\mu \\nu}-g^{\\nu \\sigma} g^{\\mu \\rho}+g^{\\mu \\sigma} g^{\\nu \\rho}) .\n\\end{align}", + "final_answer": [ + "4(g^{\\rho \\sigma} g^{\\mu \\nu}-g^{\\nu \\sigma} g^{\\mu \\rho}+g^{\\mu \\sigma} g^{\\nu \\rho})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$g^{\\rho \\sigma}$": "Component of the metric tensor with spacetime indices $\\rho$ and $\\sigma$", + "$g^{\\mu \\nu}$": "Component of the metric tensor with spacetime indices $\\mu$ and $\\nu$", + "$g^{\\nu \\sigma}$": "Component of the metric tensor with spacetime indices $\\nu$ and $\\sigma$", + "$g^{\\mu \\rho}$": "Component of the metric tensor with spacetime indices $\\mu$ and $\\rho$", + "$g^{\\mu \\sigma}$": "Component of the metric tensor with spacetime indices $\\mu$ and $\\sigma$", + "$g^{\\nu \\rho}$": "Component of the metric tensor with spacetime indices $\\nu$ and $\\rho$" + } + }, + { + "id": 76, + "context": "", + "question": "Consider two coherent states $|\\chi_{\\text {in }}\\rangle$ and $|\\vartheta_{\\text {in }}\\rangle$. The state $|\\chi_{\\text {in }}\\rangle$ is defined by the function $\\chi(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\chi_{0} \\delta(\\mathbf{k})$, and the state $|\\vartheta_{\\text {in }}\\rangle$ is defined by the function $\\vartheta(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\vartheta_{0} \\delta(\\mathbf{k})$, where $\\chi_0$ and $\\vartheta_0$ are constants, and $\\delta(\\mathbf{k})$ is the 3-dimensional Dirac delta function. Calculate the overlap $\\langle\\vartheta_{\\text {in }} \\mid \\chi_{\\text {in }}\\rangle$. \n\nThe mode operators satisfy $[a_{\\mathbf{k}},\\,a_{\\mathbf{k}'}^\\dagger]=(2\\pi)^3\\,2E_{\\mathbf{k}}\\,\\delta^{(3)}(\\mathbf{k}-\\mathbf{k}')$ with $E_{\\mathbf{k}}=\\sqrt{\\mathbf{k}^2+m^2}$, and the coherent state $|\\chi_{\\mathrm{in}}\\rangle$ is the normalized eigenstate of $a_{\\mathbf{k}}$ with eigenvalue $\\chi(\\mathbf{k})$ (and similarly for $|\\vartheta_{\\mathrm{in}}\\rangle$). Work in a finite box of volume $V$ with periodic boundary conditions so that $\\delta^{(3)}(\\mathbf{0})=V/(2\\pi)^3$, and take $\\chi_0,\\,\\vartheta_0\\in\\mathbb{R}$.", + "answer": "Consider now two such coherent states, $|\\chi_{\\text {in }}\\rangle,|\\vartheta_{\\text {in }}\\rangle$, in the special case where their defining functions have only a zero mode: $\\chi(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\chi_{0} \\delta(\\mathbf{k}), \\vartheta(\\mathbf{k}) \\equiv(2 \\pi)^{3} \\vartheta_{0} \\delta(\\mathbf{k})$. The overlap of these two states is given by\n\n\\begin{align}\n\\langle\\vartheta_{\\text {in }} \\mid \\chi_{\\text {in }}\\rangle= & \\exp (-\\frac{1}{2} \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}}[|\\chi(\\mathbf{k})|^{2}+|\\vartheta(\\mathbf{k})|^{2}]) \\\\\n& \\times\\langle 0_{\\text {in }}| \\exp (\\int \\frac{d^{3} \\mathrm{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}} \\vartheta^{*}(\\mathbf{k}) \\mathrm{a}_{\\mathbf{k}, \\text { in }}) \\exp (\\int \\frac{\\mathrm{d}^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}} \\chi(\\mathbf{k}) \\mathrm{a}_{\\mathrm{k}, \\text { in }}^{\\dagger})|0_{\\text {in }}\\rangle \\\\\n= & \\exp (-\\frac{1}{2} \\int \\frac{d^{3} \\mathbf{k}}{(2 \\pi)^{3} 2 \\mathrm{E}_{\\mathrm{k}}}|\\chi(\\mathbf{k})-\\vartheta(\\mathbf{k})|^{2}) \\\\\n= & \\exp (-\\frac{V|\\chi_{0}-\\vartheta_{0}|^{2}}{4 m})\n\\end{align}\n\n\nTherefore, spatially homogeneous coherent states (i.e. ground states of quadratic theories shifted by a uniform field), have an exponentially suppressed overlap, and the argument of the exponential is proportional to the volume. Thus, distinct coherent states of this type are orthogonal if the volume is infinite.", + "final_answer": [ + "\\exp (-\\frac{V|\\chi_{0}-\\vartheta_{0}|^{2}}{4 m})" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$V$": "Volume.", + "$\\chi_0$": "Constant amplitude for the function $\\chi(\\mathbf{k})$.", + "$\\vartheta_0$": "Constant amplitude for the function $\\vartheta(\\mathbf{k})$.", + "$m$": "Mass.", + "$\\vartheta_{0}^{*}$": "Complex conjugate of the constant $\\vartheta_0$." + } + }, + { + "id": 77, + "context": "", + "question": "Using Weyl's prescription for quantization, where a classical quantity $f(q, p)$ is mapped to an operator $F(Q,P)$ by\n\n$$ F(Q, P) \\equiv \\int \\frac{d p d q d \\mu d v}{(2 \\pi)^{2}} f(q, p) e^{i(\\mu(Q-q)+v(P-p))}, $$\n\ncalculate the quantum operator corresponding to $f(q,p) = qp$. \n(You may use the results from previous parts that for $g(q,p) = (\\alpha q + \\beta p)^k$, its Weyl map is $(\\alpha Q + \\beta P)^k$. Specifically, $q^2$ maps to $Q^2$, $p^2$ maps to $P^2$, and $(q+p)^2$ maps to $(Q+P)^2$.)", + "answer": "To obtain the Weyl mapping of $qp$, we use the polarization identity: $q p=\\frac{1}{2}((q+p)^{2}-q^{2}-p^{2})$.\nFrom the previous result $(\\alpha q+\\beta p)^{n} \\rightarrow (\\alpha Q+\\beta P)^{n}$, we have the following specific mappings:\n\nFor $q^2$: set $\\alpha=1, \\beta=0, n=2$. So, $q^{2} \\rightarrow Q^{2}$.\nFor $p^2$: set $\\alpha=0, \\beta=1, n=2$. So, $p^{2} \\rightarrow P^{2}$.\nFor $(q+p)^2$: set $\\alpha=1, \\beta=1, n=2$. So, $(q+p)^{2} \\rightarrow (Q+P)^{2}$.\n\nWe know that $(Q+P)^2 = (Q+P)(Q+P) = Q^2 + QP + PQ + P^2$, because $Q$ and $P$ do not generally commute.\n\nNow, we apply the Weyl mapping to $qp$ using its expression in terms of squares:\n\n\\text{Weyl}[qp] = \\text{Weyl}[\\frac{1}{2}((q+p)^{2}-q^{2}-p^{2})]\n\nDue to the linearity of the Weyl mapping:\n\n\\text{Weyl}[qp] = \\frac{1}{2}(\\text{Weyl}[(q+p)^{2}] - \\text{Weyl}[q^{2}] - \\text{Weyl}[p^{2}])\n\nSubstituting the mapped operators:\n\n\\text{Weyl}[qp] = \\frac{1}{2}((Q+P)^{2} - Q^{2} - P^{2})\n\n\n\\text{Weyl}[qp] = \\frac{1}{2}((Q^{2}+QP+PQ+P^{2}) - Q^{2} - P^{2})\n\n\n\\text{Weyl}[qp] = \\frac{1}{2}(QP+PQ)\n\nTherefore, the Weyl mapping of $qp$ is $\\frac{1}{2}(QP+PQ)$.", + "final_answer": [ + "\\frac{1}{2}(QP+PQ)" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$Q$": "Quantum position operator.", + "$P$": "Quantum momentum operator." + } + }, + { + "id": 78, + "context": "", + "question": "Consider the fermionic integral,\n\n$$\\langle\\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}}\\rangle \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}} \\exp (\\bar{\\chi}^{\\top} \\boldsymbol{M} \\boldsymbol{\\chi}) .$$\n\nAssuming $p=q$, compute the expectation value for $p=q=1$, i.e., find an expression for $\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$.", + "answer": "To calculate these integrals, we introduce the generating function:\n\n$$Z[\\overline{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta}) \\exp (\\bar{\\chi}^{\\top} \\mathbf{M} \\boldsymbol{\\chi})$$\n\nThis is a Gaussian Grassmann integral, which evaluates to:\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$$\n\nExpanding this exponential: \n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\mathcal{O}(\\eta^2 \\bar{\\eta}^2)$$\n\nAlternatively, by expanding the factor $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta})$ in the definition of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$:\n\n$$\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta}) = (1 + \\sum_a \\bar{\\eta}_a \\chi_a + \\dots ) (1 - \\sum_b \\bar{\\chi}_b \\eta_b + \\dots ) = 1 + \\sum_a \\bar{\\eta}_a \\chi_a - \\sum_b \\bar{\\chi}_b \\eta_b - \\sum_{a,b} \\bar{\\eta}_a \\chi_a \\bar{\\chi}_b \\eta_b + \\dots$$\n\nSo,\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\langle 1 \\rangle + \\sum_a \\bar{\\eta}_a \\langle \\chi_a \\rangle - \\sum_b \\langle \\bar{\\chi}_b \\rangle \\eta_b - \\sum_{a,b} \\bar{\\eta}_a \\langle \\chi_a \\bar{\\chi}_b \\rangle \\eta_b + \\dots$$\n\nSince $\\langle \\chi_a \\rangle = 0$ and $\\langle \\bar{\\chi}_b \\rangle = 0$ (as $p \neq q$ for these terms), we have:\n\n$$Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\langle 1 \\rangle - \\sum_{a,b} \\bar{\\eta}_a \\langle \\chi_a \\bar{\\chi}_b \\rangle \\eta_b + \\dots$$\n\n(Note: $\\langle 1 \\rangle = 1$ by normalization). Comparing the coefficients of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ in both expansions of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$:\nFrom $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$, the coefficient of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ is $M_{j_1 i_1}^{-1}$.\nFrom the series expansion in terms of expectation values, the coefficient of $\\bar{\\eta}_{j_1} \\eta_{i_1}$ is $-\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$.\nTherefore, $M_{j_1 i_1}^{-1} = -\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle$, which gives:\n\n$$\\langle \\chi_{j_1} \\bar{\\chi}_{i_1} \\rangle = -M_{j_1 i_1}^{-1}$$", + "final_answer": [ + "-M_{j_1 i_1}^{-1}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$M_{j_1 i_1}^{-1}$": "Element at row $j_1$ and column $i_1$ of the inverse matrix $\\boldsymbol{M}^{-1}$.", + "$j_1$": "Index for the first $\\chi$ field in the expectation value being computed.", + "$i_1$": "Index for the first $\\bar{\\chi}$ field in the expectation value being computed." + } + }, + { + "id": 79, + "context": "", + "question": "Consider the fermionic integral,\n$$\n\\langle\\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}}\\rangle \\equiv \\operatorname{det}^{-1}(\\boldsymbol{M}) \\int \\prod_{k=1}^{n}[d \\chi_{k} d \\bar{\\chi}_{k}] \\chi_{j_{1}} \\cdots \\chi_{j_{p}} \\bar{\\chi}_{i_{1}} \\cdots \\bar{\\chi}_{i_{q}} \\exp (\\bar{\\chi}^{\\top} \\boldsymbol{M} \\boldsymbol{\\chi}) .\n$$\nAssuming $p=q$, compute the expectation value for $p=q=2$, i.e., find an expression for $\\langle \\chi_{j_1} \\chi_{j_2} \\bar{\\chi}_{i_1} \\bar{\\chi}_{i_2} \\rangle$.", + "answer": "We use the generating function $Z[\\overline{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}]$ as defined in the previous problem. The expansion of $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{M}^{-1} \\boldsymbol{\\eta})$ to second order in $\\bar{\\eta}$ and $\\eta$ is: $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\frac{1}{2!} ( \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} )^2 + \\dots = 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} + \\frac{1}{2} \\sum_{a,b,c,d} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} \\bar{\\eta}_{c} M_{cd}^{-1} \\eta_{d} + \\dots$ Using anticommutation $\\eta_b \\bar{\\eta}_c = - \\bar{\\eta}_c \\eta_b$: $= 1 + \\sum_{a,b} \\bar{\\eta}_{a} M_{ab}^{-1} \\eta_{b} - \\frac{1}{2} \\sum_{a,b,c,d} \\bar{\\eta}_{a} \\bar{\\eta}_{c} M_{ab}^{-1} M_{cd}^{-1} \\eta_{b} \\eta_{d} + \\dots$ As noted, this can be written by antisymmetrizing the $M^{-1}$ terms: $Z[\\bar{\\boldsymbol{\\eta}}, \\boldsymbol{\\eta}] = \\dots - \\frac{1}{4} \\sum_{a,c,b,d} \\bar{\\eta}_{a} \\bar{\\eta}_{c} (M_{ab}^{-1} M_{cd}^{-1} - M_{ad}^{-1} M_{cb}^{-1}) \\eta_{b} \\eta_{d} + \\dots$ Expanding $\\exp (\\overline{\\boldsymbol{\\eta}}^{\\top} \\boldsymbol{\\chi}-\\bar{\\chi}^{\\top} \\boldsymbol{\\eta})$ gives terms with two $\\bar{\\eta}$ and two $\\eta$ variables. Comparing coefficients, the result matches $\\langle \\chi_{j_1} \\chi_{j_2} \\bar{\\chi}_{i_1} \\bar{\\chi}_{i_2} \\rangle = M_{j_1 i_2}^{-1} M_{j_2 i_1}^{-1} - M_{j_1 i_1}^{-1} M_{j_2 i_2}^{-1}$.", + "final_answer": [ + "M_{j_1 i_2}^{-1} M_{j_2 i_1}^{-1} - M_{j_1 i_1}^{-1} M_{j_2 i_2}^{-1}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$M_{j_1 i_2}^{-1}$": "Element of the inverse matrix $\\boldsymbol{M}^{-1}$ at row $j_1$ and column $i_2$.", + "$M_{j_2 i_1}^{-1}$": "Element of the inverse matrix $\\boldsymbol{M}^{-1}$ at row $j_2$ and column $i_1$.", + "$M_{j_1 i_1}^{-1}$": "Element of the inverse matrix $\\boldsymbol{M}^{-1}$ at row $j_1$ and column $i_1$.", + "$M_{j_2 i_2}^{-1}$": "Element of the inverse matrix $\\boldsymbol{M}^{-1}$ at row $j_2$ and column $i_2$." + } + }, + { + "id": 80, + "context": "", + "question": "Consider a general function of a single Grassmann variable $\\chi$, which can be written as $f(\\chi) = a + \\chi b$, where $a$ and $b$ are c-numbers (or objects that commute with Grassmann variables). Introduce a second Grassmann variable $\\eta$ that anticommutes with $\\chi$. Calculate explicitly the integral $\\widetilde{f}(\\eta) \\equiv \\int d \\chi e^{\\chi \\eta} f(\\chi)$. (Recall Grassmann properties: $e^{\\chi \\eta} = 1 + \\chi \\eta$, $\\chi^2 = 0$, and Berezin integration rules $\\int d\\chi = 0$, $\\int d\\chi \\chi = 1$).", + "answer": "Given $f(\\chi) = a + \\chi b$ and $e^{\\chi \\eta} = 1 + \\chi \\eta$ (since higher powers of $\\chi\\eta$ would involve $\\chi^2$ or $\\eta^2$ which are zero if $\\chi, \\eta$ are single components, or this is the truncated expansion for Grassmann variables).\nThe integral is:\n\\begin{align}\n\\widetilde{f}(\\eta) & =\\int d \\chi e^{\\chi \\eta}(a+\\chi b) \\\\& =\\int d \\chi(1+\\chi \\eta)(a+\\chi b) \\\\\n& =\\int d \\chi(a + \\chi b + \\chi \\eta a + \\chi \\eta \\chi b)\n\\end{align}\nSince $\\chi$ and $\\eta$ are Grassmann variables, they anticommute ($\\chi \\eta = -\\eta \\chi$), and $\\chi^2 = 0$. Thus, the term $\\chi \\eta \\chi b = -\\eta \\chi^2 b = -\\eta (0) b = 0$. \nThe integral simplifies to:$$\\widetilde{f}(\\eta) = \\int d \\chi(a+\\chi(\\eta a+b))$$\nUsing the Berezin integration rules $\\int d\\chi = 0$ and $\\int d\\chi \\chi = 1$: \n$$\\widetilde{f}(\\eta) = a \\int d\\chi (1) + (\\eta a+b) \\int d\\chi \\chi = a \\cdot 0 + (\\eta a+b) \\cdot 1 = \\eta a+b$$", + "final_answer": [ + "\\eta a+b" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\eta$": "A second Grassmann variable that anticommutes with $\\chi$.", + "$a$": "A c-number (or object that commutes with Grassmann variables).", + "$b$": "A c-number (or object that commutes with Grassmann variables)." + } + }, + { + "id": 81, + "context": "", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_3 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{-} \\frac{\\partial}{\\partial \\theta_{-}})$, find an eigenfunction corresponding to the eigenvalue $1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "answer": "From the eigenvalue equation $\\tau_3 f = \\lambda f$, setting $\\lambda=1/2$: \n$0 = \\frac{1}{2} a \\implies a=0$ \n$(\\frac{1}{2}-\\frac{1}{2})b=0 \\implies 0 \\cdot b = 0$ (no constraint on $b$) \n$(-\\frac{1}{2}-\\frac{1}{2})c=0 \\implies -c=0 \\implies c=0$ \n$0 = \\frac{1}{2} d \\implies d=0$ \nSo, eigenfunctions for $\\lambda=1/2$ are of the form $b\\theta_+$. Choosing $b=1$ for normalization, an eigenfunction is $\\theta_+$.", + "final_answer": [ + "$\\theta_{+}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\theta_+$": "One of the two Grassmann variables." + } + }, + { + "id": 82, + "context": "", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$, find an eigenfunction corresponding to the eigenvalue $1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "answer": "For $\\lambda=1/2$: $a=0, d=0$. $\\frac{1}{2}c = \\frac{1}{2}b \\implies c=b$. Eigenfunctions are $b\\theta_+ + b\\theta_- = b(\\theta_+ + \\theta_-)$. Normalized: $\\theta_+ + \\theta_-$.", + "final_answer": [ + "$\\theta_{+} + \\theta_{-}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\theta_+$": "One of the two Grassmann variables.", + "$\\theta_-$": "One of the two Grassmann variables." + } + }, + { + "id": 83, + "context": "", + "question": "Consider two Grassmann variables $\\theta_\\pm$.\n\nFor the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$, find an eigenfunction corresponding to the eigenvalue $-1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "answer": "For $\\lambda=-1/2$: $a=0, d=0$. $\\frac{1}{2}c = -\\frac{1}{2}b \\implies c=-b$. Eigenfunctions are $b\\theta_+ - b\\theta_- = b(\\theta_+ - \\theta_-)$. Normalized: $\\theta_+ - \\theta_-$.", + "final_answer": [ + "$\\theta_{+} - \\theta_{-}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\theta_+$": "Grassmann variable", + "$\\theta_-$": "Grassmann variable" + } + }, + { + "id": 84, + "context": "Consider two Grassmann variables $\\theta_\\pm$.", + "question": "For the operator $\\tau_2 \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, find an eigenfunction corresponding to the eigenvalue $-1/2$. Express it using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize.", + "answer": "For $\\lambda=-1/2$: $a=0, d=0$. $-\\frac{i}{2}c = -\\frac{1}{2}b \\implies ic=b \\implies c=-ib$. Eigenfunctions are $b\\theta_+ - ib\\theta_- = b(\\theta_+ - i\\theta_-)$. Normalized: $\\theta_+ - i\\theta_-$.", + "final_answer": [ + "$\\theta_{+} - i\\theta_{-}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\theta_+$": "Grassmann variable", + "$i$": "Imaginary unit", + "$\\theta_-$": "Grassmann variable" + } + }, + { + "id": 85, + "context": "Consider two Grassmann variables $\\theta_\\pm$.", + "question": "Given the operators $\\tau_{1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ and $\\tau_{2} \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, calculate the action of the operator $(\\tau_1 - i\\tau_2)$ on the Grassmann variable $\\theta_+$.}", + "answer": "$\\tau_1 \\theta_+ = \\frac{1{2}\\theta_-$. $\\tau_2 \\theta_+ = \\frac{i}{2}\\theta_-$. $(\\tau_1 - i\\tau_2)\\theta_+ = \\frac{1}{2}\\theta_- - i(\\frac{i}{2}\\\\theta_-) = \\frac{1}{2}\\theta_- + \\frac{1}{2}\\theta_- = \\theta_-$.}", + "final_answer": [ + "\\theta_{-}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\theta_-$": "Grassmann variable" + } + }, + { + "id": 86, + "context": "", + "question": "Assume that $[X, Y]=i c Y$ where c is a numerical constant. Use the all-orders Baker-CampbellHausdorff formula to calculate $\\ln (e^{i X} e^{i Y})$.", + "answer": "In this case, we have\n$$\n\\operatorname{ad}_{x}(Y)=-i[X, Y]=c Y, \\quad e^{\\operatorname{ad}_{X}} Y=e^{c} Y\n$$\n\nSince $Y$ commutes with itself, this implies that\n$$\ne^{\\operatorname{tad}_{Y}} e^{\\mathrm{ad}_{x}} \\mathrm{Y}=e^{\\mathrm{c}} \\mathrm{Y}\n$$\n\nUsing this in the general Baker-Campbell-Hausdorff formula gives\n\n$$\\ln (e^{i X} e^{i Y})=i X+i \\int_{0}^{1} d t \\frac{\\ln (e^{c})}{e^{c}-1} Y=i X+\\frac{i c}{e^{c}-1} Y$$\n\n(Obviously, this result goes to $i(X+Y)$ when $\\mathrm{c} \\rightarrow 0$.)", + "final_answer": [ + "i X+\\frac{i c}{e^{c}-1} Y" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$i$": "The imaginary unit.", + "$X$": "An operator.", + "$c$": "A numerical constant.", + "$e$": "Euler's number, the base of the natural logarithm.", + "$Y$": "An operator." + } + }, + { + "id": 87, + "context": "", + "question": "Calculate the one-loop $\\beta$-function of a scalar field theory with cubic interactions in six spacetime dimensions.", + "answer": "Scalar field theory with cubic interactions is renormalizable in six dimensions (the coupling constant of this cubic interaction is dimensionless in six dimensions), and the power counting indicates that the divergences are in the 2-point and 3-point functions. In order to calculate the one-loop $\\\\beta$-function, we need fist to calculate at this order the self-energy and vertex counterterms. Let us start with the self-energy, given by\n$$\n\\Sigma(p) \\equiv \\frac{\\mathfrak{i} \\lambda^{2} \\mu^{2 \\epsilon}}{2} \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{\\ell^{2}(\\ell+p)^{2}}=\\frac{\\mathfrak{i} \\lambda^{2} \\mu^{2 \\epsilon}}{2} \\int_{0}^{1} d x \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{[\\ell^{2}-\\Delta]^{2}}\n$$\nwhere $\\Delta \\equiv-x(1-x) p^{2}$ (note that the loop momentum $\\ell$ has been shifted in the last expression). Performing a Wick's rotation and using standard results for D-dimensional integral gives\n$$\n\\Sigma(p)=-\\frac{\\lambda^{2} \\mu^{2 \\epsilon}}{2(4 \\pi)^{\\mathrm{D} / 2}} \\Gamma(2-\\frac{\\mathrm{D}}{2}) \\int_{0}^{1} \\mathrm{~d} x \\Delta^{\\mathrm{D} / 2-2} .\n$$\n\nThen, define $D \\equiv 6-2 \\epsilon$, and set $\\epsilon$ to zero in all factors, except the $\\Gamma$ function that has a simple pole (with residue -1 ) and the factor $(\\mu^{2} / p^{2})^{\\epsilon}$ that carries the scale dependence. We arrive at\n$$\n\\Sigma(p)=-\\frac{\\lambda^{2} p^{2}}{12(4 \\pi)^{3} \\epsilon}[-\\frac{\\mu^{2}}{p^{2}}]^{\\epsilon}\n$$\n\nAt this order, the field renormalization constant is\n\n\\begin{equation*}\n\\mathrm{Z}=1+\\frac{\\partial \\Sigma}{\\partial \\mathrm{p}^{2}}=1-\\frac{\\lambda^{2}}{12(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon} \\tag{*}\n\\end{equation*}\n\nwhere we have chosen the renormalization point $p^{2}=-M^{2}$.\n\nLet us now turn our attention to the 3-point function. Denoting $p, q$ two of the momenta entering in the graph, the one-loop vertex correction reads\n\n\\begin{align}\n\\mu^{\\epsilon} \\Gamma(p, q) & =i \\lambda^{3} \\mu^{3 \\epsilon} \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{\\ell^{2}(\\ell+p)^{2}(\\ell-q)^{2}} \\\\\n& =2 i \\lambda^{3} \\mu^{3 \\epsilon} \\int_{\\substack{0 \\leq x, y \\leq 1 \\\\\nx+y \\leq 1}} d x d y \\int \\frac{d^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{1}{[\\ell^{2}-\\Delta]^{3}},\n\\end{align}\n\nwith $-\\Delta \\equiv x(1-x) p^{2}+y(1-y) q^{2}+2 x y p \\cdot q$. The reason why we denote the left hand side $\\mu^{\\epsilon} \\Gamma$ is that this loop integral is a correction to the 3-point coupling, that has dimension (mass) ${ }^{\\epsilon}$ in $\\mathrm{D}=6-2 \\epsilon$ dimensions. By writing it in this way, $\\Gamma$ is a correction to the dimensionless coupling $\\lambda$. After a Wick's rotation, the pole in this integral reads\n\n\\begin{equation*}\n\\Gamma(p, q)=\\frac{\\lambda^{3}}{2(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon} \\tag{**}\n\\end{equation*}\n\nwhere we have chosen the special kinematical configuration $p^{2}=q^{2}=(p+q)^{2}=-M^{2}$ as renormalization point.\n\nFrom the results $(*)$ and $(* *)$, we get the following wavefunction and vertex counterterms,\n$$\n\\delta_{z}=-\\frac{\\lambda^{2}}{12(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}, \\quad \\delta_{\\lambda}=-\\frac{\\lambda^{3}}{2(4 \\pi)^{3} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}\n$$\nand the $\\beta$-function in six dimensions is then given by\n$$\n\\beta=\\lim _{\\epsilon \\rightarrow 0} M \\frac{\\partial}{\\partial M}[\\frac{3 \\lambda^{3}}{8(4 \\pi)^{3} \\epsilon}(1-2 \\epsilon \\ln (\\frac{M}{\\mu})+\\mathcal{O}(\\epsilon^{2}))]=-\\frac{3 \\lambda^{3}}{4(4 \\pi)^{3}} .\n$$", + "final_answer": [ + "\\beta=-\\frac{3 \\lambda^{3}}{4(4 \\pi)^{3}}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\beta$": "The one-loop beta-function of the scalar field theory.", + "$\\lambda$": "The coupling constant of the cubic interaction." + } + }, + { + "id": 88, + "context": "", + "question": "Calculate the one-loop $\\beta$ function in quantum electrodynamics. How does the electromagnetic coupling strength vary with distance? What is the physical interpretation of this behaviour?", + "answer": "In order to follow the same procedure as in Exercise 10.1 for calculating the $\\beta$ function in QED, we would need to calculate the one-loop electron self-energy, the one-loop photon self-energy, and the one-loop correction to the electron-photon vertex. An alternative is to use the relation $Z_{1} Z_{2}^{-1} Z_{3}^{-1 / 2} e_{r}=e_{b}$ and the fact that $Z_{1}=Z_{2}$ due to Ward identities. Therefore, the relationship between the bare and renormalized couplings can also be obtained from the photon wavefunction renormalization, i.e. from a unique loop diagram. At one-loop, the photon polarization tensor is given by\n\n$$\\Pi^{\\mu u}(q)=-i e^{2} \\mu^{2 \\epsilon} \\int \\frac{d^{D} \\ell}{(2 \\pi)^{D}} \\frac{\\operatorname{tr}(\\gamma^{\\mu} \\ell \\gamma^{v}(\\ell+ot q))}{\\ell^{2}(\\ell+q)^{2}}$$\nFrom gauge invariance, we expect the tensor structure to be $\\Pi^{\\mu v}(q) \\equiv(g^{\\mu v} q^{2}-q^{\\mu} q^{v}) \\Pi(q^{2})$, and the function $\\Pi(q^{2})$ can be obtained from the trace, since $(D-1) q^{2} \\Pi(q^{2})=\\Pi_{\\mu}^{\\mu}(q)$, that reads\n\n\\begin{align}\n\\Pi_{\\mu}^{\\mu}(\\mathrm{q}) & =4(\\mathrm{D}-2) i e^{2} \\mu^{2 \\epsilon} \\int \\frac{\\mathrm{~d}^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{\\ell \\cdot(\\ell+\\mathrm{q})}{\\ell^{2}(\\ell+\\mathrm{q})^{2}} \\\\\n& =4(\\mathrm{D}-2) i e^{2} \\mu^{2 \\epsilon} \\int_{0}^{1} \\mathrm{dx} \\int \\frac{\\mathrm{~d}^{\\mathrm{D}} \\ell}{(2 \\pi)^{\\mathrm{D}}} \\frac{\\ell^{2}+\\Delta+\\text { terms odd in } \\ell}{[\\ell^{2}-\\Delta]^{2}}\n\\end{align}\n\nwhere $\\Delta \\equiv-x(1-x) q^{2}$, and the $\\ell$ in the last expression is a shifted integration variable. Performing a Wick's rotation and integrating over $\\ell$ gives\n\n\\begin{align}\n\\Pi(q^{2}) & =e^{2} \\mu^{2 \\epsilon} \\frac{4(\\mathrm{D}-2)}{(\\mathrm{D}-1) \\mathrm{q}^{2}(4 \\pi)^{\\mathrm{D} / 2}}[\\Gamma(1-\\frac{\\mathrm{D}}{2})-2 \\Gamma(2-\\frac{\\mathrm{D}}{2})] \\int_{0}^{1} \\mathrm{~d} x \\Delta^{\\mathrm{D} / 2-1} \\\\\n& =\\frac{4 \\mathrm{e}^{2}}{3(4 \\pi)^{2} \\epsilon}[-\\frac{\\mu^{2}}{\\mathrm{q}^{2}}]^{\\epsilon}\n\\end{align}\n\n\nThe photon wavefunction renormalization factor is\n$$\nZ_{3}=\\frac{1}{1+\\Pi}=1-\\Pi+\\mathcal{O}(e^{4})=1-\\frac{4 e^{2}}{3(4 \\pi)^{2} \\epsilon}[\\frac{\\mu^{2}}{M^{2}}]^{\\epsilon}+\\mathcal{O}(e^{4})\n$$\nthe last equality being at the point $q^{2}=-M^{2}$. From the relation $Z_{3}^{-1 / 2} e_{r}=e_{b}$, the scale dependence of the renormalized coupling constant can be determined from that of $Z_{3}$ (since the bare coupling is by definition scale independent),\n$$\n\\beta(e)=\\frac{e}{2 Z_{3}} M \\frac{\\partial Z_{3}}{\\partial M}=\\frac{e^{3}}{12 \\pi^{2}},\n$$\nwhich is often written as\n$$\n\\beta(\\alpha) \\equiv M \\frac{\\partial \\alpha}{\\partial M}=\\frac{2 \\alpha^{2}}{3 \\pi} .\n$$\nSince the $\\beta$ function of QED is positive, the coupling strength increases with the energy scale, or at shorter distances. Conversely, it decreases at larger distances. This can be understood physically as a screening phenomenon: a test charge polarizes the vacuum around it by arranging virtual electron-antielectron pairs in such a way that they screen the Coulomb potential seen at a distance. This is very similar to Debye screening in plasma physics (but there it is due to the polarization of on-shell charged particles, instead of vacuum fluctuations). As the probe of the electrical field approaches the test charge, the charge it sees is closer and closer to the naked charge of the test particle. For a point-like charge, the latter is infinite since it is not hidden by any screening.", + "final_answer": [ + "\\beta(e)=\\frac{e^{3}}{12 \\pi^{2}}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\beta(e)$": "Beta function for the electromagnetic coupling constant $e$, describing its running with the renormalization scale $M$.", + "$e$": "Electromagnetic coupling constant.", + "$\\pi$": "Pi, mathematical constant." + } + }, + { + "id": 89, + "context": "", + "question": "The special conformal transformation is given by $y^{\\mu}=\\frac{x^{\\mu}+b^{\\mu} x^{2}}{1+2 b \\cdot x+b^{2} x^{2}}$. For an infinitesimal 4-vector $b^{\\mu}$, this transformation can be expanded to first order in $b^{\\mu}$ as $y^{\\mu} \\approx x^{\\mu}+\\delta x^{\\mu}$, where $\\delta x^{\\mu} = (x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) b_{\\rho}$ (using $g^{\\mu\\rho}$ to raise the index of $b$). The generator $T_{\\rho}$ (associated with parameter $b^{\\rho}$) is defined such that the transformation on coordinates is $x'^{\\mu} = \\exp (i b^{\\rho} T_{\\rho}) x^{\\mu} \\approx x^{\\mu} + i b^{\\rho} (T_{\\rho} x^{\\mu})$, from which $\\delta x^{\\mu} = i b^{\\rho} (T_{\\rho} x^{\\mu})$. The provided text then gives an expression for an object $T^{\\mu}$ (note the upper index) as $T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}$. What is this expression for $T^{\\mu}$?", + "answer": "Given the infinitesimal transformation for $y^{\\mu}$:\n$$y^{\\mu}=\\frac{x^{\\mu}+b^{\\mu} x^{2}}{1+2 b \\cdot x+b^{2} x^{2}}$$\nFor infinitesimal $b^{\\mu}$, we expand $(1+2 b \\cdot x+b^{2} x^{2})^{-1} \\approx 1 - (2 b \\cdot x+b^{2} x^{2}) + \\mathcal{O}(b^2) \\approx 1 - 2 b \\cdot x + \\mathcal{O}(b^2)$.\nSo, $y^{\\mu} \\approx (x^{\\mu}+b^{\\mu} x^{2})(1-2b \\cdot x) + \\mathcal{O}(b^2)$\n$y^{\\mu} \\approx x^{\\mu} - 2(b \\cdot x)x^{\\mu} + b^{\\mu}x^2 + \\mathcal{O}(b^2)$.\nLet $b^{\\mu} = g^{\\mu\\sigma}b_{\\sigma}$. Then $b \\cdot x = b_{\\sigma}x^{\\sigma}$.\n$y^{\\mu} \\approx x^{\\mu} + (g^{\\mu\\sigma}x^2 - 2x^{\\mu}x^{\\sigma})b_{\\sigma} + \\mathcal{O}(b^2)$.\nSo, $\\delta x^{\\mu} = (x^2 g^{\\mu \\sigma} - 2x^{\\sigma}x^{\\mu})b_{\\sigma}$. (The problem statement uses $\\rho$ for the summed index of $b$).\n$\\delta x^{\\mu} = (x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) b_{\\rho}$.\nFrom the definition of how generators act on fields $\\Phi(x)$, an infinitesimal transformation $\\delta \\Phi = i \\epsilon^a (G_a \\Phi)$ involves the generator $G_a$. For coordinate transformations $x_\\mu \\rightarrow x_\\mu + \\delta x_\\mu$, where $\\delta x_\\mu = \\epsilon^a \\xi_{a \\mu}(x)$, the generators are often written as $G_a = -i \\xi_a^\nu (x) \\partial_\nu$. \nThe text states that from $\\delta x^{\\mu}$, we can read off the generator $T_{\\rho}$ such that $\\exp (i b^{\\rho} T_{\\rho}) x^{\\mu}}= x^{\\mu}+\\delta x^{\\mu}+\\cdots$.\nIt then provides the expression: $T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}$.\nThis expression has a free index $\\mu$ on $T$ and summation over $\\rho$ on the right hand side. This represents a set of four differential operators, one for each value of $\\mu=0,1,2,3$. This $T^{\\mu}$ is the specific quantity requested, as derived/stated in the source material.", + "final_answer": [ + "T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$T^{\\mu}$": "The specific differential operator representing a component of the special conformal generator, whose expression is requested.", + "$\\mathfrak{i}$": "Imaginary unit.", + "$x^{2}$": "Shorthand for the squared magnitude of the 4-vector $x$, defined as $x_{\\nu}x^{\\nu}$.", + "$g^{\\mu \\rho}$": "Metric tensor, used to raise and lower indices in Minkowski spacetime.", + "$x^{\\rho}$": "Original coordinate 4-vector.", + "$x^{\\mu}$": "Original coordinate 4-vector.", + "$\\partial_{\\rho}$": "Partial derivative operator with respect to the coordinate $x^{\\rho}$." + } + }, + { + "id": 90, + "context": "", + "question": "In Yang-Mills theory in the temporal gauge $A^{0}=0$ (with coupling $g=1$ for simplicity), what is the expression for the conjugate momentum $\\Pi_a^i$ of the gauge field component $A_a^i$?", + "answer": "By using the temporal gauge $A^{0}=0$, we circumvent the problem that $A_{a}^{0}$ has a vanishing conjugate momentum, because $A^{0}$ is not a dynamical variable in this gauge. Regarding the other components of the gauge potential, the conjugate momentum $\\Pi^{i}$ of $A^{i}$ is given by\n\n$$\\Pi_{a}^{i} \\equiv \\frac{\\partial \\mathcal{L}_{\\mathrm{YM}}}{\\partial \\partial_{0} \\mathcal{A}_{\\mathrm{a}}^{i}}=\\partial_{0} A_{\\mathrm{a}}^{i}$$", + "final_answer": [ + "\\Pi_{a}^{i} = \\partial_{0} A_{\\mathrm{a}}^{i}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\Pi_{a}^{i}$": "Conjugate momentum of the gauge field component $A_a^i$, defined as $\\Pi_{a}^{i} \\equiv \\frac{\\partial \\mathcal{L}_{\\mathrm{YM}}}{\\partial \\partial_{0} \\mathcal{A}_{\\mathrm{a}}^{i}}$.", + "$\\partial_{0}$": "Partial derivative with respect to the time coordinate.", + "$A_{\\mathrm{a}}^{i}$": "Gauge field component." + } + }, + { + "id": 91, + "context": "", + "question": "For Yang-Mills theory in the temporal gauge $A^{0}=0$ (with $g=1$), derive the Hamiltonian density $\\mathcal{H}$. Express it first in terms of the chromo-electric fields $E_a^i = \\Pi_a^i$ and chromo-magnetic fields $B_a^i = \\frac{1}{2} \\epsilon_{ijk} F_a^{jk}$, and then in terms of $\\Pi_a^i$ and $F_a^{ij}$.", + "answer": "It is convenient to introduce the chromo-electric and chromo-magnetic fields by\n$$\n\\mathrm{E}_{\\mathrm{a}}^{i} \\equiv \\mathrm{~F}_{\\mathrm{a}}^{0 i}=\\Pi_{a}^{i} \\quad(\\text { in } A^{0}=0 \\text { gauge }), \\quad B_{a}^{i} \\equiv \\frac{1}{2} \\epsilon_{i j k} F_{a}^{j k}\n$$\nin terms of which the Lagrangian density can be written as follows,\n\n\\begin{align}\n-\\frac{1}{4} F_{a}^{\\mu \\nu} F_{\\mu v}^{a} & =-\\frac{1}{4}(-F_{a}^{O i} F_{a}^{O i}-F_{a}^{i 0} F_{a}^{i O}+F_{a}^{i j} F_{a}^{i j})=\\frac{1}{4}(2 E_{a}^{i} E_{a}^{i}-\\epsilon_{i j k} \\epsilon_{i j l} B_{a}^{k} B_{a}^{l}) \\\n& =\\frac{1}{4}(2 E_{a}^{i} E_{a}^{i}-(\\delta_{j j} \\delta_{k l}-\\delta_{j k} \\delta_{j l}) B_{a}^{k} B_{a}^{l})=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}-B_{a}^{i} B_{a}^{i}) .\n\\end{align}\n\nThen, the Hamiltonian is given by\n\n$$\\mathcal{H}=\\underbrace{\\Pi_{a}^{i}(\\partial_{0} A_{a}^{i})}_{E_{a}^{i} E_{a}^{i}}-\\mathcal{L}=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}+B_{a}^{i} B_{a}^{i})=\\frac{1}{2} \\Pi_{a}^{i} \\Pi_{a}^{i}+\\frac{1}{4} F_{a}^{i j} F_{a}^{i j} .$$", + "final_answer": [ + "\\mathcal{H}=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}+B_{a}^{i} B_{a}^{i}) = \\frac{1}{2} \\Pi_{a}^{i} \\Pi_{a}^{i}+\\frac{1}{4} F_{a}^{i j} F_{a}^{i j}" + ], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathcal{H}$": "Hamiltonian density of the Yang-Mills theory.", + "$E_a^i$": "Chromo-electric field, defined as $E_a^i \\equiv F_a^{0i} = \\Pi_a^i$ in the $A^0=0$ gauge.", + "$B_a^i$": "Chromo-magnetic field, defined as $B_a^i \\equiv \\frac{1}{2} \\epsilon_{ijk} F_a^{jk}$.", + "$\\Pi_a^i$": "Canonical momentum conjugate to the spatial component of the gauge field $A_a^i$, identified with the chromo-electric field $E_a^i$.", + "$F_a^{ij}$": "Spatial components of the Yang-Mills field strength tensor." + } + }, + { + "id": 92, + "context": "", + "question": "A point charge $e$ is located at point $O$ near a system of grounded conductors, inducing charges $e_{a}$ on these conductors. If the charge $e$ is absent and one of the conductors (the $a$-th) has a potential $\\varphi_{a}^{\\prime}$ (with the remaining conductors still grounded), then the potential at point $O$ is $\\varphi_{0}^{\\prime}$. Express the charge $e_{a}$ in terms of $\\varphi_{a}^{\\prime}$ and $\\varphi_{0}^{\\prime}$.", + "answer": "If the charge $e_{a}$ on the conductor gives the conductor a potential $\\varphi_{a}$, while the charge $e_{a}^{\\prime}$ gives the conductor a potential $\\varphi_{a}^{\\prime}$, then we have:\n\n$$\\sum_{a} \\varphi_{a} e_{a}^{\\prime}=\\sum_{a, b} \\varphi_{a} C_{a b} \\varphi_{b}^{\\prime}=\\sum_{a} \\varphi_{a}^{\\prime} e_{a}$$\n\n\nWe apply this relationship to the two states of a system consisting of all conductors and a point charge $e$ (the latter viewed as a limiting case of a small-sized conductor). In one state, there is charge $e$ with charge on the conductors being $e_{a}$ and the potential $\\varphi_{a}=0$. In another state, the charge $e=0$, but one conductor has a potential $\\varphi_{a}^{\\prime} \\neq 0$. Thus we get $e \\varphi_{0}^{\\prime}+e_{a} \\varphi_{a}^{\\prime}=0$, from which follows\n\n$$e_{a}=-\\frac{e \\varphi_{0}^{\\prime}}{\\varphi_{a}^{\\prime}}$$\n\n\nFor example, if the charge $e$ is at a distance $r(r>a)$ from the center of a grounded spherical conductor of radius $a$, then $\\varphi_{0}^{\\prime}=\\varphi_{a}^{\\prime} \\frac{a}{r}$, and the induced charge on the sphere is\n$$\ne_{a}=-\\frac{e a}{r}\n$$\n\nAs another example, consider a charge $e$ between two grounded concentric spheres of radii $a$ and $b$ respectively (the charge is at a distance $r$ from the center, where $a\\Delta$, where $\\Delta$ is a distance such that $a \\ll \\Delta \\ll b$. Hence, when $r<\\Delta$, it can be assumed that this section of the ring is straight, yielding\n$$\n\\int_{\\Delta>r} \\frac{\\mathrm{~d} l}{r}=\\int_{-\\Delta}^{\\Delta} \\frac{\\mathrm{d} l}{\\sqrt{l^{2}+a^{2}}} \\approx 2 \\ln \\frac{2 \\Delta}{a} .\n$$\n\nIn the region $r>\\Delta$, the thickness of the wire can be neglected, which is to say $r$ is assumed to be the distance between two points along the axis of the ring. Thus\n$$\n\\int_{r>\\Delta} \\frac{\\mathrm{d} l}{r}=2 \\int_{\\varphi_{0}}^{\\pi} \\frac{b \\mathrm{~d} \\varphi}{2 b \\sin (\\varphi / 2)}=-2 \\ln \\tan \\frac{\\varphi_{0}}{4}\n$$\n\nIn this expression, $\\varphi$ is the angle subtended by chord $r$ at the center of the ring, and the lower limit of integration is derived from $2 b \\sin (\\frac{\\varphi_{0}}{2})=\\Delta$, yielding $\\varphi_{0} \\approx \\Delta / b$. Adding the two parts of the integral, the quantity $\\Delta$ cancels automatically, and finally, the expression for the capacitance $C$ of the ring is given by\n\n$$C=\\frac{\\pi b}{\\ln (8 b / a)}$$", + "final_answer": [ + "C=\\frac{\\pi b}{\\ln (8 b / a)}" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$C$": "Capacitance of the circular ring", + "$b$": "Radius of the circular ring", + "$a$": "Radius of the wire cross-section" + } + }, + { + "id": 94, + "context": "", + "question": "An infinitely long cylindrical conductor with radius $R$ is immersed in a uniform transverse electric field with strength $\\mathfrak{C}$. Find the potential distribution $\\varphi(r, \\theta)$ outside the cylinder.", + "answer": "Using polar coordinates in the plane perpendicular to the axis of the cylinder. Thus, the solution to the two-dimensional Laplace equation involving a constant vector is\n$$\n\\varphi_{1}=\\text { const } \\cdot \\mathfrak{C} \\cdot \\nabla \\ln r=\\text { const } \\cdot \\frac{\\mathfrak{C} \\cdot \\boldsymbol{r}}{r^{2}} .\n$$\n\nAdding the above with $\\varphi_{0}=-\\boldsymbol{r} \\cdot \\mathfrak{C}$ and letting const $=R^{2}$, we obtain\n$$\n\\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}) .\n$$", + "final_answer": [ + "\\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}})" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "Total potential distribution outside the cylinder.", + "$\\mathfrak{C}$": "Uniform transverse electric field vector.", + "$r$": "Radial coordinate in polar coordinates.", + "$\\theta$": "Angular coordinate in polar coordinates.", + "$R$": "Radius of the infinitely long cylindrical conductor." + } + }, + { + "id": 95, + "context": "", + "question": "An infinitely long conducting cylinder with a radius $R$ is immersed in a uniform transverse electric field with a strength of $\\mathfrak{C}$. Find the induced surface charge density $\\sigma(\\theta)$ on the surface of the cylinder.", + "answer": "The potential is \n\\begin{equation}\n \\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}).\n\\end{equation}\n\nThus, the surface charge density is\n\n$$\\sigma= -\\frac{1}{4\\pi} \\frac{\\partial \\varphi}{\\partial r} = \\frac{\\mathfrak{C}}{2 \\pi} \\cos \\theta. $$", + "final_answer": [ + "\\sigma=\\frac{\\mathfrak{C}}{2 \\pi} \\cos \\theta" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Induced surface charge density on the surface of the cylinder", + "$\\mathfrak{C}$": "Strength of the uniform transverse electric field", + "$\\pi$": "Mathematical constant pi", + "$\\theta$": "Angular position (polar angle) in cylindrical coordinates" + } + }, + { + "id": 96, + "context": "", + "question": "An infinitely long conducting cylinder with a radius of $R$ is immersed in a uniform transverse electric field with a strength of $\\mathfrak{C}$. Find the induced dipole moment $\\mathscr{P}$ per unit length of the cylinder.", + "answer": "The potential is \n\\begin{equation*}\n \\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}}).\n\\end{equation*}\n\nThe dipole moment $\\mathscr{P}$ per unit length of the cylinder can be determined by comparing $\\varphi$ with the potential of a two-dimensional dipole field, which is given by\n\n$$2 \\mathscr{P} \\cdot \\nabla \\ln r=\\frac{2 \\mathscr{P} \\cdot \\boldsymbol{r}}{r^{2}},$$\n\n\nTherefore, $\\mathscr{P}=\\mathfrak{C} R^{2} / 2$.", + "final_answer": [ + "\\mathscr{P}=\\mathfrak{C} R^{2} / 2" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{P}$": "Induced dipole moment per unit length of the cylinder.", + "$\\mathfrak{C}$": "Strength of the uniform transverse electric field.", + "$R$": "Radius of the infinitely long conducting cylinder." + } + }, + { + "id": 97, + "context": "", + "question": "Determine the attraction energy between an electric dipole and a planar conductor surface.", + "answer": "Choose the $x$-axis perpendicular to the conductor surface, passing through the point where the dipole is located; let the dipole moment vector $\\mathscr{P}$ lie in the $xy$ plane. The 'image' of the dipole is at the point $-x$ and has a dipole moment $\\mathscr{P}_{x}^{\\prime}=\\mathscr{P}_{x}, \\mathscr{P}_{y}^{\\prime}=-\\mathscr{P}_{y}$. Therefore, the required attraction energy is half the interaction energy between the dipole and its 'mirror image', and it equals\n$$\\mathscr{U}= \\frac{1}{2} \\left[ -\\frac{1}{(2x)^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2}) \\right] = -\\frac{1}{16 x^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2}) .$$", + "final_answer": [ + "\\mathscr{U}=-\\frac{1}{16 x^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2})" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{U}$": "Attraction energy between the electric dipole and the planar conductor surface.", + "$x$": "Distance from the electric dipole to the planar conductor surface.", + "$\\mathscr{P}_{x}$": "x-component of the dipole moment vector $\\mathscr{P}$.", + "$\\mathscr{P}_{y}$": "y-component of the dipole moment vector $\\mathscr{P}$." + } + }, + { + "id": 98, + "context": "", + "question": "Try to find an expression for the electric dipole moment of a conductive thin cylindrical rod (length $2l$, radius $a$, where $a \\ll l$) placed in an electric field $\\mathfrak{C}$, expressed in terms of the parameter $L=\\ln (2 l / a)-1$. The electric field is parallel to the axis of the rod. \n\nYou should work in Gaussian units and only keep the leading and next-to-leading terms in the small parameter $1 / L$.", + "answer": "Let $\\tau(z)$ be the charge induced per unit length on the rod's surface; $z$ is the coordinate along the axis of the cylindrical rod, with the origin at the midpoint of the rod axis. The condition of constant potential on the conductor surface takes the form\n$$-\\mathfrak{C} z+\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\int_{-l}^{l} \\frac{\\tau(z^{\\prime}) \\mathrm{d} z^{\\prime} \\mathrm{d} \\varphi}{R}=0, \\quad R=[(z-z)^{2}+4 a^{2} \\sin ^{2} \\frac{\\varphi}{2}]^{1 / 2},$$\n\nwhere $\\varphi$ is the angle between planes through the rod axis and points on its surface separated by distance $R$. In the integral, using the identity $\\tau(z^{\\prime})=\\tau(z)+[\\tau(z^{\\prime})-\\tau(z)]$, the integration is divided into two parts. In the first part of the integration, noting $l \\gg a$, for points not very close to the ends of the rod, we have\n$$\\frac{\\tau(z)}{2 \\pi} \\iint \\frac{\\mathrm{~d} z^{\\prime} \\mathrm{d} \\varphi}{R} \\approx \\frac{\\tau(z)}{2 \\pi} \\int_{0}^{2 \\pi} \\ln \\frac{l^{2}-z^{2}}{a^{2} \\sin ^{2}(\\varphi / 2)} \\mathrm{d} \\varphi=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}$$\n\n(using the known integral value $\\int_{0}^{\\pi} \\ln \\sin \\varphi \\mathrm{d} \\varphi=-\\pi \\ln 2$). In the second part of the integral, containing the difference $\\tau(z^{\\prime})-\\tau(z)$, terms containing $a^{2}$ in $R$ can be neglected as they do not cause the integral to diverge. Thus,\n$$\n\\mathfrak{C} z=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}+\\int_{-l}^{l} \\frac{\\tau(z^{\\prime})-\\tau(z)}{|z^{\\prime}-z|} \\mathrm{d} z^{\\prime}.\n$$\nThe dependence of $\\tau$ on $z$ is essentially proportional to $z$; in this approximation, the integral in the above equation gives $-2 \\tau(z)$, resulting in\n$$\n\\tau(z)=\\frac{\\mathfrak{C} z}{\\ln [4(l^{2}-z^{2}) / a^{2}]-2}\n$$\n\nThis expression is not applicable near the end points of the rod, but the region of $z$ values for calculating the requested dipole moment is not significant. At the precision adopted here, we have\n\n\\begin{align}\n\\mathscr{P} & =\\int_{-l}^{l} \\tau(z) z \\mathrm{~d} z=\\frac{\\mathfrak{C}}{L} \\int_{0}^{l}[z^{2}-\\frac{z^{2}}{2 L} \\ln (1-\\frac{z^{2}}{l^{2}})] \\mathrm{d} z \\\\\n& =\\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]\n\\end{align}\n\n(where $L=\\ln (2 l / a)-1$ is a large number).", + "final_answer": [ + "\\mathscr{P} = \\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{P}$": "Electric dipole moment of the rod.", + "$\\mathfrak{C}$": "External electric field, parallel to the axis of the rod.", + "$l$": "Half-length of the cylindrical rod.", + "$L$": "Dimensionless parameter related to the rod's geometry, defined as $L=\\ln (2 l / a)-1$." + } + }, + { + "id": 99, + "context": "", + "question": "Attempt to find another approximate expression for the electric dipole moment of a conductive thin cylindrical rod (length $2l$, radius $a$, and $a \\ll l$) in an electric field $\\mathfrak{C}$, expressed directly in the logarithmic form of $l$ and $a$ $(\\ln (4l/a))$. The field is parallel to the rod's axis. \n\nYou should work in Gaussian units and only keep the leading and next-to-leading terms in the small parameter $1 / \\ln (4 l / a)$.", + "answer": "Let $\\tau(z)$ be the induced charge per unit length on the surface of the rod; $z$ is the coordinate along the axis of the cylinder, with the origin chosen at the midpoint of the rod. The condition of constant potential on the conductor's surface is\n\n$$-\\mathfrak{C} z+\\frac{1}{2 \\pi} \\int_{0}^{2 \\pi} \\int_{-l}^{l} \\frac{\\tau(z^{\\prime}) \\mathrm{d} z^{\\prime} \\mathrm{d} \\varphi}{R}=0, \\quad R=[(z-z)^{2}+4 a^{2} \\sin ^{2} \\frac{\\varphi}{2}]^{1 / 2},\n$$\n\nwhere $\\varphi$ is the angle between the plane through the rod's axis and two points on its surface separated by distance $R$. In the integral, use the identity $\\tau(z^{\\prime})=\\tau(z)+[\\tau(z^{\\prime})-\\tau(z)]$, splitting the integral into two parts. In the first part, noting that $l \\gg a$, for points not very close to the ends of the rod, we have\n$$\n\\frac{\\tau(z)}{2 \\pi} \\iint \\frac{\\mathrm{~d} z^{\\prime} \\mathrm{d} \\varphi}{R} \\approx \\frac{\\tau(z)}{2 \\pi} \\int_{0}^{2 \\pi} \\ln \\frac{l^{2}-z^{2}}{a^{2} \\sin ^{2}(\\varphi / 2)} \\mathrm{d} \\varphi=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}\n$$\n(using the known integral result $\\int_{0}^{\\pi} \\ln \\sin \\varphi \\mathrm{d} \\varphi=-\\pi \\ln 2$). In the second part of the integral, containing the difference $\\tau(z^{\\prime})-\\tau(z)$, terms involving $a^{2}$ in $R$ can be omitted as they do not cause divergence of the integral. Thus,\n$$\n\\mathfrak{C} z=\\tau(z) \\ln \\frac{4(l^{2}-z^{2})}{a^{2}}+\\int_{-l}^{l} \\frac{\\tau(z^{\\prime})-\\tau(z)}{|z^{\\prime}-z|} \\mathrm{d} z^{\\prime} .\n$$\nThe dependence of $\\tau$ on $z$ essentially reduces to being proportional to $z$; in this approximation, the integral in the above expression gives $-2 \\tau(z)$, resulting in\n$$\n\\tau(z)=\\frac{\\mathfrak{C} z}{\\ln [4(l^{2}-z^{2}) / a^{2}]-2}\n$$\n\nNear the endpoints of the rod, this expression is not applicable, but for the calculation of the desired dipole moment, this region of $z$-values is not important. To the precision adopted here, we have\n\n\\begin{aligned}\n\\mathscr{P} & =\\int_{-l}^{l} \\tau(z) z \\mathrm{~d} z=\\frac{\\mathfrak{C}}{L} \\int_{0}^{l}[z^{2}-\\frac{z^{2}}{2 L} \\ln (1-\\frac{z^{2}}{l^{2}})] \\mathrm{d} z \\\\\n& =\\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]\n\\end{aligned}\n\n(where $L=\\ln (2 l / a)-1$ is a large number), or to the same precision\n$$\n\\mathscr{P}=\\frac{\\mathfrak{C} l^{3}}{3[\\ln (4 l / a)-7 / 3]} .\n$$\nOnly preserving the leading and next-to-leading terms in the small parameter $1 / \\ln (4 l / a)$, we arrive at the final expression\n$$\n\\mathscr{P}=\\frac{\\mathfrak{C} l^{3}}{3 \\ln (4 l / a)} (1 + \\frac{7}{3 \\ln (4 l / a)})\n$$", + "final_answer": [ + "\\mathscr{P}=\\frac{\\mathfrak{C} l^{3}}{3 \\ln (4 l / a)} (1 + \\frac{7}{3 \\ln (4 l / a)})" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{P}$": "Electric dipole moment of the conductive thin cylindrical rod.", + "$\\mathfrak{C}$": "Electric field, parallel to the rod's axis.", + "$l$": "Half the length of the conductive thin cylindrical rod.", + "$a$": "Radius of the conductive thin cylindrical rod." + } + }, + { + "id": 100, + "context": "", + "question": "Under the influence of a uniform external electric field, consider an uncharged ellipsoid. When the external electric field is only along the $x$ axis of the ellipsoid, find the charge distribution on its surface $\\sigma$.", + "answer": "We first have\n$$\n\\sigma=-\\frac{1}{4 \\pi} \\frac{\\partial \\varphi}{\\partial n}|_{\\xi=0}=-(\\frac{1}{4 \\pi h_{1}}-\\frac{\\partial \\varphi}{\\partial \\xi})_{\\xi=0}\n$$\n(According to equation \n\\begin{align}\nd l^2 &= h_1^2 d \\xi^2 + h_2^2 d \\eta^2 + h_3^2 d \\zeta^2, \\\\\nh_1 &= \\frac{\\sqrt{(\\xi - \\eta)(\\xi - \\zeta)}}{2 R_\\xi}, \\quad \nh_2 = \\frac{\\sqrt{(\\eta - \\zeta)(\\eta - \\xi)}}{2 R_\\eta}, \\quad \nh_3 = \\frac{\\sqrt{(\\zeta - \\xi)(\\xi - \\eta)}}{2 R_\\zeta},\n\\end{align}\nthe length element along the normal direction of the ellipsoid surface is $h_{1} \\mathrm{~d} \\xi$.) With the help of equation \n\\begin{align}\n\\varphi = \\varphi_0 \\left\\{ 1 - \\frac{\\displaystyle \\int_{\\xi}^{\\infty} \\frac{ds}{(s + a^2) R_s}}{\\displaystyle \\int_{0}^{\\infty} \\frac{ds}{(s + a^2) R_s}}, \\right\\}\n\\end{align}\nand considering\n$$\n\\nu_{x}=\\frac{1}{h_{1}} \\frac{\\partial x}{\\partial \\xi}|_{\\xi=0}=\\frac{x}{2 a^{2} h_{1}}|_{\\xi=0},\n$$\n\nwhen the external electric field is along the x-axis direction of the ellipsoid, we obtain\n$$\n\\sigma=\\mathfrak{C} \\frac{\\nu_{x}}{4 \\pi n^{(x)}}\n$$", + "final_answer": [ + "\\sigma=\\mathfrak{C} \\frac{\\nu_{x}}{4 \\pi n^{(x)}}" + ], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Charge distribution on the surface of the ellipsoid.", + "$\\mathfrak{C}$": "A constant introduced in the final expression for $\\sigma$.", + "$\\nu_x$": "A quantity related to the x-component of the normal vector, defined as $\\nu_{x}=\\frac{1}{h_{1}} \\frac{\\partial x}{\\partial \\xi}|_{\\xi=0}$.", + "$\\pi$": "Mathematical constant pi.", + "$n^{(x)}$": "A normalization factor or a specific component related to the normal vector in the x-direction, appearing in the final expression for $\\sigma$." + } + }, + { + "id": 101, + "context": "", + "question": "Consider an uncharged ellipsoid subjected to a uniform external electric field. When the external electric field is oriented in any direction relative to the ellipsoid's $x, y, z$ axes, find the charge distribution $\\sigma$ on its surface.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Surface charge distribution on the ellipsoid.", + "$\\nu_i$": "Components of a vector representing the normal direction or related geometric factors.", + "$n^{-1}_{ik}$": "Inverse of a tensor related to depolarization factors.", + "$\\mathfrak{C}_k$": "Components of the external electric field vector, used in the final answer.", + "$\\nu_x$": "A geometric factor representing the x-component of the normal direction, defined as $\\left.\\frac{1}{h_1}\\frac{\\partial x}{\\partial \\xi}\\right|_{\\xi=0}$.", + "$n^{(x)}$": "A depolarization factor (or similar geometric factor) specific to the x-direction.", + "$\\mathfrak{C}_x$": "x-component of the external electric field vector, used in the final answer.", + "$\\nu_y$": "A geometric factor representing the y-component of the normal direction.", + "$n^{(y)}$": "A depolarization factor (or similar geometric factor) specific to the y-direction.", + "$\\mathfrak{C}_y$": "y-component of the external electric field vector, used in the final answer.", + "$\\nu_z$": "A geometric factor representing the z-component of the normal direction.", + "$n^{(z)}$": "A depolarization factor (or similar geometric factor) specific to the z-direction.", + "$\\mathfrak{C}_z$": "z-component of the external electric field vector, used in the final answer." + } + }, + { + "id": 102, + "context": "", + "question": "For a prolate rotational ellipsoid conductor, find the external potential $\\varphi$ when its symmetry axis is perpendicular to the external field (specifically referring to the scenario described in the solution where the field is in the $z$ axis direction).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "External potential around the conductor.", + "$\\mathfrak{C}$": "A constant related to the strength of the uniform external electric field.", + "$z$": "The Cartesian coordinate along the axis of the external electric field.", + "$\\xi$": "A coordinate in the prolate spheroidal coordinate system.", + "$a$": "Semi-major axis of the prolate ellipsoid.", + "$b$": "Semi-minor axis of the prolate ellipsoid." + } + }, + { + "id": 103, + "context": "", + "question": "For an oblate rotating ellipsoidal conductor, when its symmetry axis is perpendicular to the external field (specifically referring to the case described in the solution where the field is in the $x$ axis direction), find the potential $\\varphi$ outside the conductor.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "Potential outside the conductor.", + "$\\mathfrak{c}$": "A constant related to the magnitude of the external electric field.", + "$x$": "Cartesian coordinate, specifically the direction of the external field.", + "$a$": "Equatorial semi-axis length of the oblate ellipsoid.", + "$c$": "Polar semi-axis length of the oblate ellipsoid.", + "$\\xi$": "Ellipsoidal coordinate, one of the coordinates in the oblate spheroidal coordinate system." + } + }, + { + "id": 104, + "context": "", + "question": "Consider a uniform electric field of magnitude \\( \\mathfrak{C} \\) existing along the positive z-axis in the half-space \\( z<0 \\) (i.e., at \\( z \\to -\\infty \\), the electric field is \\( \\vec{E} = \\mathfrak{C}\\hat{k} \\), corresponding to the potential \\( \\varphi = -\\mathfrak{C}z \\)). This electric field is constrained by a grounded conductive plane with a circular hole of radius \\( a \\) centered at the origin, \\( z=0 \\). Determine the expression for the potential \\( \\varphi \\) throughout the entire space (which can be expressed in either oblate spheroidal coordinates \\( \\xi, \\eta \\) or Cartesian coordinates \\( z \\)).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "Electric potential.", + "$\\mathfrak{C}$": "Magnitude of the uniform electric field.", + "$z$": "Cartesian coordinate along the z-axis.", + "$a$": "Radius of the circular hole.", + "$\\xi$": "Oblate spheroidal coordinate.", + "$\\eta$": "Oblate spheroidal coordinate." + } + }, + { + "id": 105, + "context": "", + "question": "In the same physical scenario as the previous sub-question (that is, in the half-space $z<0$, there exists a uniform electric field $\\mathfrak{C}$ along the positive $z$-axis, constrained by a grounded conductive plane at $z=0$ with a circular hole of radius $a$), the expression for the electric potential $\\varphi$ is known to be $\\varphi=-\\mathfrak{C} \\frac{z}{\\pi}[\\arctan \\frac{a}{\\sqrt{\\xi}}-\\frac{a}{\\sqrt{\\xi}}]$. Try to find the expression for the surface charge density $\\sigma$ on the lower side of the conductive plane ($z=0^-, \\rho > a$) in terms of the cylindrical radial distance $\\rho$ and hole radius $a$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Surface charge density.", + "$\\mathfrak{C}$": "Magnitude of the uniform electric field along the positive z-axis.", + "$a$": "Radius of the circular hole.", + "$\\rho$": "Cylindrical radial distance." + } + }, + { + "id": 106, + "context": "", + "question": "Assume a charged spherical conductor is cut in half, try to determine the mutual repulsive force between the two hemispheres.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$F$": "Mutual repulsive force between the two hemispheres", + "$a$": "Radius of the sphere", + "$C$": "Constant appearing in the final answer", + "$\\varphi$": "Electric potential appearing in the final answer", + "$r$": "Distance or radius appearing in the final answer" + } + }, + { + "id": 107, + "context": "", + "question": "A spherical conductor is cut into two halves, determine the mutual repulsive force between the two hemispheres. The conductor sphere is uncharged and is in a uniform external electric field $\\mathfrak{C}$ perpendicular to the interface.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$F$": "Mutual repulsive force between the two hemispheres", + "$a$": "Radius of the spherical conductor", + "$\\mathfrak{C}$": "Uniform external electric field" + } + }, + { + "id": 108, + "context": "", + "question": "For waves propagating on the charged surface of a liquid conductor in a gravitational field, determine the stability conditions of this surface.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma_0$": "Surface charge density of the liquid conductor", + "$g$": "Gravitational acceleration", + "$\\rho$": "Density of the liquid conductor", + "$\\alpha$": "Surface tension coefficient of the liquid", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 109, + "context": "", + "question": "Find the stability condition for a charged spherical droplet with respect to small deformations.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$e$": "Total charge of the spherical droplet.", + "$a$": "Radius of the spherical droplet.", + "$\\alpha$": "Surface tension coefficient." + } + }, + { + "id": 110, + "context": "", + "question": "Find the stability condition (Rayleigh, 1882) of a charged spherical droplet relative to splitting into two identical smaller droplets (large deformation). Assume the original droplet has a charge of $e$ and a radius of $a$, while each smaller droplet after splitting has a charge of $e/2$ and a radius of $a/2^{1/3}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$e$": "Charge of the original spherical droplet.", + "$a$": "Radius of the original spherical droplet.", + "$\\alpha$": "Surface tension coefficient." + } + }, + { + "id": 111, + "context": "", + "question": "An infinitely long straight charged wire (with charge per unit length of $e$) is parallel to the interface between two media with different dielectric constants ($\\varepsilon_1$ and $\\varepsilon_2$ respectively), and the distance from the interface is $h$. Determine the potential $\\varphi_1$ in medium 1 ($\\varepsilon_1$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi_1$": "Potential in medium 1.", + "$e$": "Charge per unit length of the infinitely long straight charged wire.", + "$\\varepsilon_1$": "Dielectric constant of medium 1.", + "$r$": "Distance from the observation point to the original wire.", + "$\\varepsilon_2$": "Dielectric constant of medium 2.", + "$r^{\\prime}$": "Distance from the observation point to the image wire." + } + }, + { + "id": 112, + "context": "", + "question": "An infinitely long straight conductor (with a linear charge of $e$) is parallel to the interface between two media with different dielectric constants ($\\varepsilon_1$ and $\\varepsilon_2$ respectively) and at a distance $h$ from the interface. Determine the potential $\\varphi_2$ within medium 2 ($\\varepsilon_2$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi_2$": "Potential within medium 2", + "$e$": "Linear charge density of the infinitely long straight conductor", + "$\\varepsilon_1$": "Dielectric constant of the first medium", + "$\\varepsilon_2$": "Dielectric constant of the second medium", + "$r$": "Distance from the observation point to the original wire" + } + }, + { + "id": 113, + "context": "", + "question": "Find the torque $K$ acting on a rotational ellipsoid in a uniform electric field $\\mathfrak{C}$, with a dielectric constant of $\\varepsilon$. The volume of the ellipsoid is $V$, $\\alpha$ is the angle between the direction of $\\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization coefficient along this axis.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$K$": "Torque acting on a rotational ellipsoid.", + "$\\varepsilon$": "Dielectric constant of the ellipsoid.", + "$n$": "Depolarization coefficient along the symmetry axis of the ellipsoid.", + "$V$": "Volume of the ellipsoid.", + "$\\alpha$": "Angle between the direction of the electric field $\\mathfrak{C}$ and the symmetry axis of the ellipsoid.", + "$\\mathfrak{C}$": "Uniform electric field." + } + }, + { + "id": 114, + "context": "", + "question": "For a conductive rotating ellipsoid ($\\varepsilon \\rightarrow \\infty$), in a uniform electric field $\\mathfrak{C}$, find the torque $K$ acting on it. The volume of the ellipsoid is $V$, $\\alpha$ is the angle between the direction of $\\mathfrak{C}$ and the symmetry axis of the ellipsoid, and $n$ is the depolarization factor along this axis.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$K$": "Torque acting on the ellipsoid.", + "$n$": "Depolarization factor along the symmetry axis of the ellipsoid.", + "$V$": "Volume of the ellipsoid.", + "$\\mathfrak{C}$": "Uniform external electric field.", + "$\\alpha$": "Angle between the direction of the electric field $\\mathfrak{C}$ and the symmetry axis of the ellipsoid." + } + }, + { + "id": 115, + "context": "", + "question": "A hollow dielectric sphere (dielectric constant $\\varepsilon$, with inner and outer radii $b$ and $a$, respectively) is placed in a uniform external electric field $\\mathfrak{E}$. Determine the field inside the cavity of the sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\boldsymbol{E}_3$": "Electric field inside the cavity of the sphere", + "$\\mathfrak{C}$": "Symbol used in the final answer, representing the external electric field (likely a typo for $\\mathfrak{E}$)", + "$\\varepsilon$": "Dielectric constant of the hollow dielectric sphere", + "$b$": "Inner radius of the hollow dielectric sphere", + "$a$": "Outer radius of the hollow dielectric sphere" + } + }, + { + "id": 116, + "context": "", + "question": "Determine the height $h$ by which the liquid surface inside a vertical parallel-plate capacitor rises.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$h$": "Height by which the liquid surface inside a vertical parallel-plate capacitor rises", + "$\\varepsilon$": "Dielectric constant (or relative permittivity) of the liquid", + "$\\rho$": "Density of the liquid", + "$g$": "Acceleration due to gravity", + "$E$": "Electric field strength between the capacitor plates", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 117, + "context": "", + "question": "If the object is not in a vacuum, but in a medium with a dielectric constant of $\\varepsilon^{(e)}$, find the formula of $\\mathscr{F}-\\mathscr{F}_{0}$.\nIf the answer exists in an integral, then find the integrand", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathfrak{C}$": "A vector field, used in the integral expression for the free energy difference.", + "$\\boldsymbol{D}$": "Electric displacement field.", + "$\\varepsilon^{(e)}$": "Dielectric constant of the medium.", + "$\\boldsymbol{E}$": "Electric field." + } + }, + { + "id": 118, + "context": "", + "question": "Consider a capacitor composed of two conducting surfaces separated by a distance $h$, with $h$ being smaller than the dimensions of the capacitor plates. The space between the capacitor plates is filled with a material of dielectric constant $\\varepsilon_{1}$. A small sphere with radius $a \\ll h$ and dielectric constant $\\varepsilon_{2}$ is placed inside the capacitor. Determine the change in the capacitance of the capacitor.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$C$": "Total capacitance of the capacitor with the sphere.", + "$C_{0}$": "Original capacitance of the capacitor in the absence of the sphere.", + "$a$": "Radius of the small sphere.", + "$h$": "Distance between the capacitor plates.", + "$\\varepsilon_{1}$": "Dielectric constant of the material filling the space between the capacitor plates.", + "$\\varepsilon_{2}$": "Dielectric constant of the small sphere." + } + }, + { + "id": 119, + "context": "", + "question": "Try to determine the potential $\\varphi$ produced by a point charge $e$ inside an anisotropic homogeneous medium (with the point charge located at the origin and the principal axes of the dielectric tensor $\\varepsilon_{ik}$ aligned along the $x, y, z$ axes), and express it using the principal dielectric constants $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "Electric potential produced by the point charge.", + "$e$": "Magnitude of the point charge.", + "$\\varepsilon^{(x)}$": "Principal dielectric constant along the x-axis.", + "$\\varepsilon^{(y)}$": "Principal dielectric constant along the y-axis.", + "$\\varepsilon^{(z)}$": "Principal dielectric constant along the z-axis.", + "$x$": "Spatial coordinate along the x-axis.", + "$y$": "Spatial coordinate along the y-axis.", + "$z$": "Spatial coordinate along the z-axis." + } + }, + { + "id": 120, + "context": "", + "question": "Determine the potential $\\varphi$ generated by a point charge $e$ in an anisotropic homogeneous medium, using tensor notation that does not depend on the choice of coordinate system.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi$": "Potential generated by a point charge.", + "$e$": "Point charge.", + "$|\\varepsilon|$": "Determinant of the dielectric tensor $\\varepsilon_{i k}$, given by $|\\varepsilon| = \\varepsilon^{(x)}\\varepsilon^{(y)}\\varepsilon^{(z)}$.", + "$\\varepsilon_{i k}^{-1}$": "Inverse of the dielectric tensor $\\varepsilon_{i k}$.", + "$x_i$": "i-th component of the position vector.", + "$x_k$": "k-th component of the position vector." + } + }, + { + "id": 121, + "context": "", + "question": "An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes, respectively) is in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ (in vacuum). Determine the $x$ component of the torque $K_x$ acting on the sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$K_x$": "x component of the torque acting on the sphere", + "$a$": "Radius of the anisotropic dielectric sphere", + "$\\mathfrak{C}_y$": "y-component of the uniform external electric field", + "$\\mathfrak{C}_z$": "z-component of the uniform external electric field", + "$\\varepsilon^{(y)}$": "Principal value of the dielectric tensor along the y-axis", + "$\\varepsilon^{(z)}$": "Principal value of the dielectric tensor along the z-axis" + } + }, + { + "id": 122, + "context": "", + "question": "An anisotropic dielectric sphere with a radius of $a$ (the principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ directions, respectively) is placed in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ (in vacuum). Determine the $y$ component of the torque $K_y$ acting on the sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$K_y$": "$y$-component of the torque acting on the sphere", + "$a$": "Radius of the anisotropic dielectric sphere", + "$\\mathfrak{C}_z$": "$z$-component of the uniform external electric field", + "$\\mathfrak{C}_x$": "$x$-component of the uniform external electric field", + "$\\varepsilon^{(z)}$": "Principal value of the dielectric tensor along the $z$-direction", + "$\\varepsilon^{(x)}$": "Principal value of the dielectric tensor along the $x$-direction" + } + }, + { + "id": 123, + "context": "", + "question": "An anisotropic dielectric sphere with a radius of $a$ (principal values of the dielectric tensor are $\\varepsilon^{(x)}, \\varepsilon^{(y)}, \\varepsilon^{(z)}$, with principal axes along the $x, y, z$ axes respectively) is placed in a uniform external electric field $\\boldsymbol{\\mathfrak{C}} = (\\mathfrak{C}_x, \\mathfrak{C}_y, \\mathfrak{C}_z)$ in vacuum. Determine the $z$ component $K_z$ of the torque acting on the sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$K_z$": "z component of the torque acting on the sphere", + "$a$": "Radius of the anisotropic dielectric sphere", + "$\\mathfrak{C}_x$": "x-component of the uniform external electric field", + "$\\mathfrak{C}_y$": "y-component of the uniform external electric field", + "$\\varepsilon^{(x)}$": "Principal value of the dielectric tensor along the x-axis", + "$\\varepsilon^{(y)}$": "Principal value of the dielectric tensor along the y-axis" + } + }, + { + "id": 124, + "context": "", + "question": "Consider a dielectric sphere (with radius $a$) placed in a uniform external electric field $\\mathfrak{C}$, sliced into two halves by a plane perpendicular to the field direction. Find the attraction force between the two hemispheres.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$F$": "Attraction force between the two hemispheres", + "$\\varepsilon$": "Dielectric constant of the sphere", + "$a$": "Radius of the dielectric sphere", + "$\\mathfrak{C}$": "Uniform external electric field" + } + }, + { + "id": 125, + "context": "", + "question": "Try to determine the shape change of a dielectric sphere in a uniform external electric field.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$a$": "Semi-axis length of the deformed sphere along the x-axis", + "$b$": "Semi-axis length of the deformed sphere along the y-axis (or perpendicular to x-axis)", + "$R$": "Original radius of the sphere", + "$\\mathfrak{C}$": "Uniform external electric field", + "$\\pi$": "Mathematical constant pi", + "$\\mu$": "Shear modulus of the material", + "$\\varepsilon_{0}$": "Dielectric constant of the undeformed material", + "$a_{1}$": "Material constant related to electrostriction" + } + }, + { + "id": 126, + "context": "", + "question": "Determine the Young's modulus of a non-pyroelectric piezoelectric material parallel plate thin slab under the following conditions (the ratio of tensile stress to relative tensile strain): The slab is under tensile stress between the plates of a short-circuited capacitor.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E$": "Young's modulus, defined as the ratio of tensile stress to relative tensile strain.", + "$\\mu_{zzzz}$": "Component of the elastic compliance tensor, defined by $u_{zz}=\\mu_{zzzz} \\sigma_{zz}$." + } + }, + { + "id": 127, + "context": "", + "question": "Try to derive the expression or equation for the velocity of sound within a piezoelectric medium.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\rho$": "Density of the medium.", + "$\\omega$": "Angular frequency.", + "$\\delta_{i k}$": "Kronecker delta, defined as $\\delta_{i k} = 1$ if $i=k$ and $0$ if $i \\neq k$.", + "$\\lambda_{i l k m}$": "Elastic stiffness tensor components.", + "$k_{l}$": "Wave vector component.", + "$k_{m}$": "Wave vector component.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$\\beta_{l, m i}$": "Piezoelectric stress tensor components (or piezoelectric coefficients).", + "$k_{\\pi m}$": "Wave vector component, likely a typo for $k_m$.", + "$\\beta_{p, \\phi}$": "Piezoelectric stress tensor components, likely a typo for $\\beta_{p, qr}$ or similar, where $\\phi$ is an index.", + "$k_{p}$": "Wave vector component.", + "$k_{q}$": "Wave vector component.", + "$\\varepsilon_{r s}$": "Permittivity tensor components.", + "$k_{r}$": "Wave vector component.", + "$k_{s}$": "Wave vector component." + } + }, + { + "id": 128, + "context": "", + "question": "The piezoelectric crystals belonging to the $C_{6 v}$ crystal class are constrained by the surface plane ( $xz$ plane) of the symmetry axis ($z$ axis). Try to determine the velocity of surface waves propagating perpendicular to the symmetry axis (along the $x$ axis) which undergo displacement $u_{z}$ and potential $\\varphi$ oscillations", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\omega$": "Angular frequency.", + "$k$": "Wave number.", + "$\\bar{\\lambda}$": "Effective elastic constant, $\\bar{\\lambda}=\\lambda+\\frac{4 \\pi \\beta^{2}}{\\varepsilon}$.", + "$\\rho$": "Mass density of the medium.", + "$\\Lambda$": "Dimensionless parameter, $\\Lambda = \\frac{4 \\pi \\beta^{2}}{\\bar{\\lambda} \\varepsilon(1+\\varepsilon)}$." + } + }, + { + "id": 129, + "context": "", + "question": "Given the second-order tensor $\\sigma_{ik}$, with its symmetric part $s_{ik}$ and its antisymmetric part formed by the axial vector $\\boldsymbol{a}$ (specific definitions are found in the symbols table), express the determinant $|\\sigma|$ of the tensor $\\sigma_{ik}$ in terms of the components of $s_{ik}$ and $\\boldsymbol{a}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$|\\sigma|$": "Determinant of the second-order tensor $\\sigma_{ik}$.", + "$|s|$": "Determinant of the symmetric part $s_{ik}$.", + "$s_{ik}$": "Symmetric part of the second-order tensor $\\sigma_{ik}$.", + "$a_i$": "Component of the axial vector $\\boldsymbol{a}$ along the $i$-th axis." + } + }, + { + "id": 130, + "context": "", + "question": "Given a second-order tensor $\\sigma_{ik}$, its symmetric part is $s_{ik}$, and the antisymmetric part is formed by the axial vector $\\boldsymbol{a}$. Given its determinant as $|\\sigma|=|s|+s_{i k} a_{i} a_{k}$. Try to express the axial vector $b_i$ of the antisymmetric part of its inverse tensor $\\sigma_{ik}^{-1}$ using the components of $s_{i k}$ and $\\boldsymbol{a}$ (i.e., $\\sigma_{ik}^{-1} = \\rho_{ik} + \\epsilon_{ikl}b_l$, where $\\epsilon_{ikl}b_l$ is the antisymmetric part).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$b_i$": "Components of the axial vector $\\boldsymbol{b}$, which forms the antisymmetric part of the inverse tensor $\\sigma_{ik}^{-1}$.", + "$|\\sigma|$": "Determinant of the tensor $\\sigma_{ik}$.", + "$s_{ik}$": "Symmetric part of the second-order tensor $\\sigma_{ik}$.", + "$a_k$": "Components of the axial vector $\\boldsymbol{a}$." + } + }, + { + "id": 131, + "context": "", + "question": "Assuming two parallel planar plates (made of the same metal $A$) are immersed in an electrolyte solution $AX$. In the case of a very small current $j$, derive the expression for the effective resistivity of the solution $\\frac{\\mathscr{E}}{l j}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{E}$": "Potential difference between the two plates.", + "$l$": "Distance between the parallel plates.", + "$j$": "Current density.", + "$\\sigma$": "Conductivity of the solution.", + "$\\rho$": "A parameter representing a property of the electrolyte solution, appearing in the integral expressions.", + "$D$": "Diffusion coefficient.", + "$\\zeta$": "A potential that varies with concentration, often referred to as zeta potential.", + "$c$": "Concentration (used as an integration variable and as a general concentration in the approximation).", + "$\\beta$": "A constant or parameter in the integral expressions.", + "$m$": "A constant or parameter in the integral expressions.", + "$e$": "Elementary charge." + } + }, + { + "id": 132, + "context": "", + "question": "Consider a circular line current with a radius $a$. Try to find the radial component $B_r$ of the magnetic field in cylindrical coordinates.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$B_r$": "Radial component of the magnetic induction.", + "$A_{\\varphi}$": "Azimuthal component of the magnetic vector potential.", + "$z$": "Axial coordinate in the cylindrical coordinate system.", + "$J$": "Magnitude of the current in the circular line.", + "$c$": "Speed of light.", + "$r$": "Radial coordinate in the cylindrical coordinate system.", + "$a$": "Radius of the circular line current.", + "$K$": "Complete elliptic integral of the first kind.", + "$E$": "Complete elliptic integral of the second kind." + } + }, + { + "id": 133, + "context": "", + "question": "Consider a circular line current with a radius of $a$. Try to find the axial component $B_z$ of the magnetic field it produces in cylindrical coordinates.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$B_z$": "Axial component of the magnetic field.", + "$r$": "Radial cylindrical coordinate.", + "$A_{\\varphi}$": "Azimuthal component of the magnetic vector potential.", + "$J$": "Current flowing in the circular line.", + "$c$": "Speed of light.", + "$a$": "Radius of the circular line current.", + "$z$": "Axial cylindrical coordinate.", + "$K$": "Complete elliptic integral of the first kind, $K=\\int_{0}^{\\pi / 2} \\frac{\\mathrm{~d} \\theta}{\\sqrt{1-k^{2} \\sin ^{2} \\theta}}$.", + "$E$": "Complete elliptic integral of the second kind, $E=\\int_{0}^{\\pi / 2} \\sqrt{1-k^{2} \\sin ^{2} \\theta} \\mathrm{~d} \\theta$." + } + }, + { + "id": 134, + "context": "", + "question": "Try to find the 'internal' part of the self-inductance $L_i$ of a closed thin wire with a circular cross-section.\n\n\\footnotetext{\n(1) The assertion in the main text that the self-inductance does not depend on the current distribution actually applies not only to the approximation (34.1), but also to subsequent approximations that do not contain large logarithmic terms (this corresponds to considering the coefficient in front of $l / a$ in the argument of the logarithm); see the exercises in this section.\n(2) In exercises $1-6$, it is assumed that the susceptibility of the medium is $\\mu_{e}=1$.\n}", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$L_i$": "Internal part of the self-inductance of a closed thin wire", + "$l$": "Length of the closed wire", + "$\\mu_i$": "Permeability of the internal medium of the wire" + } + }, + { + "id": 135, + "context": "", + "question": "Try to determine the self-inductance of a thin circular ring (radius $b$) made from a wire with a circular cross-section (radius $a$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$L$": "Total self-inductance of the circular ring.", + "$\\pi$": "Mathematical constant pi.", + "$b$": "Radius of the thin circular ring.", + "$a$": "Radius of the wire's circular cross-section.", + "$\\mu_i$": "Internal permeability of the wire material." + } + }, + { + "id": 136, + "context": "", + "question": "A current flows through a wire loop $(\\mu_{i}=1)$; try to determine the elongation of the loop under the magnetic field generated by this current.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\Delta b$": "Elongation or change in the radius of the wire loop.", + "$b$": "Radius of the wire loop.", + "$J$": "Current flowing through the wire loop.", + "$a$": "Radius of the wire's cross-section.", + "$c$": "Speed of light in vacuum.", + "$E$": "Young's modulus of the wire material.", + "$\\sigma$": "Poisson's ratio of the wire material." + } + }, + { + "id": 137, + "context": "", + "question": "Seek the first-order correction value of the cylindrical helical tube's self-inductance, due to the field distortion near both ends of the cylindrical helical tube when $l / h$ (with $\\mu_{e}=1$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$L$": "Self-inductance of the helical tube", + "$b$": "Radius of the helical tube", + "$n$": "Turns per unit length", + "$h$": "Length of the helical tube" + } + }, + { + "id": 138, + "context": "", + "question": "If a planar circuit is placed on the surface of a semi-infinite medium with a permeability of $\\mu_{e}$, find the factor by which the self-inductance of the planar circuit changes. We neglect the internal part of the conductor's self-inductance.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mu_{e}$": "Permeability of the semi-infinite medium" + } + }, + { + "id": 139, + "context": "", + "question": "Given a straight wire carrying a current $J$ parallel to an infinitely long cylindrical conductor with a radius $a$ (permeability $\\mu$) at a distance $l$ from the axis of the cylinder, determine the force on the straight wire.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$F$": "Force per unit length on the conductor", + "$J$": "Current carried by the straight wire", + "$a$": "Radius of the infinitely long cylindrical conductor", + "$\\mu$": "Permeability of the cylindrical conductor", + "$b$": "Distance from the axis of the cylinder to the straight wire", + "$c$": "Speed of light" + } + }, + { + "id": 140, + "context": "", + "question": "Try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \\gg 4 \\pi M)$, where the microcrystals have uniaxial symmetry.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\bar{M}$": "Average magnetization intensity of the polycrystal", + "$M$": "Magnitude of the magnetization intensity of a single microcrystal", + "$\\beta$": "Material parameter related to anisotropy", + "$H$": "Magnitude of the strong magnetic field" + } + }, + { + "id": 141, + "context": "", + "question": "For cubic symmetric microcrystals, try to determine the average magnetization intensity of a polycrystal in a strong magnetic field $(H \\gg 4 \\pi M)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\bar{M}$": "Average magnetization intensity of a polycrystal", + "$M$": "Magnitude of the magnetization vector", + "$\\beta$": "Anisotropy constant parameter for cubic symmetric microcrystals", + "$H$": "Magnitude of the magnetic field vector" + } + }, + { + "id": 142, + "context": "", + "question": "Try to find out the relative elongation of a ferromagnetic cubic crystal depending on the magnetization direction $\\boldsymbol{m}$ and the measurement direction $\\boldsymbol{n}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\frac{\\delta l}{l}$": "Relative elongation of the crystal.", + "$a_1$": "Material constant related to magnetostriction, specifically for terms involving squares of magnetization and measurement direction components along the same axis.", + "$m_x$": "$x$-component of the magnetization direction $\\boldsymbol{m}$.", + "$n_x$": "$x$-component of the measurement direction unit vector $\\boldsymbol{n}$.", + "$m_y$": "$y$-component of the magnetization direction $\\boldsymbol{m}$.", + "$n_y$": "$y$-component of the measurement direction unit vector $\\boldsymbol{n}$.", + "$m_z$": "$z$-component of the magnetization direction $\\boldsymbol{m}$.", + "$n_z$": "$z$-component of the measurement direction unit vector $\\boldsymbol{n}$.", + "$a_2$": "Material constant related to magnetostriction, specifically for terms involving products of different components of magnetization and measurement directions." + } + }, + { + "id": 143, + "context": "", + "question": "The easy magnetization axes of a cubic ferromagnet align along the three edges of the cube (specifically the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$ axis, and the domain walls are distributed parallel to the (100) plane. Determine the surface tension of the domain wall in this case.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\Delta_{(100)}$": "Surface tension of the domain wall for the (100) plane", + "$\\alpha$": "Exchange stiffness constant", + "$\\beta$": "Anisotropy constant" + } + }, + { + "id": 144, + "context": "", + "question": "The easy magnetization axes of a cubic ferromagnet are along the three edges of the cube (namely the $x, y, z$ axes). The magnetic domains are magnetized parallel or antiparallel to the $z$-axis, and the domain walls are distributed parallel to the (110) plane. Determine the surface tension of the domain walls in this scenario.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\Delta_{(110)}$": "Surface tension of the domain walls for the (110) plane.", + "$\\alpha$": "Exchange stiffness constant.", + "$\\beta$": "Anisotropy constant." + } + }, + { + "id": 145, + "context": "", + "question": "If the transition between magnetic domains is not achieved through the rotation of $M$ but by changing the magnitude of $M$ (i.e., when $M$ changes sign after passing through zero), determine the surface tension of the domain wall in a uniaxial crystal. The free energy's dependence on $M$ (at $\\boldsymbol{H}=0$) takes the expanded form corresponding to the situation near the Curie point as given by Equation \\begin{align*}\n\\tilde{\\Phi} = \\Phi_0 + AM^2 + BM^4 - MH - \\frac{H^2}{8\\pi},\n\\end{align*}.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\Delta$": "Surface tension of the domain wall", + "$\\alpha_1$": "Coefficient related to inhomogeneity in free energy density", + "$|A|$": "Absolute value of coefficient A" + } + }, + { + "id": 146, + "context": "", + "question": "The parallel plane magnetic domains extend perpendicularly to the surface of the ferromagnetic material without changing the direction of magnetization . Try to derive and present the exact mathematical expression for the magnetic field energy per unit surface area near the surface of the ferromagnet (expressed in terms of $\\zeta(3)$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$a$": "Domain width", + "$M$": "Magnetization (magnitude of surface charge density)", + "$\\pi$": "Mathematical constant pi", + "$\\zeta(3)$": "Riemann zeta function evaluated at 3" + } + }, + { + "id": 147, + "context": "", + "question": "Find the magnetic moment ${ }^{(1)}$ of a superconducting disk perpendicular to the external magnetic field.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathscr{M}$": "Magnetic moment of the superconducting disk.", + "$a$": "Semi-axis of the spheroid, representing the radius of the disk in the limit.", + "$\\pi$": "Mathematical constant pi.", + "$\\mathfrak{H}$": "External magnetic field." + } + }, + { + "id": 148, + "context": "", + "question": "Seek the heat capacity of a superconducting ellipsoid in the intermediate state.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathscr{C}_{t}$": "Heat capacity of the object in the intermediate state.", + "$\\mathscr{C}_{s}$": "Heat capacity of the object in the superconducting state.", + "$V$": "Volume of the object (superconducting ellipsoid).", + "$T$": "Temperature.", + "$\\pi$": "Mathematical constant pi.", + "$n$": "Demagnetization factor.", + "$H_{\\mathrm{cr}}^{\\prime}$": "First derivative of the critical magnetic field with respect to temperature, $H_{\\mathrm{cr}}^{\\prime} = \\frac{dH_{\\mathrm{cr}}}{dT}$.", + "$H_{\\mathrm{cr}}$": "Critical magnetic field.", + "$H_{\\mathrm{cr}}^{\\prime \\prime}$": "Second derivative of the critical magnetic field with respect to temperature, $H_{\\mathrm{cr}}^{\\prime \\prime} = \\frac{d^2H_{\\mathrm{cr}}}{dT^2}$.", + "$\\mathfrak{H}$": "External magnetic field." + } + }, + { + "id": 149, + "context": "", + "question": "An isotropic conducting sphere with radius $a$ is in a uniform periodic external magnetic field. Determine the expression for its magnetic polarizability $\\alpha$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha$": "Magnetic polarizability of the sphere.", + "$\\pi$": "Mathematical constant pi.", + "$a$": "Radius of the isotropic conducting sphere.", + "$k$": "Wave number or propagation constant inside the sphere, defined as $k=\\frac{1+\\mathrm{i}}{\\delta}$." + } + }, + { + "id": 150, + "context": "", + "question": "An isotropic conducting sphere with a radius of $a$ is in a uniform periodic external magnetic field, with its magnetic susceptibility given by $\\alpha = \\alpha^{\\prime} + \\mathrm{i} \\alpha^{\\prime \\prime}$. Determine the expression for the real part of its magnetic susceptibility $\\alpha^{\\prime}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha^{\\prime}$": "Real part of the magnetic susceptibility", + "$\\pi$": "Mathematical constant pi", + "$\\delta$": "Skin depth", + "$a$": "Radius of the isotropic conducting sphere" + } + }, + { + "id": 151, + "context": "", + "question": "An isotropic conductive sphere with a radius of $a$ is in a uniform periodic external magnetic field. Its magnetic susceptibility is $\\alpha = \\alpha^{\\prime} + \\mathrm{i} \\alpha^{\\prime \\prime}$. Determine the expression for the imaginary part of its magnetic susceptibility $\\alpha^{\\prime \\prime}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha^{\\prime \\prime}$": "Imaginary part of the magnetic susceptibility $\\alpha$.", + "$\\delta$": "Skin depth of the conductive sphere.", + "$a$": "Radius of the isotropic conductive sphere.", + "$\\pi$": "Mathematical constant pi.", + "$\\phi$": "A symbol appearing in the final answer, likely a typo for the radius $a$." + } + }, + { + "id": 152, + "context": "", + "question": "Find the smallest value of the attenuation coefficient for the magnetic field inside a conducting sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\gamma_1$": "Smallest value of the attenuation coefficient for the magnetic field", + "$\\pi$": "Pi, a mathematical constant", + "$c$": "Speed of light", + "$\\sigma$": "Electrical conductivity of the conducting sphere", + "$a$": "Radius of the conducting sphere" + } + }, + { + "id": 153, + "context": "", + "question": "Two inductively coupled circuits respectively contain self-inductances $L_{1}$ and $L_{2}$ and capacitances $C_{1}$ and $C_{2}$. Determine the intrinsic frequencies of electric oscillations within these coupled circuits (we neglect resistances $R_{1}$ and $R_{2}$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\omega_{1,2}^{2}$": "Squared intrinsic frequencies of electric oscillations for the coupled circuits.", + "$c$": "Speed of light.", + "$L_1$": "Self-inductance of the first circuit.", + "$C_1$": "Capacitance of the first circuit.", + "$L_2$": "Self-inductance of the second circuit.", + "$C_2$": "Capacitance of the second circuit.", + "$L_{12}$": "Mutual inductance between the two circuits." + } + }, + { + "id": 154, + "context": "", + "question": "A uniformly magnetized sphere rotates uniformly around an axis parallel to the magnetization direction. Determine the unipolar induced electromotive force between one pole of the uniformly magnetized sphere and the equator.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{E}$": "Unipolar induced electromotive force (EMF) between one pole and the equator of the sphere.", + "$\\Omega$": "Angular velocity of the sphere.", + "$\\mathscr{M}$": "Total magnetic moment of the sphere.", + "$a$": "Radius of the sphere.", + "$c$": "Speed of light." + } + }, + { + "id": 155, + "context": "", + "question": "Determine the current generated inside the superconducting ring when its uniform rotation stops.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$J$": "Current generated inside the superconducting ring", + "$m$": "Mass (likely electron mass)", + "$c$": "Speed of light", + "$b$": "Radius of the superconducting ring", + "$\\Omega$": "Angular velocity of the uniform rotation of the ring", + "$e$": "Elementary charge (likely electron charge)", + "$a$": "Radius of the wire's circular cross-section" + } + }, + { + "id": 156, + "context": "", + "question": "Try to determine the absorption coefficient of Alfven waves in an incompressible fluid (assuming this coefficient is very small).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\gamma$": "Absorption coefficient of the wave.", + "$\\omega$": "Angular frequency of the wave.", + "$u_{\\mathrm{A}}$": "Alfven speed, $u_{\\mathrm{A}} = \\frac{|H_x|}{\\sqrt{4\\pi\\rho}}$.", + "$\\eta$": "Dynamic viscosity.", + "$\\rho$": "Mass density of the fluid.", + "$c$": "Speed of light in vacuum.", + "$\\sigma$": "Electrical conductivity." + } + }, + { + "id": 157, + "context": "", + "question": "Try to find the law of rotational discontinuity expanding with time.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$c$": "Speed of light.", + "$\\pi$": "Mathematical constant pi.", + "$\\sigma$": "Electrical conductivity of the medium.", + "$\\nu$": "Kinematic viscosity of the fluid.", + "$t$": "Time." + } + }, + { + "id": 158, + "context": "", + "question": "A dielectric sphere in vacuum rotates in a constant magnetic field $\\mathfrak{H}$, determine the electric field produced around the sphere.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$D_{i k}$": "Electric quadrupole moment tensor of the sphere.", + "$a$": "Radius of the sphere.", + "$c$": "Speed of light in vacuum.", + "$\\varepsilon$": "Permittivity of the dielectric sphere.", + "$\\mu$": "Permeability of the dielectric sphere.", + "$\\mathfrak{H}_{i}$": "i-th component of the constant magnetic field $\\mathfrak{H}$.", + "$\\Omega_{k}$": "k-th component of the angular velocity $\\boldsymbol{\\Omega}$.", + "$\\mathfrak{H}_{k}$": "k-th component of the constant magnetic field $\\mathfrak{H}$.", + "$\\Omega_{i}$": "i-th component of the angular velocity $\\boldsymbol{\\Omega}$.", + "$\\delta_{i k}$": "Kronecker delta.", + "$\\mathfrak{H}$": "Constant magnetic field vector in vacuum.", + "$\\Omega$": "Angular velocity vector (used as a shorthand for $\\boldsymbol{\\Omega}$ in scalar products)." + } + }, + { + "id": 159, + "context": "", + "question": "A magnetized dielectric sphere (with dielectric constant $\\varepsilon$) rotates uniformly in a vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with angular velocity $\\Omega$. This rotation generates an electric field around the sphere. To describe this electric field, it is necessary to calculate the electric quadrupole moment. Determine the $D_{zz}$ component of the electric quadrupole tensor generated by this rotating sphere. The sphere has a radius $a$, total magnetic moment $\\mathscr{M}$, and the speed of light in vacuum $c$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$D_{zz}$": "$zz$-component of the electric quadrupole tensor.", + "$\\varepsilon$": "Dielectric constant of the sphere.", + "$c$": "Speed of light in vacuum.", + "$a$": "Radius of the sphere.", + "$\\Omega$": "Angular velocity of the sphere's rotation.", + "$\\mathscr{M}$": "Total magnetic moment of the sphere." + } + }, + { + "id": 160, + "context": "", + "question": "A magnetized metallic sphere (considered as the case of dielectric constant $\\varepsilon \\rightarrow \\infty$) rotates uniformly in vacuum around its own axis parallel to the magnetization direction (the $z$-axis) with an angular velocity $\\Omega$. This rotation will generate an electric field around the sphere. Determine the $D_{zz}$ component of the electric quadrupole moment tensor produced by the metallic sphere to describe its external electric field. The sphere radius is $a$, the total magnetic moment is $\\mathscr{M}$, and the speed of light in vacuum is $c$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$D_{zz}$": "$zz$ component of the electric quadrupole moment tensor.", + "$c$": "Speed of light in vacuum.", + "$\\Omega$": "Angular velocity of the sphere's rotation.", + "$\\mathscr{M}$": "Total magnetic moment of the sphere.", + "$a$": "Radius of the sphere.", + "$\\varepsilon$": "Dielectric constant of the sphere, which approaches infinity for a metallic sphere." + } + }, + { + "id": 161, + "context": "", + "question": "Try to find the dispersion relation of magnetostatic oscillations in an unbounded medium.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\omega$": "Angular frequency of magnetostatic oscillations.", + "$\\gamma$": "Gyromagnetic ratio.", + "$M$": "Magnetization of the medium.", + "$\\beta$": "A parameter in the permeability tensor, related to the medium's properties.", + "$\\theta$": "Angle between the wave vector $\\boldsymbol{k}$ and the easy magnetization axis (z-axis)." + } + }, + { + "id": 162, + "context": "", + "question": "The surface of an infinite parallel plate is perpendicular to the easy magnetization axis, and an external magnetic field $\\mathfrak{H}$ is applied along this axis direction. Determine the non-uniform resonance frequency within this plate.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\omega$": "Non-uniform resonance frequency (vibration frequency).", + "$\\gamma$": "Gyromagnetic ratio.", + "$M$": "Magnetization.", + "$\\beta$": "A parameter in the expression for $\\mu_{xx}$.", + "$\\mathfrak{H}$": "External magnetic field applied along the easy magnetization axis.", + "$\\pi$": "Mathematical constant pi.", + "$\\theta$": "Angle between the wave vector $\\boldsymbol{k}$ and the $z$ axis." + } + }, + { + "id": 163, + "context": "", + "question": "Calculate the reflection coefficient when light is almost grazing the surface of a material with $\\varepsilon$ close to 1 from a vacuum.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$R_{\\perp}$": "Reflection coefficient for perpendicular polarization", + "$R_{\\|}$": "Reflection coefficient for parallel polarization", + "$\\varphi_{0}$": "Grazing angle, defined as $\\pi / 2 - \\theta_0$", + "$\\varepsilon$": "Permittivity of the material" + } + }, + { + "id": 164, + "context": "", + "question": "Find the reflection coefficient $R_{\\perp}$ when a wave is incident from vacuum onto the surface of a medium where both $\\varepsilon$ and $\\mu$ are different from 1, with the electric field vector perpendicular to the plane of incidence.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$R_{\\perp}$": "Reflection coefficient when the electric field vector is perpendicular to the plane of incidence.", + "$\\mu$": "Relative permeability of the medium.", + "$\\theta_0$": "Angle of incidence from vacuum.", + "$\\varepsilon$": "Relative permittivity of the medium." + } + }, + { + "id": 165, + "context": "", + "question": "Find the reflection coefficient $R_{\\|}$ when a wave is incident on the surface of a medium with both $\\varepsilon$ and $\\mu$ different from 1, and when the electric field vector is parallel to the plane of incidence.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$R_{\\|}$": "Reflection coefficient when the electric field vector is parallel to the plane of incidence.", + "$\\varepsilon$": "Permittivity of the medium.", + "$\\mu$": "Permeability of the medium.", + "$\\theta_{0}$": "Angle of incidence." + } + }, + { + "id": 166, + "context": "", + "question": "For metals with impedance determined by formula \\begin{align*}\n\\zeta = (1 - i)\\sqrt{\\frac{\\omega\\mu}{8\\pi\\sigma}}\n\\end{align*} (a special case with a flat surface having low impedance), and assuming its permeability $\\mu=1$, try to determine the ratio of its thermal radiation intensity to the absolute blackbody surface radiation intensity ($I/I_0$). The ratio should be expressed in terms of the angular frequency of radiation $\\omega$ and the conductivity of the metal $\\sigma$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$I$": "Thermal radiation intensity from the metal surface", + "$I_0$": "Absolute blackbody surface radiation intensity", + "$\\omega$": "Angular frequency of radiation", + "$\\pi$": "Mathematical constant pi", + "$\\sigma$": "Conductivity of the metal" + } + }, + { + "id": 167, + "context": "", + "question": "Determine the dependence of the radiation intensity of a dipole emitter immersed in a homogeneous isotropic medium on the medium's permittivity $\\varepsilon$ and magnetic permeability $\\mu$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$I$": "Radiation intensity of the dipole emitter in the medium.", + "$I_{0}$": "Radiation intensity of the dipole emitter in vacuum.", + "$\\mu$": "Magnetic permeability of the homogeneous isotropic medium.", + "$\\varepsilon$": "Permittivity of the homogeneous isotropic medium." + } + }, + { + "id": 168, + "context": "", + "question": "For the E wave in a circular waveguide with a radius of $a$, provide the expression for its attenuation coefficient $\\alpha$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha$": "Attenuation coefficient of the E wave in the circular waveguide", + "$\\omega$": "Angular frequency of the E wave", + "$\\zeta^{\\prime}$": "Surface impedance of the waveguide material", + "$c$": "Speed of light in vacuum", + "$a$": "Radius of the circular waveguide", + "$k_z$": "Longitudinal wave number (propagation constant) of the E wave" + } + }, + { + "id": 169, + "context": "", + "question": "Provide the expression for the attenuation coefficient $\\alpha$ of H modes in a circular (radius $a$) cross-section waveguide.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha$": "Attenuation coefficient of H modes in the waveguide.", + "$c$": "Speed of light in vacuum.", + "$\\varkappa$": "Transverse wave number (used interchangeably with $\\kappa$).", + "$\\zeta'$": "Surface impedance, representing losses in the waveguide walls.", + "$\\omega$": "Angular frequency of the electromagnetic wave.", + "$k_z$": "Longitudinal wave number.", + "$a$": "Radius of the circular waveguide.", + "$n$": "Azimuthal mode index for H modes." + } + }, + { + "id": 170, + "context": "", + "question": "Linearly polarized light is scattered by small particles with random orientations, where the particles' polarizability tensor has three distinct principal values. It is known that the scalar constants describing the average of the electric dipole moment tensor are $A = \\frac{1}{15}(2 \\alpha_{i i} \\alpha_{k k}^{*}-\\alpha_{i k} \\alpha_{i k}^{*})$ and $B = \\frac{1}{30}(3 \\alpha_{i k} \\alpha_{i k}^{*}-\\alpha_{i i} \\alpha_{k k}^{*})$. Determine the expression for the depolarization ratio $\\frac{I_y}{I_x}$ of the scattered light, where $\\theta$ is the angle between the incident light electric field $\\boldsymbol{E}$ and the scattering direction $\\boldsymbol{n}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$I_y$": "Intensity of scattered light in the y-direction.", + "$I_x$": "Intensity of scattered light in the x-direction.", + "$B$": "Scalar constant describing the average of the electric dipole moment tensor, $B = \\frac{1}{30}(3 \\alpha_{i k} \\alpha_{i k}^{*}-\\alpha_{i i} \\alpha_{k k}^{*})$", + "$A$": "Scalar constant describing the average of the electric dipole moment tensor, $A = \\frac{1}{15}(2 \\alpha_{i i} \\alpha_{k k}^{*}-\\alpha_{i k} \\alpha_{i k}^{*})$", + "$\\theta$": "Angle between the incident light electric field $\\boldsymbol{E}$ and the scattering direction $\\boldsymbol{n}$." + } + }, + { + "id": 171, + "context": "", + "question": "Try to express the components of the ray vector $s$ in terms of the components of $\\boldsymbol{n}$ within the principal dielectric axes.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$s_x$": "x-component of the ray vector $\\boldsymbol{s}$", + "$n_x$": "x-component of the wave vector $\\boldsymbol{n}$", + "$\\varepsilon^{(x)}$": "Principal dielectric constant (permittivity) along the x-axis", + "$\\varepsilon^{(y)}$": "Principal dielectric constant (permittivity) along the y-axis", + "$\\varepsilon^{(z)}$": "Principal dielectric constant (permittivity) along the z-axis", + "$n_y$": "y-component of the wave vector $\\boldsymbol{n}$", + "$n_z$": "z-component of the wave vector $\\boldsymbol{n}$" + } + }, + { + "id": 172, + "context": "", + "question": "Find the polarization of the reflected light when linearly polarized light is perpendicularly incident from vacuum onto the surface of an anisotropic object induced by a magnetic field.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$g$": "A parameter related to the magnetic field induced anisotropy, appearing in the approximation for $E_{1y}$ and the final ratio.", + "$\\theta$": "Angle between the incident direction and the vector $\\boldsymbol{g}$.", + "$n_{0}$": "Base refractive index of the anisotropic medium." + } + }, + { + "id": 173, + "context": "", + "question": "Attempt to find the limiting law of the dependence of the surface tension coefficient $\\alpha$ of liquid nitrogen near absolute zero on temperature", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\alpha$": "Surface tension coefficient of liquid nitrogen; free energy per unit surface area of a liquid.", + "$\\alpha_0$": "Surface tension coefficient at absolute zero temperature ($T=0$).", + "$T$": "Absolute temperature. (Note: The symbol $\\tau$ is used inconsistently in some intermediate integral expressions, but it represents temperature.)", + "$\\rho$": "Liquid density.", + "$\\hbar$": "Reduced Planck's constant. (Note: The symbols $h$ and $n$ are used inconsistently in some intermediate integral expressions, but they represent the reduced Planck's constant.)" + } + }, + { + "id": 174, + "context": "", + "question": "Try to find the dispersion relation of impurity particles in a moving superfluid $\\varepsilon_{\\text {imp}}(p)$, given that in a stationary fluid the dispersion relation is $\\varepsilon_{\\text {imp}}^{(0)}(p)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\varepsilon_{\\text {imp}}$": "Dispersion relation of impurity particles in a moving superfluid.", + "$\\varepsilon_{\\text {imp}}^{(0)}$": "Dispersion relation of impurity particles in a stationary fluid.", + "$p$": "Momentum of the impurity atom in the moving superfluid's reference frame, defined as $p=p_{0}+m v$.", + "$m$": "Mass of the impurity atom.", + "$v$": "Velocity of the moving superfluid." + } + }, + { + "id": 175, + "context": "", + "question": "Try to find the dispersion relation of small oscillations of a rectilinear vortex line (W. Thomson, 1880).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\omega$": "Angular frequency of the oscillation.", + "$\\kappa$": "Vortex strength (circulation).", + "$k$": "Wave number of the oscillation.", + "$a$": "Vortex core radius." + } + }, + { + "id": 176, + "context": "", + "question": "A neutron with an initial velocity $v$ scatters within a liquid. Determine the conditions under which an excitation with momentum $p$ and energy $\\varepsilon(p)$ can be produced.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$V$": "Magnitude of the initial velocity of the neutron", + "$\\varepsilon(p)$": "Energy of the excitation, as a function of its momentum $p$", + "$p$": "Magnitude of the momentum of the excitation", + "$m$": "Mass of the neutron" + } + }, + { + "id": 177, + "context": "", + "question": "Try to find the magnetic moment of a superconducting sphere of radius $R \\ll \\delta$ in a magnetic field under the London situation.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\boldsymbol{M}$": "Magnetic moment", + "$R$": "Radius of the superconducting sphere", + "$\\delta$": "London penetration depth", + "$\\mathfrak{G}$": "External magnetic field" + } + }, + { + "id": 178, + "context": "", + "question": "For superconductors with parameter $\\kappa \\ll 1$, find the first-order correction to the penetration depth in a weak magnetic field.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\delta_{\\text{eff}}$": "Effective penetration depth of the magnetic field into the superconductor.", + "$\\delta$": "London penetration depth, the uncorrected penetration depth.", + "$\\kappa$": "Ginzburg-Landau parameter, a dimensionless parameter characterizing superconductors.", + "$\\mathfrak{S}$": "Magnetic field strength at the surface of the superconductor, $B(x=0)$.", + "$H_{\\mathrm{c}}$": "Thermodynamic critical field of the superconductor." + } + }, + { + "id": 179, + "context": "", + "question": "Attempt to find the critical field of a superconducting small sphere with radius $R \\ll \\delta$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$H_{\\mathrm{c}}^{\\text {(sphere)}}$": "Critical magnetic field of the small superconducting sphere", + "$H_{\\mathrm{c}}$": "Critical magnetic field", + "$\\delta$": "Superconducting penetration depth", + "$R$": "Radius of the small superconducting sphere" + } + }, + { + "id": 180, + "context": "", + "question": "Calculate the interaction energy of two vortices separated by $d \\gg \\xi$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\varepsilon_{12}$": "Interaction energy per unit length between the two vortices.", + "$\\phi_0$": "Magnetic flux quantum.", + "$\\delta$": "Penetration depth.", + "$d$": "Separation distance between two vortices." + } + }, + { + "id": 181, + "context": "", + "question": "A thin film (thickness $d \\ll \\xi(T)$) is placed in a weak magnetic field perpendicular to its plane. Find the magnetic moment of the film when the temperature $T$ $>T_{\\mathrm{c}}, T-T_{\\mathrm{c}} \\ll T_{\\mathrm{c}}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$M$": "Magnetic moment of the film.", + "$S$": "Area of the film.", + "$e$": "Elementary charge.", + "$T_{\\mathrm{c}}$": "Critical temperature.", + "$\\mathfrak{S}$": "Magnetic field, appearing in the calculation of the number of eigenfunctions, the derived free energy formula, and the provided final answer.", + "$\\pi$": "Mathematical constant pi.", + "$m$": "Mass of the charge carrier (e.g., electron mass).", + "$c$": "Speed of light.", + "$\\alpha$": "A constant, such that $a = \\alpha(T-T_c)$.", + "$T$": "Temperature of the film." + } + }, + { + "id": 182, + "context": "", + "question": "Under the conditions of the previous question, determine the magnetic moment of a small sphere with radius $R \\ll \\xi(T)$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$e$": "Elementary charge", + "$T_{\\mathrm{c}}$": "Critical temperature", + "$R$": "Radius of the small sphere", + "$\\mathscr{S}_{\\mathrm{g}}$": "Magnetic field strength (or flux density)", + "$m$": "Mass of the charge carrier (e.g., electron)", + "$c$": "Speed of light in vacuum", + "$\\alpha$": "A constant related to the temperature dependence", + "$T$": "Temperature" + } + }, + { + "id": 183, + "context": "", + "question": "Find the energy spectrum of spin wave quanta in an uniaxial ferromagnet of the 'easy magnetization plane' type $(K<0)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varepsilon$": "Energy of spin wave quanta", + "$k$": "Magnitude of the wave vector", + "$\\beta$": "A constant related to the equations of motion for magnetization", + "$M$": "Magnetization, representing the equilibrium magnetization $M_0$", + "$\\alpha$": "Exchange stiffness constant", + "$|K|$": "Magnitude of the anisotropy constant", + "$\\pi$": "Mathematical constant pi", + "$\\theta$": "Polar angle of the wave vector $k$ with respect to the direction of $M_0$", + "$\\varphi$": "Azimuthal angle of the wave vector $k$, measured from the $xz$ plane" + } + }, + { + "id": 184, + "context": "", + "question": "Calculate the spin wave quantum part of thermodynamic quantities (energy) at temperature $T \\ll \\varepsilon(0)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E_{\\operatorname{mag}}$": "Spin wave quantum part of the thermodynamic energy.", + "$V$": "Volume.", + "$K$": "Anisotropy constant.", + "$T$": "Temperature.", + "$\\pi$": "Mathematical constant pi.", + "$A$": "Coefficient related to exchange stiffness, where for cubic crystals $A=2 \\beta M \\alpha$, and for 'easy magnetization axis'-type uniaxial crystals $A=2 \\beta M \\alpha_{2}$.", + "$\\beta$": "A coefficient appearing in the energy expression and related to material properties.", + "$M$": "Magnetization." + } + }, + { + "id": 185, + "context": "", + "question": "In the exchange approximation, determine the spatial correlation function of magnetization fluctuations at distances $r \\gg a$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varphi_{i k}(\\boldsymbol{r})$": "Spatial correlation function of magnetization fluctuations.", + "$\\beta$": "Constant related to spin or magnetic moment.", + "$M$": "Magnetization.", + "$\\mathrm{e}$": "Base of the natural logarithm.", + "$\\varepsilon(k)$": "Energy of a spin-wave quantum state, dependent on the magnitude of the wave vector $k$.", + "$k$": "Magnitude of the wave vector $\\boldsymbol{k}$. Also used as a subscript index for magnetization components (e.g., x or y).", + "$T$": "Temperature.", + "$\\delta_{i k}$": "Kronecker delta.", + "$i$": "Subscript index for magnetization components (e.g., x or y)." + } + }, + { + "id": 186, + "context": "", + "question": "Given $S \\gg 1$, try to find the correction terms related to the interaction of heat capacity of a cubic lattice. In this lattice, only the exchange integral between a pair of neighboring atoms (along the cubic axes) is non-zero.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$C_{\\text{int}}$": "Interaction heat capacity, $C_{\\text{int}}=\\frac{15 \\pi \\zeta^{2}(5 / 2) N}{S}(\\frac{T}{4 \\pi S J_{0}})^{4}$", + "$\\pi$": "Mathematical constant pi", + "$\\zeta$": "Riemann zeta function", + "$N$": "Number of atoms in the lattice", + "$S$": "Spin quantum number, given $S \\gg 1$", + "$T$": "Temperature", + "$J_0$": "Exchange integral between a pair of nearest neighbor atoms" + } + }, + { + "id": 187, + "context": "", + "question": "Try to determine the interaction law between an atom and a metal wall at 'large' distances.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$U(L)$": "Interaction energy between a single atom and the wall as a function of distance $L$.", + "$\\alpha_{2}$": "A coefficient related to the interaction energy between a single atom and the wall.", + "$\\hbar$": "Reduced Planck's constant.", + "$c$": "Speed of light in vacuum.", + "$\\pi$": "Mathematical constant pi.", + "$L$": "Distance between the atom and the wall, specifically for the single atom-wall interaction energy." + } + }, + { + "id": 188, + "context": "", + "question": "Determine the correlation function $\\nu(r)$ of a Bose liquid at temperature $T \\ll T_{\\lambda}$, at distances $r \\gtrsim \\hbar u / T$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\nu(r)$": "Correlation function of a Bose liquid at distance $r$.", + "$T$": "Temperature of the Bose liquid.", + "$m$": "Mass of the particles in the Bose liquid.", + "$u$": "Speed of sound (or phonon velocity) in the liquid.", + "$\\hbar$": "Reduced Planck's constant.", + "$r$": "Distance." + } + }, + { + "id": 189, + "context": "", + "question": "In a Bose superfluid, the condensate wave function exhibits fluctuations. Try to find the asymptotic form of this fluctuation correlation function at large distances.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$G(r)$": "Simultaneous correlation function.", + "$T$": "Temperature.", + "$n_0$": "Condensate density.", + "$m$": "Mass of a boson.", + "$\\pi$": "Mathematical constant pi.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\rho_s$": "Superfluid density.", + "$r$": "Distance." + } + }, + { + "id": 190, + "context": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral.", + "question": "Calculate the bandwidth of the body-centered cubic lattice;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\Delta E$": "Bandwidth of the body-centered cubic lattice", + "$J$": "Overlap integral" + } + }, + { + "id": 191, + "context": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral.", + "question": "Calculate the effective mass of electrons at the bottom of the band;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$m_{\\mathrm{b}}^{*}$": "Effective mass of electrons at the bottom of the band, defined as $m_{\\mathrm{b}}^{*}=\\frac{h^{2}}{8 \\pi^{2} a^{2} J}$.", + "$h$": "Planck's constant.", + "$\\pi$": "Mathematical constant pi.", + "$a$": "Lattice constant of the body-centered cubic lattice.", + "$J$": "Overlap integral, a parameter in the tight-binding approximation." + } + }, + { + "id": 192, + "context": "If only the interaction between nearest neighbors is considered, the energy band of $s$-state electrons in a body-centered cubic lattice derived by the tight-binding approximation method is\n\\begin{align*}\nE(\\mathbf{k}) = E_0 - A - \\delta J(\\cos\\pi a k_x \\cos\\pi a k_y \\cos\\pi a k_z)，\n\\end{align*}\nwhere $J$ is the overlap integral.", + "question": "Calculate the effective mass of electrons at the top of the band;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$m_{\\mathrm{t}}^{*}$": "Effective mass of electrons at the top of the band, $m_{\\mathrm{t}}^{*}=-\\frac{h^{2}}{8 \\pi^{2} a^{2} J}$.", + "$h$": "Planck's constant.", + "$a$": "Lattice constant.", + "$J$": "Overlap integral." + } + }, + { + "id": 193, + "context": "By considering only the nearest-neighbor interactions and using the tight-binding method, the energy band of s-state electrons in a simple cubic crystal is derived as\n$$\nE(\\boldsymbol{k})=E_{0}-A-2 J(\\cos 2 \\pi a k_{x}+\\cos 2 \\pi a k_{y}+\\cos 2 \\pi a k_{z})\n$$", + "question": "Find the bandwidth $(\\Delta E)$ ;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\Delta E$": "Bandwidth of the energy band", + "$J$": "Hopping integral (or overlap integral) between nearest neighbors" + } + }, + { + "id": 194, + "context": "", + "question": "The energy $E$ near the top of the valence band of a semiconductor crystal can be expressed as: $E(k)=E_{\\text {max }}-10^{26} k^{2}(\\mathrm{erg})$. Now, remove an electron with the wave vector $k=10^{7} \\mathrm{i} / \\mathrm{cm}$, and find the speed of the hole left by this electron.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "i": "Unit vector in the x-direction" + } + }, + { + "id": 195, + "context": "", + "question": "In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:\n$\\psi_{k}(x)=\\sin \\frac{\\pi}{a} x$: \nTry to use Bloch's theorem to point out the wave vector $\\mathbf{k}$ values within the reduced Brillouin zone.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$k$": "Wave vector.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$a$": "Lattice constant or period of the one-dimensional periodic potential." + } + }, + { + "id": 196, + "context": "", + "question": "In a one-dimensional periodic potential, the wavefunctions of electrons take the following form:\n$\\psi_{k}(x)=i \\cos \\frac{3 \\pi}{a} x$ ；\nTry to use Bloch's theorem to point out the wave vector $\\mathbf{k}$ values within the reduced Brillouin zone.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$k$": "Wave number, representing the magnitude of the wave vector in one dimension.", + "$\\pi$": "Mathematical constant pi.", + "$a$": "Periodicity length of the one-dimensional potential (lattice constant)." + } + }, + { + "id": 197, + "context": "", + "question": "The electron wave function moving in a one-dimensional periodic potential field has the following form: $\\psi_{k}(x)=\\sum_{l=-\\infty}^{\\infty}(-1)^{l} f(x-l a)$ . Here $a$ is the lattice constant of the one-dimensional lattice, $f(x)$ is a certain function, try using Bloch's theorem to indicate the values of the wave vector $\\boldsymbol{k}$ within the reduced Brillouin zone.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$k$": "Wave vector", + "$\\pi$": "Mathematical constant pi", + "$a$": "Lattice constant of the one-dimensional lattice" + } + }, + { + "id": 198, + "context": "It is known that the electronic energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nWhere $a$ is the lattice constant.", + "question": "Try to find: the width of the energy band;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\Delta E$": "Width of the energy band", + "$h$": "Planck's constant", + "$m_0$": "Mass of the electron", + "$a$": "Lattice constant of the one-dimensional crystal" + } + }, + { + "id": 199, + "context": "It is known that the electron energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nwhere $a$ is the lattice constant.", + "question": "Try to find the velocity of the electron at wave vector $k$;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$v(\\boldsymbol{k})$": "Velocity of the electron at wave vector $\\boldsymbol{k}$", + "$h$": "Planck's constant", + "$m_0$": "Mass of the electron", + "$a$": "Lattice constant", + "$k$": "Wave vector" + } + }, + { + "id": 200, + "context": "It is known that the electronic energy band of a one-dimensional crystal can be expressed as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nwhere $a$ is the lattice constant. Find:", + "question": "Find the effective mass of electrons at the bottom of the band.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$m_0$": "Rest mass of the electron." + } + }, + { + "id": 201, + "context": "It is known that the electronic band of a one-dimensional crystal can be written as\n$$\nE(k)=\\frac{h^{2}}{m_{0} a^{2}}(\\frac{7}{8}-\\cos 2 \\pi k a+\\frac{1}{8} \\cos 6 \\pi k a)\n$$\n\nIn this equation, $a$ is the lattice constant.", + "question": "Try to find the effective mass of electrons at the top of the band.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$m_0$": "Mass of the electron (rest mass)" + } + }, + { + "id": 202, + "context": "", + "question": "Given the lattice constant of a two-dimensional square lattice is $a$, if the electron energy can be expressed as\n$$\nE(k)=\\frac{h^{2}(k_{x}^{2}+k_{y}^{2})}{2 m_{\\mathrm{n}}^{*}}\n$$\n\nTry to find the density of states.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$g(E)$": "Number of states per unit energy interval, or the density of states, $g(E)=\\frac{\\mathrm{d} Z(E)}{\\mathrm{d} E}$", + "$\\pi$": "Mathematical constant pi", + "$S$": "Area of the crystal", + "$m_{\\mathrm{n}}^{*}$": "Effective mass of the electron", + "$h$": "Planck's constant" + } + }, + { + "id": 203, + "context": "", + "question": "Calculate the number of quantum states per unit volume between the energy $E=E_{\\mathrm{c}}$ and $E=E_{\\mathrm{c}}+100(\\frac{h^{2}}{8 m_{\\mathrm{n}}^{*} L^{2}})$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\pi$": "Mathematical constant pi", + "$L$": "Characteristic length, representing the dimension of the system" + } + }, + { + "id": 204, + "context": "For two pieces of n-type silicon material, at a certain temperature $T$, the ratio of the electron densities of the first piece to the second piece is $n_{1} / n_{2}=\\mathrm{e}$ (e is the base of the natural logarithm).", + "question": "If the Fermi level of the first piece of material is $3 k_{0} T$ below the conduction band edge, find the position of the Fermi level in the second piece of material.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$E_{\\mathrm{F} 2}$": "Fermi level of the second piece of material", + "$E_{\\mathrm{c}}$": "Conduction band edge", + "$k_0$": "Boltzmann constant", + "$T$": "Temperature" + } + }, + { + "id": 205, + "context": "", + "question": "For a p-type semiconductor, in the ionization region of impurities, the known relation is $\\frac{p_{0}(p_{0}+N_{\\mathrm{D}})}{N_{\\mathrm{A}}-N_{\\mathrm{D}}-p_{0}}=\\frac{N_{\\mathrm{v}}}{g} \\exp (-\\frac{E_{\\mathrm{A}}-E_{\\mathrm{v}}}{k_{0} T})$. When the condition $p_{0} \\ll N_{\\mathrm{D}}$ is satisfied, find the expression for hole density $p_{0}$. In the formula, $g$ is the spin degeneracy of the acceptor level.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$p_0$": "Hole density in the ionization region of impurities.", + "$N_{\\mathrm{A}}$": "Acceptor impurity concentration.", + "$N_{\\mathrm{D}}$": "Donor impurity concentration.", + "$N_{\\mathrm{v}}$": "Effective density of states in the valence band.", + "$g$": "Spin degeneracy of the acceptor level.", + "$E_{\\mathrm{A}}$": "Energy level of the acceptor impurities.", + "$E_{\\mathrm{v}}$": "Energy level of the valence band edge.", + "$k_0$": "Boltzmann constant.", + "$T$": "Absolute temperature." + } + }, + { + "id": 206, + "context": "", + "question": "There is an n-type semiconductor, in addition to the donor concentration $N_{\\mathrm{D}}$, it also contains a small amount of acceptors, with a concentration of $N_{\\mathrm{A}}$. Find the expression for the electron concentration under weak ionization conditions.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$n_0$": "Electron concentration in the conduction band", + "$N_{\\mathrm{c}}$": "Effective density of states in the conduction band", + "$N_{\\mathrm{D}}$": "Donor concentration", + "$N_{\\mathrm{A}}$": "Acceptor concentration", + "$E_{\\mathrm{c}}$": "Conduction band minimum energy", + "$E_{\\mathrm{D}}$": "Donor level energy", + "$k_0$": "Boltzmann constant", + "$T$": "Temperature" + } + }, + { + "id": 207, + "context": "", + "question": "Please explain why at room temperature, for a certain semiconductor, the electron concentration $n=n_{i} \\sqrt{\\mu_{\\mathrm{p}} / \\mu_{\\mathrm{n}}}$ results in the minimum electrical conductivity $\\sigma$. In this equation, $n_{\\mathrm{i}}$ is the intrinsic carrier concentration, and $\\mu_{\\mathrm{p}}, ~ \\mu_{\\mathrm{n}}$ are the mobilities of holes and electrons respectively. Find the hole concentration under the above condition.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$p_{0}$": "Hole concentration under minimum conductivity condition, $p_{0}=n_{\\mathrm{i}} \\sqrt{\\frac{\\mu_{\\mathrm{n}}}{\\mu_{\\mathrm{p}}}}$", + "$n_{\\mathrm{i}}$": "Intrinsic carrier concentration", + "$\\mu_{\\mathrm{n}}$": "Mobility of electrons", + "$\\mu_{\\mathrm{p}}$": "Mobility of holes" + } + }, + { + "id": 208, + "context": "", + "question": "Suppose a semiconductor crystal is subjected to an electric field $\\boldsymbol{E}$ and a magnetic field $\\boldsymbol{B}$, with $\\boldsymbol{E}$ in the $x-y$ plane and $\\boldsymbol{B}$ along the $z$ direction, try to derive the distribution function of semiconductor electrons in the electromagnetic field considering the multiple interactions of the magnetic field with electrons.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$f$": "Distribution function of semiconductor electrons.", + "$f_0$": "Equilibrium distribution function.", + "$\\tau$": "Relaxation time.", + "$v_x$": "x-component of electron velocity.", + "$f_1$": "Auxiliary function, $f_1=\\frac{\\partial f_{0}}{\\partial x}-q \\mathscr{E}_{x} \\frac{\\partial f_{0}}{\\partial E}$.", + "$k$": "Dimensionless parameter related to the magnetic field, $k=\\frac{q l}{m v} B$.", + "$f_2$": "Auxiliary function, $f_2=\\frac{\\partial f_{0}}{\\partial y}-q \\mathscr{E}_{y} \\frac{\\partial f_{0}}{\\partial E}$.", + "$v_y$": "y-component of electron velocity." + } + }, + { + "id": 209, + "context": "", + "question": "Assuming $\\tau_{\\mathrm{n}}=\\tau_{\\mathrm{p}}=\\tau_{0}$ is a constant that does not change with the doping density in the sample, find the value of conductivity when the small-signal lifetime of the sample reaches its maximum.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\tau_{\\max }$": "Maximum small-signal lifetime.", + "$\\tau_{0}$": "A constant lifetime that does not change with the doping density in the sample.", + "$n_{1}$": "Electron concentration when the Fermi level is at the trap energy level.", + "$p_{1}$": "Hole concentration when the Fermi level is at the trap energy level.", + "$n_{\\mathrm{i}}$": "Intrinsic carrier concentration.", + "$E_{\\mathrm{t}}$": "Trap energy level.", + "$E_{\\mathrm{i}}$": "Intrinsic Fermi energy level.", + "$k$": "Boltzmann constant.", + "$T$": "Absolute temperature." + } + }, + { + "id": 210, + "context": "", + "question": "Assume $\\tau_{\\mathrm{p}}=\\tau_{\\mathrm{n}}=\\tau_{0}$, and based on the small signal lifetime formula\n\n\\tau=\\tau_{\\mathrm{p}} \\frac{n_{0}+n_{1}}{n_{0}+p_{0}}+\\tau_{\\mathrm{n}} \\frac{p_{0}+p_{1}}{n_{0}+p_{0}}\n\n\ndiscuss the relationship between the lifetime $\\tau$ and the position of the recombination center level $E_{\\mathrm{t}}$ in the band gap, and briefly explain its physical significance.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\tau$": "Small signal lifetime.", + "$\\tau_{0}$": "A characteristic lifetime constant, equal to the lifetime for holes and electrons under the given assumption.", + "$n_{\\mathrm{i}}$": "Intrinsic carrier concentration.", + "$n_{0}$": "Equilibrium electron concentration.", + "$p_{0}$": "Equilibrium hole concentration.", + "$E_{\\mathrm{t}}$": "Energy level of the recombination center.", + "$E_{\\mathrm{i}}$": "Intrinsic Fermi level.", + "$k$": "Boltzmann constant.", + "$T$": "Absolute temperature." + } + }, + { + "id": 211, + "context": "Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by an electron, $N_{\\mathrm{c}}$ and $N_{\\mathrm{v}}$ are the effective density of states of the conduction band and the valence band, respectively.", + "question": "Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by an electron, and $N_{\\mathrm{c}}$ be the effective density of states of the conduction band. Consider the rate equation for electrons:\n \\frac{\\mathrm{d} n}{\\mathrm{~d} t}=-\\frac{n(1-f_{\\mathrm{t}})}{\\tau_{\\mathrm{n}}}+\\frac{N_{\\mathrm{c}} f_{\\mathrm{t}}}{\\tau_{\\mathrm{n}}^{\\prime}} \nwhere $\\tau_{\\mathrm{n}}$ and $\\tau_{\\mathrm{n}}^{\\prime}$ are the characteristic time constants related to electron capture and emission, respectively. Under thermal equilibrium conditions, $\\tau_{\\mathrm{n}}^{\\prime}$ can be expressed in terms of $\\tau_{\\mathrm{n}}$, $N_{\\mathrm{c}}$, and parameter $n_1$ (where $n_1 = N_{\\mathrm{c}} \\exp (-\\frac{E_{\\mathrm{c}}-E_{\\mathrm{t}}}{k_{0} T})$, $E_c$ is the conduction band edge energy, $k_0$ is the Boltzmann constant, and $T$ is the temperature). Find the expression for $\\tau_{\\mathrm{n}}^{\\prime}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\tau_{\\mathrm{n}}$": "Characteristic time constant related to electron capture.", + "$N_{\\mathrm{c}}$": "Effective density of states of the conduction band.", + "$n_{1}$": "Parameter related to electron emission, defined as $n_1 = N_{\\mathrm{c}} \\exp (-(E_{\\mathrm{c}}-E_{\\mathrm{t}})/(k_{0} T))$." + } + }, + { + "id": 212, + "context": "Let $f_{\\mathrm{t}}$ be the probability that the composite center energy level $E_{\\mathrm{t}}$ is occupied by electrons, and $N_{\\mathrm{c}}$ and $N_{\\mathrm{v}}$ be the effective density of states for the conduction band and the valence band, respectively.", + "question": "Derive the small-signal lifetime formula.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\tau$": "Small-signal lifetime.", + "$\\tau_{\\mathrm{p}}$": "Hole lifetime.", + "$n_{0}$": "Equilibrium electron concentration.", + "$n_{1}$": "Electron concentration when the Fermi level is at the trap level.", + "$p_{0}$": "Equilibrium hole concentration.", + "$\\tau_{\\mathrm{n}}$": "Electron lifetime.", + "$p_{1}$": "Hole concentration when the Fermi level is at the trap level." + } + }, + { + "id": 213, + "context": "A square pulse with an appropriate frequency is irradiated onto an n-type semiconductor sample and is uniformly absorbed inside the sample to generate nonequilibrium carriers at a generation rate of $g_{\\mathrm{p}}$. The lifetime of nonequilibrium holes is $\\tau_{\\mathrm{p}}$, and the pulse width is $\\Delta t=3 \\tau_{\\mathrm{p}}$.", + "question": "Assume the moment the pulse light starts irradiating is $t=0$, find the expression for the concentration of nonequilibrium holes $\\Delta p(\\Delta t)$ at the moment the light pulse ends ($t=\\Delta t$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$g_{\\mathrm{p}}$": "Generation rate of nonequilibrium carriers.", + "$\\tau_{\\mathrm{p}}$": "Lifetime of nonequilibrium holes.", + "$\\mathrm{e}$": "Base of the natural logarithm." + } + }, + { + "id": 214, + "context": "", + "question": "Using the bipolar diffusion theory, consider the case of intrinsic semiconductors (i.e., the electron concentration $n$ is equal to the hole concentration $p$). It is known that the general expression for the bipolar diffusion coefficient is $D^{*}=\\frac{(n+p) D_{\\mathrm{n}} D_{\\mathrm{p}}}{n D_{\\mathrm{n}}+p D_{\\mathrm{p}}}$. Try to derive the specific expression for the bipolar diffusion coefficient $D^{*}$ in intrinsic semiconductors.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$D^{*}$": "bipolar diffusion coefficient", + "$D_{\\mathrm{n}}$": "electron diffusion coefficient", + "$D_{\\mathrm{p}}$": "hole diffusion coefficient" + } + }, + { + "id": 215, + "context": "For a silicon pn junction, the doping concentrations of the p and n regions are $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$, respectively; the hole and electron mobilities in the p region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively, while in the n region, they are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Assume the lifetime of the non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}$, and the cross-sectional area of the pn junction is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, calculate:", + "question": "Determine the expression for the hole diffusion current variation with $x$ in the n region.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$I_{\\mathrm{pD}}$": "Hole diffusion current.", + "$x$": "Position." + } + }, + { + "id": 216, + "context": "There is a silicon pn junction, with doping concentrations in the p-region and n-region of $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively, and in the n-region, the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively; assume the lifetime of non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}, \\mathrm{pn}$ junction cross-sectional area is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$ . When the applied forward voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$, try to find:", + "question": "Determine the expression for the electron diffusion current variation with $x$ in the n-region.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$I_{\\mathrm{nD}}$": "Electron diffusion current", + "$x$": "Position coordinate" + } + }, + { + "id": 217, + "context": "There is a silicon pn junction, with doping concentrations in the p-region and n-region being $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$, respectively. The hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively, and in the n-region, the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively. Assume the lifetime of non-equilibrium carriers in both regions is $1 \\mu \\mathrm{~s}$, the area of the pn junction is $10^{-2} \\mathrm{~cm}^{2}$; $\\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward bias voltage of $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, determine:", + "question": "Determine the expression for the variation of electron drift current with $x$ in the n-region.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$I_{\\mathrm{nt}}$": "Electron drift current", + "$\\mathrm{e}$": "Base of the natural logarithm, approximately $2.71828$", + "$x$": "Position coordinate" + } + }, + { + "id": 218, + "context": "There is a silicon pn junction, with the doping concentrations of the p-region and n-region being $N_{\\mathrm{A}}=9 \\times 10^{15} \\mathrm{~cm}^{-3}$ and $N_{\\mathrm{D}}=2 \\times 10^{16} \\mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively, while in the n-region the hole and electron mobilities are $300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ respectively; assuming the minority carrier lifetime in both regions is $1 \\mu \\mathrm{~s}, \\mathrm{pn}$ junction cross-sectional area is $10^{-2} \\mathrm{~cm}^{2} ; \\frac{q}{k_{0} T}=38.7(\\frac{1}{V})$. When a forward bias voltage $V_{\\mathrm{F}}=0.65 \\mathrm{~V}$ is applied, find:", + "question": "Determine the expression for the total electron current in the n-region as a function of $x$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$I_{\\mathrm{n}}$": "Total electron current, defined as $I_{\\mathrm{n}}=I_{\\mathrm{nt}}+I_{\\mathrm{nD}}$", + "$x$": "Position coordinate" + } + }, + { + "id": 219, + "context": "", + "question": "A metal contacts uniformly doped $n-Si$ material to form a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$, find the relationship curve of $1 / C^{2}$ versus $(V_{\\mathrm{D}}-V)$ under a 5V reverse bias voltage.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$C$": "Unit area barrier capacitance.", + "$V_{\\mathrm{D}}$": "Built-in potential (diffusion voltage) on the semiconductor side.", + "$V$": "Applied bias voltage.", + "$\\varepsilon_{0}$": "Permittivity of free space.", + "$\\varepsilon_{\\mathrm{rs}}$": "Relative static permittivity (dielectric constant) of the semiconductor (given as 10 in the text, but 12 is used in calculations).", + "$q$": "Elementary charge.", + "$N$": "Dopant concentration, used in formulas for maximum electric field and capacitance, representing $N_{\\mathrm{D}}$." + } + }, + { + "id": 220, + "context": "A metal plate is 0.4 $\\mu \\mathrm{~m}$ away from n-type silicon, forming a parallel plate capacitor, with dry air in between having a relative permittivity $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state.", + "question": "Find the expression for depletion layer width $X_{\\mathrm{d}}$ when $V_{\\mathrm{s}}=0.4 \\mathrm{~V}$;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$X_{\\mathrm{d}}$": "Depletion layer width", + "$\\varepsilon_{\\mathrm{rs}}$": "Relative permittivity of n-type silicon", + "$\\varepsilon_{0}$": "Permittivity of free space", + "$V_{\\mathrm{s}}$": "Surface potential (or voltage applied to the semiconductor surface)", + "$q$": "Elementary charge", + "$N_{\\mathrm{D}}$": "Donor concentration in n-type silicon" + } + }, + { + "id": 221, + "context": "A metal plate is separated from n-type silicon by a distance of $0.4 \\mu \\mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air in between is $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal end, the semiconductor is in a depletion state.", + "question": "Find the expression for the maximum depletion layer width $X_{\\mathrm{dm}}$ when $V_{\\mathrm{s}}=0.4 \\mathrm{~V}$;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$X_{\\mathrm{dm}}$": "Maximum depletion layer width.", + "$\\varepsilon_{\\mathrm{rs}}$": "Relative permittivity of the semiconductor (silicon).", + "$\\varepsilon_{0}$": "Permittivity of free space.", + "$V_{\\mathrm{sm}}$": "Surface potential at maximum depletion layer width, defined as $V_{\\mathrm{sm}} = \\frac{2 k_{0} T}{q} \\ln (\\frac{N_{\\mathrm{D}}}{n_{\\mathrm{i}}})$.", + "$q$": "Elementary charge.", + "$N_{\\mathrm{D}}$": "Donor doping concentration.", + "$k_{0}$": "Boltzmann constant.", + "$T$": "Absolute temperature.", + "$n_{\\mathrm{i}}$": "Intrinsic carrier concentration." + } + }, + { + "id": 222, + "context": "For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is applied in the $x$ direction, and a magnetic field of 0.5 T is applied in the $z$ direction, resulting in a Hall voltage of -0.4 mV, find:", + "question": "Given the resistivity of the material is $1.5 \\times 10^{-3} \\Omega \\cdot \\mathrm{~cm}$, find the carrier mobility.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\mu_{\\mathrm{H}}$": "Hall mobility of the carriers, defined as $\\mu_{\\mathrm{H}}= |R_{\\mathrm{H}} \\sigma_{0}|=\\frac{1}{\\rho_{0}}|R_{\\mathrm{H}}|$" + } + }, + { + "id": 223, + "context": "", + "question": "Assume the relaxation time $\\tau$ is constant, and try to calculate the Hall coefficient of an n-type semiconductor.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$R$": "Hall coefficient.", + "$n$": "Electron concentration (number density).", + "$e$": "Elementary charge of an electron." + } + }, + { + "id": 224, + "context": "In the experiment of measuring the Hall coefficient of a semiconductor, the current $J_{x}$ induced by the external electric field $\\varepsilon_{x}$ is called the original current. The transverse current generated under the action of the Lorentz force and the Hall electric field $\\varepsilon_{y}$ includes the electron current component $J_{\\mathrm{n} y}$ and the hole current component $J_{\\mathrm{p} y}$.", + "question": "Find the ratio $f_{\\mathrm{c}}=J_{\\mathrm{ny}} / J_{x}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$f_{\\mathrm{c}}$": "Ratio of transverse electron current to the original current, $f_{\\mathrm{c}}=J_{\\mathrm{ny}} / J_{x}$", + "$B_{z}$": "Magnetic field along the $z$ axis", + "$\\mu_{\\mathrm{p}}$": "Mobility of holes", + "$\\mu_{\\mathrm{n}}$": "Mobility of electrons", + "$\\sigma_{\\mathrm{p}}$": "Conductivity of holes, $\\sigma_{\\mathrm{p}} = p e \\mu_{\\mathrm{p}}$", + "$\\sigma_{\\mathrm{n}}$": "Conductivity of electrons, $\\sigma_{\\mathrm{n}} = n e \\mu_{\\mathrm{n}}$" + } + }, + { + "id": 225, + "context": "Try to prove that in the Hall effect under the conditions of simultaneous presence of two types of charge carriers and a weak magnetic field, the Hall angle $\\theta$ and the Hall coefficient $R$ can be expressed as\n\n\\begin{aligned}\n& \\theta=\\arctan \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}}} B_{z} \\\\\n& R=\\frac{1}{q} \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{(p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}})^{2}}\n\\end{aligned}", + "question": "If the Hall angle of a sample is measured to be $\\theta=0$, find the corresponding electrical conductivity;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\sigma$": "Electrical conductivity, defined as $\\sigma =q(p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}})$", + "$q$": "Elementary charge", + "$n$": "Concentration of n-type charge carriers (electrons)", + "$\\mu_{\\mathrm{n}}$": "Mobility of n-type charge carriers (electrons)", + "$\\mu_{\\mathrm{p}}$": "Mobility of p-type charge carriers (holes)" + } + }, + { + "id": 226, + "context": "In the experiment of measuring the Hall coefficient of semiconductors, the current induced by the external electric field $\\mathscr{E}_{x}$ is called the primary current. Under the action of Lorentz force and Hall electric field, a transverse electron current $J_{\\mathrm{ey}}$ and a transverse hole current $J_{\\mathrm{py}}$ are generated.", + "question": "Find the ratio of the transverse electron current to the primary current at equilibrium $f_{\\mathrm{e}}=\\frac{J_{\\mathrm{ey}}}{J_{x}}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$f_{\\mathrm{e}}$": "Ratio of the transverse electron current to the primary current, $f_{\\mathrm{e}}=\\frac{J_{\\mathrm{ey}}}{J_{x}}$.", + "$B$": "Magnetic field along the positive z-axis.", + "$\\mu_{\\mathrm{h}}$": "Hole mobility.", + "$\\mu_{\\mathrm{e}}$": "Electron mobility.", + "$\\sigma_{\\mathrm{h}}$": "Hole conductivity.", + "$\\sigma_{\\mathrm{e}}$": "Electron conductivity." + } + }, + { + "id": 227, + "context": "N atoms form a two-dimensional square lattice, with each atom contributing one electron to form a two-dimensional free electron gas. The electron energy expression is\n$$\nE(k)=\\frac{\\hbar^{2} k_{x}^{2}}{2 m}+\\frac{\\hbar^{2} k_{y}^{2}}{2 m}\n$$", + "question": "Derive the formula for the density of st ates of a two-dimensional free gas.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$g(E)$": "Density of states of the two-dimensional free electron gas per unit area", + "$m$": "Mass of the electron", + "$\\pi$": "Mathematical constant pi", + "$\\hbar$": "Reduced Planck's constant" + } + }, + { + "id": 228, + "context": "A two-dimensional square lattice composed of N atoms, each contributing 1 electron to form a two-dimensional free electron gas. The expression for the electron energy is\n$$\nE(k)=\\frac{\\hbar^{2} k_{x}^{2}}{2 m}+\\frac{\\hbar^{2} k_{y}^{2}}{2 m}\n$$", + "question": "At this time, a magnetic field B is applied perpendicular to the square lattice. The energy levels of the free electron gas will condense into Landau levels. What is the degeneracy of these levels?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$D$": "Degeneracy of the Landau levels", + "$e$": "Elementary charge", + "$B$": "Applied magnetic field perpendicular to the square lattice", + "$\\hbar$": "Reduced Planck's constant" + } + }, + { + "id": 229, + "context": "", + "question": "A particle is incident with kinetic energy $E$, subjected to the following double $\\delta$ potential barriers:\n\nV(x)=V_{0}[\\delta(x)+\\delta(x-a)]\n\n\nFind the expression for the conditions under which complete transmission occurs. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mathrm{e}$": "Base of the natural logarithm.", + "$\\mathrm{i}$": "Imaginary unit.", + "$a$": "Distance between the two delta potential barriers.", + "$k$": "Wave number, defined as $k=\\sqrt{2 m E} / \\hbar$.", + "$\\theta$": "Dimensionless parameter, defined as $\\theta=\\frac{2 a k}{C}=\\frac{k \\hbar^{2}}{m V_{0}}$." + } + }, + { + "id": 230, + "context": "", + "question": "For the energy eigenstate $|n\\rangle$ of the harmonic oscillator, calculate the expression for the uncertainty product $\\Delta x \\cdot \\Delta p$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\Delta x$": "Uncertainty in position.", + "$\\Delta p$": "Uncertainty in momentum.", + "$n$": "Quantum number representing the energy level of the harmonic oscillator.", + "$\\hbar$": "Reduced Planck's constant." + } + }, + { + "id": 231, + "context": "", + "question": "In the coherent state $|\\alpha\\rangle$, calculate the uncertainty product $\\Delta x \\cdot \\Delta p$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\Delta x \\cdot \\Delta p$": "Uncertainty product of position and momentum.", + "$\\hbar$": "Reduced Planck's constant." + } + }, + { + "id": 232, + "context": "", + "question": "Define the radial momentum operator\n\n\\begin{equation*}\n\\boldsymbol{p}_{r}=\\frac{1}{2}(\\frac{\\boldsymbol{r}}{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\frac{\\boldsymbol{r}}{r}) \\tag{1}\n\\end{equation*}\n\n\nFind the commutation relation $[r, p_{r}]$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$r$": "Magnitude of the position vector, also known as the radial coordinate", + "$p_{r}$": "Radial momentum operator, defined as $\\frac{1}{2}(\\frac{\\boldsymbol{r}}{r} \\cdot \\boldsymbol{p}+\\boldsymbol{p} \\cdot \\frac{\\boldsymbol{r}}{r})$", + "$\\mathrm{i}$": "Imaginary unit", + "$\\hbar$": "Reduced Planck's constant" + } + }, + { + "id": 233, + "context": "", + "question": "A particle with mass $\\mu$ moves in a central potential field\n\n\\begin{equation*}\nV(r)=\\lambda r^{\\nu}, \\quad-2<\\nu<\\infty \\tag{1}\n\\end{equation*}\n\n\nWe only discuss the case where bound states can exist, i.e., when $\\lambda \\nu > 0$. The radial wave function $u(r)=rR(r)$ satisfies the following radial Schrödinger equation:\n\n\\begin{equation*}\n\\frac{\\hbar^{2}}{2 \\mu} \\frac{\\mathrm{~d}^{2} u}{\\mathrm{~d} r^{2}}+[E-\\lambda r^{\\nu}-l(l+1) \\frac{\\hbar^{2}}{2 \\mu r^{2}}] u=0 \\tag{2}\n\\end{equation*}\n\nBy introducing the dimensionless radial distance $\\rho$ and energy $\\varepsilon$, and denoting the radial function as $w(\\rho)=u(r)$, the above equation can be non-dimensionalized. Please write down the dimensionless radial equation in terms of $w(\\rho)$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$w$": "Dimensionless radial function, $w(\\rho)=u(r)$", + "$\\rho$": "Dimensionless radial distance, defined as $\\rho=r(\\frac{2 \\mu|\\lambda|}{\\hbar^{2}})^{\\frac{1}{2 \\hbar \\imath}}$ (Note: `2 \\hbar \\imath` is likely a typo for `2+\\nu`)", + "$\\varepsilon$": "Dimensionless energy, defined as $\\varepsilon=E \\frac{2 \\mu}{\\hbar^{2}}(\\frac{\\hbar^{2}}{2 \\mu|\\lambda|})^{\\frac{2}{2 \\hbar \\imath}}$ (Note: `2 \\hbar \\imath` is likely a typo for `2+\\nu`)", + "$\\lambda$": "Parameter in the central potential field, $V(r)=\\lambda r^{\\nu}$", + "$\\nu$": "Exponent in the central potential field, $V(r)=\\lambda r^{\\nu}$", + "$l$": "Orbital angular momentum quantum number" + } + }, + { + "id": 234, + "context": "", + "question": "For the hydrogen atom's s states $(n l m=n 00)$, calculate the expression for the uncertainty product $\\Delta x \\cdot \\Delta p_{x}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\Delta x$": "Uncertainty in the position in the x-direction, defined as $\\sqrt{\\langle x^{2}\\rangle}$ given that $\\langle x\\rangle=0$.", + "$\\Delta p_{x}$": "Uncertainty in the momentum in the x-direction, defined as $\\sqrt{\\langle p_{x}^{2}\\rangle}$ given that $\\langle p_{x}\\rangle=0$.", + "$n$": "Principal quantum number for the hydrogen atom.", + "$\\hbar$": "Reduced Planck's constant." + } + }, + { + "id": 235, + "context": "", + "question": "For electrons and other spin $1 / 2$ particles, the eigenstates of $s_{z}$ are often denoted by $\\alpha$ and $\\beta$, where $\\alpha$ is equivalent to $\\chi_{\\frac{1}{2}}$, and $\\beta$ is equivalent to $\\chi_{-\\frac{1}{2}}$. Given the electron wave function\n\n\\begin{equation*}\n\\psi(r, \\theta, \\varphi, s_{z})=\\alpha Y_{l 0}(\\theta, \\varphi) R(r) \n\\end{equation*}\n\n\nfind the only possible measurement value of the total angular momentum $j_{z}$ (taking $\\hbar=1$).", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 236, + "context": "", + "question": "If the operator $\\hat{f}(x)$ commutes with $\\hat{D}_{x}(a)$, find the general solution for $\\hat{f}(x)$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\hat{f}(x)$": "An operator, a function of position $x$.", + "$x$": "Position coordinate.", + "$a$": "The period of the function $\\hat{f}(x)$." + } + }, + { + "id": 237, + "context": "", + "question": "For a spin $1 / 2$ particle, find the effect of the operator $\\sigma_{r}=\\boldsymbol{\\sigma} \\cdot \\boldsymbol{r} / \\boldsymbol{r}$ acting on the common eigenfunctions $\\phi_{l j m_{j}}$ of $(l^{2}, j^{2}, j_{z})$ (taking $\\hbar=1$). You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\sigma_r$": "Operator defined as $\\boldsymbol{\\sigma} \\cdot \\boldsymbol{r} / r$", + "$\\phi_{j m_{j}}^{A}$": "Specific form of eigenfunction for $j=l+1/2$", + "$\\phi_{j m_{j}}^{B}$": "Specific form of eigenfunction for $j=l'-1/2$" + } + }, + { + "id": 238, + "context": "", + "question": "For a system composed of two spin $1/2$ particles, where $\\boldsymbol{s}_{1}, ~ \\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{s}_{2}, ~ \\boldsymbol{\\sigma}_{2}$ represent the spin angular momentum and Pauli operators for particles 1 and 2, respectively, $\\boldsymbol{s}_{1}=\\frac{1}{2} \\boldsymbol{\\sigma}_{1}, \\boldsymbol{s}_{2}=\\frac{1}{2} \\boldsymbol{\\sigma}_{2}$ (taking $\\hbar=1$). Find the simplest algebraic equation satisfied by $\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2}$": "Dot product of the Pauli operators for particle 1 and particle 2, which is the quantity for which an algebraic equation is sought." + } + }, + { + "id": 239, + "context": "Majorana fermions. One can write a relativistic equation for a massless 2 -component fermion field that transforms as the upper two components of a Dirac spinor $(\\psi_{L})$. Call such a 2-component field $\\chi_{a}(x), a=1,2.$", + "question": "Let us write a 4 -component Dirac field as\n\n\\psi(x)=\\binom{\\psi_{L}}{\\psi_{R}}\n\nand recall that the lower components of $\\psi$ transform in a way equivalent by a unitary transformation to the complex conjugate of the representation $\\psi_{L}$. In this way, we can rewrite the 4 -component Dirac field in terms of two 2 -component spinors:\n\n\\psi_{L}(x)=\\chi_{1}(x), \\quad \\psi_{R}(x)=i \\sigma^{2} \\chi_{2}^{*}(x)\n\n\nFrom the Dirac Lagrangian $\\mathcal{L} = \\bar{\\psi}(\\mathrm{i} \\not \\partial-m) \\psi$ rewritten in terms of $\\chi_{1}$ and $\\chi_{2}$, identify and state the form of the mass term.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{i}$": "The imaginary unit.", + "m": "The mass parameter.", + "$\\chi_{2}^{T}$": "Transpose of the 2-component spinor $\\chi_2$.", + "$\\sigma^{2}$": "The second Pauli matrix.", + "$\\chi_{1}$": "A 2-component spinor, defined as $\\psi_L$.", + "$\\chi_{1}^{\\dagger}$": "Hermitian conjugate of the 2-component spinor $\\chi_1$.", + "$\\chi_{2}^{*}$": "Complex conjugate of the 2-component spinor $\\chi_2$." + } + }, + { + "id": 240, + "context": "Fierz transformations. Let $u_{i}, i=1, \\ldots, 4$, be four 4 -component Dirac spinors. In the text, we proved the Fierz rearrangement formulaes. The first of these formulae can be written in 4 -component notation as\n$$\n\\bar{u}_{1} \\gamma^{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{2} \\bar{u}_{3} \\gamma_{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{4}=-\\bar{u}_{1} \\gamma^{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{4} \\bar{u}_{3} \\gamma_{\\mu}(\\frac{1+\\gamma^{5}}{2}) u_{2} .\n$$\n\nIn fact, there are similar rearrangement formulae for any product\n$$\n(\\bar{u}_{1} \\Gamma^{A} u_{2})(\\bar{u}_{3} \\Gamma^{B} u_{4}),\n$$\nwhere $\\Gamma^{A}, \\Gamma^{B}$ are any of the 16 combinations of Dirac matrices.", + "question": "To begin, normalize the 16 matrices $\\Gamma^{A}$ to the convention\n$$\n\\operatorname{tr}[\\Gamma^{A} \\Gamma^{B}]=4 \\delta^{A B} .\n$$\n\nThis gives $\\Gamma^{A}={1, \\gamma^{0}, i \\gamma^{j}, \\ldots}$; write all 16 elements of this set. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$1$": "Identity matrix, representing the unit element in the set of Dirac matrices.", + "$\\gamma^0$": "Time-like Dirac gamma matrix.", + "$i\\gamma^i$": "Normalized spatial Dirac gamma matrix, where `i` is the imaginary unit and $i$ is a spatial index.", + "$i\\sigma^{0i}$": "Normalized component of the $\\sigma^{\\mu \\nu}$ matrix, where `i` is the imaginary unit, $\\mu=0$ and $\\nu=i$ (spatial index).", + "$\\sigma^{ij}$": "Normalized component of the $\\sigma^{\\mu \\nu}$ matrix, where $i, j$ are spatial indices.", + "$\\gamma^5$": "Chiral gamma matrix, defined as $\\gamma^{5}=-\\mathrm{i} \\gamma^{0} \\gamma^{1} \\gamma^{2} \\gamma^{3}$.", + "$i\\gamma^5\\gamma^0$": "Normalized product of chiral gamma matrix and time-like gamma matrix, where `i` is the imaginary unit.", + "$\\gamma^5\\gamma^i$": "Normalized product of chiral gamma matrix and spatial gamma matrix, where $i$ is a spatial index." + } + }, + { + "id": 241, + "context": "This problem concerns the discrete symmetries $P, C$, and $T$.", + "question": "Compute the transformation property under $C$ of the antisymmetric tensor fermion bilinear $\\bar{\\psi} \\sigma^{\\mu \\nu} \\psi$, with $\\sigma^{\\mu \\nu}=\\frac{i}{2}[\\gamma^{\\mu}, \\gamma^{\\nu}]$. This completes the table of the transformation properties of bilinears at the end of the chapter. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$C$": "Discrete symmetry operator for Charge Conjugation.", + "$\\bar{\\psi}$": "Dirac adjoint of the fermion field.", + "$t$": "Time coordinate.", + "$\\mathbf{x}$": "Spatial coordinate vector.", + "$\\sigma^{\\mu \\nu}$": "Antisymmetric tensor, defined as $\\sigma^{\\mu \\nu}=\\frac{\\mathrm{i}}{2}[\\gamma^{\\mu}, \\gamma^{\\nu}]$.", + "$\\psi$": "Fermion field.", + "$\\mu$": "Lorentz index.", + "$\\nu$": "Lorentz index." + } + }, + { + "id": 242, + "context": "Exotic contributions to $\\boldsymbol{g} \\mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles.\n\nThe unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to\n\\begin{align*}\nH_{\\text{int}} = \\int d^3 x \\frac{\\lambda}{\\sqrt{2}} h \\bar{\\psi} \\psi.\n\\end{align*}\nOne can study the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\\lambda$ and the mass $m_h$ of the Higgs boson.", + "question": "QED accounts extremely well for the electron's anomalous magnetic moment. If $a=(g-2) / 2$,\n$$\n|a_{\\text {expt. }}-a_{\\mathrm{QED}}|<1 \\times 10^{-10}\n$$\n\nWhat limits does this place on $\\lambda$ and $m_{h}$ ? In the simplest version of the electroweak theory, $\\lambda=3 \\times 10^{-6}$ and $m_{h}>60 \\mathrm{GeV}$. Show that these values are not excluded. \n\nHint: You can find the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\\lambda$ and the mass $m_h$ of the Higgs boson and check its value with $1 \\times 10^{-10}$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$\\lambda$": "Coupling constant between the Higgs boson and the electron", + "$\\pi$": "Mathematical constant pi", + "$m_h$": "Mass of the Higgs boson", + "$m$": "Mass of the electron", + "$\\delta a_h$": "Contribution of a virtual Higgs boson to the electron's anomalous magnetic moment", + "$m_e$": "Mass of the electron" + } + }, + { + "id": 243, + "context": "Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This \"final project\" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections. \n\nStrongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.\n\nLet us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:\n\n\\Delta H=\\int d^{3} x g \\bar{\\psi}_{f i} \\gamma^{\\mu} \\psi_{f i} B_{\\mu}\n\n\nHere $f$ labels the type (\"flavor\") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\\alpha$, let us define\n\n\\alpha_{g}=\\frac{g^{2}}{4 \\pi}\n\n\nIn this exercise, we will compute the radiative corrections to quark pair production proportional to $\\alpha_{g}$.\n\nThis model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also\ncorrect in QCD with the replacement\n$$\n\\alpha_{g} \\rightarrow \\frac{4}{3} \\alpha_{s} .\n$$\n\nThroughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\\mu$, which can be taken to zero only at the end of the calculation. However, it is consistent to sum over polarization states of this massive boson by the replacement:\n$$\n\\sum \\epsilon^{\\mu} \\epsilon^{\\nu *} \\rightarrow-g^{\\mu \\nu}\n$$\nthis also implies that we may use the propagator\n$$\n\\widehat{B^{\\mu} B^{\\nu}}=\\frac{-i g^{\\mu \\nu}}{k^{2}-\\mu^{2}+i \\epsilon}\n$$", + "question": "In the analysis of the 3-body final state process $e^{+} e^{-} \\rightarrow \\bar{q} q g$, the total 4-momentum is $q$. The final quark (4-momentum $k_1$) and antiquark (4-momentum $k_2$) are massless, while the gluon (4-momentum $k_3$) has mass $\\mu$. Dimensionless energy fractions are defined as $x_i = \\frac{2 k_i \\cdot q}{q^{2}}$. The physical integration region for $x_1$ and $x_2$ is bounded. One of these boundaries corresponds to the kinematic configuration where the 3-momenta of the quark ($\\mathbf{k}_1$) and the antiquark ($\\mathbf{k}_2$) are parallel. Determine the equation for this specific boundary in terms of $x_1$, $x_2$, $\\mu^2$, and $q^2$ (the square of the total 4-momentum). You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$x_1$": "Dimensionless energy fraction of the final quark, defined as $x_1 = \\frac{2 k_1 \\cdot q}{q^{2}}$.", + "$x_2$": "Dimensionless energy fraction of the final antiquark, defined as $x_2 = \\frac{2 k_2 \\cdot q}{q^{2}}$.", + "$\\mu$": "Small nonzero mass of the gluon.", + "$q$": "Total 4-momentum of the $e^+ e^-$ annihilation system.", + "$s$": "Mandelstam variable, representing the square of the total 4-momentum, $s = q^2$." + } + }, + { + "id": 244, + "context": "", + "question": "Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,\n$$\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{\\lambda}{4!} \\phi^{4}+\\bar{\\psi}(i \\not \\partial) \\psi-i g \\bar{\\psi} \\gamma^{5} \\psi \\phi,\n$$\ncompute the Callan-Symanzik $\\beta$ function for $g$:\n$$\n\\beta_{g}(\\lambda, g),\n$$\nto leading order in coupling constants, assuming that $\\lambda$ and $g^{2}$ are of the same order.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\beta_{g}$": "Callan-Symanzik beta function for the coupling constant $g$.", + "$g$": "Yukawa coupling constant.", + "$\\pi$": "Mathematical constant pi." + } + }, + { + "id": 245, + "context": "", + "question": "Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,\n$$\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu} \\phi)^{2}-\\frac{\\lambda}{4!} \\phi^{4}+\\bar{\\psi}(i \\not \\partial) \\psi-i g \\bar{\\psi} \\gamma^{5} \\psi \\phi,\n$$\ncompute the Callan-Symanzik $\\beta$ function for $\\lambda$:\n\n\\beta_{\\lambda}(\\lambda, g),\n\nto leading order in coupling constants, assuming that $\\lambda$ and $g^{2}$ are of the same order.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\beta_{\\lambda}$": "Callan-Symanzik beta function for the coupling constant $\\lambda$.", + "$\\lambda$": "Quartic coupling constant for the $\\phi^4$ interaction.", + "$g$": "Yukawa coupling constant." + } + }, + { + "id": 246, + "context": "Asymptotic symmetry. Consider the following Lagrangian, with two scalar fields $\\phi_{1}$ and $\\phi_{2}$ :\n\n\\mathcal{L}=\\frac{1}{2}((\\partial_{\\mu} \\phi_{1})^{2}+(\\partial_{\\mu} \\phi_{2})^{2})-\\frac{\\lambda}{4!}(\\phi_{1}^{4}+\\phi_{2}^{4})-\\frac{2 \\rho}{4!}(\\phi_{1}^{2} \\phi_{2}^{2}) .\n\n\nNotice that, for the special value $\\rho=\\lambda$, this Lagrangian has an $O(2)$ invariance rotating the two fields into one another.", + "question": "For a theory with two scalar fields $\\phi_1$ and $\\phi_2$ described by the Lagrangian:\n$$\n\\mathcal{L}=\\frac{1}{2}((\\partial_{\\mu} \\phi_{1})^{2}+(\\partial_{\\mu} \\phi_{2})^{2})-\\frac{\\lambda}{4!}(\\phi_{1}^{4}+\\phi_{2}^{4})-\\frac{2\\rho}{4!} \\phi_{1}^{2} \\phi_{2}^{2}\n$$\n\nWorking in four dimensions, what is the $\\beta$ function for the coupling constant $\\lambda$, denoted as $\\beta_{\\lambda}$, to leading order in the coupling constants?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\lambda$": "Coupling constant.", + "$\\rho$": "Coupling constant.", + "$\\pi$": "Mathematical constant pi." + } + }, + { + "id": 247, + "context": "", + "question": "State the leading term in $\\gamma(\\lambda)$ for $\\phi^{4}$ theory.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\gamma$": "Anomalous dimension, defined as $\\gamma = \\frac{1}{2}M\\frac{\\partial}{\\partial M}\\delta_{Z}$", + "$\\lambda$": "Coupling constant in $\\phi^{4}$ theory", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 248, + "context": "", + "question": "Compute the anomalous dimension \\(\\gamma\\) in an \\(O(N)\\)-symmetric \\(\\phi^{4}\\) theory for \\(N = 3\\) and coupling constant \\(\\lambda = 0.5\\), using the formula\n\\[\\gamma = (N+2)\\,\\frac{\\lambda^{2}}{(4\\pi)^{4}}.\\]", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": { + "$\\gamma$": "Anomalous dimension", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 249, + "context": "Brute-force computations in $\\boldsymbol{S U ( 3 )}$. The standard basis for the fundamental representation of $S U(3)$ is\n\n\\begin{array}{rlrl}\nt^{1} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 1 & 0 \\\\\n1 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\quad t^{2}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & -i & 0 \\\\\ni & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\quad t^{3}=\\frac{1}{2}(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & -1 & 0 \\\\\n0 & 0 & 0\n\\end{array}), \\\\\nt^{4} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 0 & 1 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0\n\\end{array}), & t^{5}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & 0 & -i \\\\\n0 & 0 & 0 \\\\\ni & 0 & 0\n\\end{array}), \\\\\nt^{6} & =\\frac{1}{2}(\\begin{array}{lll}\n0 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{array}), \\quad t^{7}=\\frac{1}{2}(\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & -i \\\\\n0 & i & 0\n\\end{array}), \\quad t^{8}=\\frac{1}{2 \\sqrt{3}}(\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & -2\n\\end{array}) .\n\\end{array}", + "question": "Write down the dimension $d$ of $S U(N)$ group. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$d$": "Dimension of the $SU(N)$ group", + "$N$": "Degree of the Special Unitary group" + } + }, + { + "id": 250, + "context": "Matrix element for proton decay. Some advanced theories of particle interactions include heavy particles $X$ whose couplings violate the conservation of baryon number. Integrating out these particles produces an effective interaction that allows the proton to decay to a positron and a photon or a pion. This effective interaction is most easily written using the definite-helicity components of the quark and electron fields: If $u_{L}, d_{L}, u_{R}, e_{R}$ are two-component spinors, then this effective interaction is\n\n\\Delta \\mathcal{L}=\\frac{2}{m_{X}^{2}} \\epsilon_{a b c} \\epsilon^{\\alpha \\beta} \\epsilon^{\\gamma \\delta} e_{R \\alpha} u_{R a \\beta} u_{L b \\gamma} d_{L c \\delta} .\n\n\nA typical value for the mass of the $X$ boson is $m_{X}=10^{16} \\mathrm{GeV}$.", + "question": "Estimate, in order of magnitude, the value of the proton lifetime if the proton is allowed to decay through this interaction.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 251, + "context": "A model with two Higgs fields.", + "question": "Assume that the two Higgs fields couple to quarks by the set of fundamental couplings\n\n\\mathcal{L}_{m}=-\\lambda_{d}^{i j} \\bar{Q}_{L}^{i} \\cdot \\phi_{1} d_{R}^{j}-\\lambda_{u}^{i j} \\epsilon^{a b} \\bar{Q}_{L a}^{i} \\phi_{2 b}^{\\dagger} u_{R}^{j}+\\text { h.c. }\n\n\nFind the couplings of the physical charged Higgs boson of part (c) to the mass eigenstates of quarks. These couplings depend only on the values of the quark masses and $\\tan \\beta$ and on the elements of the CKM matrix.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "v": "Combined vacuum expectation value, $v=\\sqrt{v_{1}^{2}+v_{2}^{2}}$.", + "H^+": "Physical charged Higgs boson (positive charge).", + "$\\tan\\beta$": "Tangent of the angle $\\beta$, related to the ratio of vacuum expectation values of the two Higgs doublets.", + "$\\bar{u}_L^i$": "Left-handed up-type quark anti-field with flavor index i.", + "$(V_{\\text{CKM}})_{ij}$": "Element of the CKM matrix with indices i, j.", + "m_{d_j}": "Mass of the j-th down-type quark.", + "d_R^j": "Right-handed down-type quark field with flavor index j.", + "$\\cot\\beta$": "Cotangent of the angle $\\beta$.", + "$\\bar{d}_L^i$": "Left-handed down-type quark anti-field with flavor index i.", + "$(V_{\\text{CKM}}^\\dagger)_{ij}$": "Element of the Hermitian conjugate of the CKM matrix with indices i, j.", + "m_{u_j}": "Mass of the j-th up-type quark.", + "u_R^j": "Right-handed up-type quark field with flavor index j.", + "h.c.": "Hermitian conjugate." + } + }, + { + "id": 252, + "context": "Dependence of radiative corrections on the Higgs boson mass.", + "question": "In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\\Pi_{WW}(q^2)$ (specifically the part proportional to $g^{\\mu\\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_W^2$. Provide the derivation steps and the final expression for $\\Pi_{WW}(q^2)$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$\\Pi_{WW}$": "Vacuum polarization amplitude for W bosons", + "$q^2$": "Squared four-momentum transfer", + "$g$": "Gauge coupling constant", + "$\\pi$": "Mathematical constant pi", + "$m_h$": "Higgs boson mass", + "$m_W$": "W boson mass" + } + }, + { + "id": 253, + "context": "Dependence of radiative corrections on the Higgs boson mass.", + "question": "In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\\Pi_{ZZ}(q^2)$ (specifically the part proportional to $g^{\\mu\\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_Z^2$. Provide the final expression for $\\Pi_{ZZ}(q^2)$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$\\Pi_{ZZ}$": "Vacuum polarization amplitude for Z bosons", + "$q^2$": "Squared momentum transfer", + "$g$": "Electroweak coupling constant", + "$\\cos\\theta_w$": "Cosine of the Weinberg angle", + "$m_h$": "Higgs boson mass", + "$m_Z$": "Z boson mass" + } + }, + { + "id": 254, + "context": "", + "question": "Consider the wave equation $\\nabla^{2} u+k^{2} n(r)^{2} u=0$ with slowly varying $n(r)$. If we introduce the eikonal function $S(r)$ by substituting $u = \\mathrm{e}^{\\frac{2 \\pi i}{\\lambda} S(r)}$ (where $\\lambda=2 \\pi / k$) into the wave equation, what is the resulting differential equation for $S(r)$ before any series expansion of $S(r)$ is performed (this is known as the Riccati equation in this context)? You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\lambda$": "Wavelength, specifically deBroglie wavelength, defined as $\\lambda=\\frac{2 \\pi \\hbar}{p}$.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$i$": "Imaginary unit, where $i^2 = -1$.", + "$\\nabla^{2}$": "Laplacian operator.", + "$S$": "Eikonal function, introduced by substituting $u = \\mathrm{e}^{\\frac{2 \\pi i}{\\lambda} S(r)}$ into the wave equation.", + "$\\nabla$": "Gradient operator.", + "$n$": "Refractive index, which is a function of position $r$, defined as $n(r)=\\sqrt{1-\\frac{V(r)}{E}}$." + } + }, + { + "id": 255, + "context": "", + "question": "The tensor force between two particles 1 and 2 of spin $1/2$ is associated with the operator $T_{12}$ given by:\n\n T_{12}=\\frac{(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{r})(\\boldsymbol{\\sigma}_{2} \\cdot \\boldsymbol{r})}{r^{2}}-\\frac{1}{3}(\\boldsymbol{\\sigma}_{1} \\cdot \\boldsymbol{\\sigma}_{2})\n\nwhere $\\boldsymbol{\\sigma}_{1}$ and $\\boldsymbol{\\sigma}_{2}$ are the Pauli spin matrices for particle 1 and 2 respectively, and $\\boldsymbol{r}$ is the relative position vector between the particles.\nIf $\\chi_{0,0}$ represents the spin singlet state (an eigenfunction of total spin $S=0$, defined as $\\chi_{0,0}=\\frac{1}{\\sqrt{2}}(\\alpha_{1} \\beta_{2}-\\beta_{1} \\alpha_{2})$) for the two-particle system, calculate the result of applying the operator $T_{12}$ to $\\chi_{0,0}$. In other words, find $T_{12} \\chi_{0,0}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 256, + "context": "", + "question": "In a neutral helium atom, one electron is in the $1s$ ground state and the other is in the $2p$ excited state ($n=2, l=1$). Using a theoretical model based on hydrogen-like wave functions with screening of one nuclear charge by the $1s$ electron, calculate the ionization energy (in eV) for the $2p$ electron if the atom is in the parahelium state (give a number and keep three decimal places).", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 257, + "context": "", + "question": "The function\n\\begin{equation*}\n\\tilde{\\varphi}(x)=\\frac{1}{(1+\\alpha x)^{2}} \\tag{176.1}\n\\end{equation*}\nwith a suitable value of $\\alpha$, independent of $Z$, may be used as a fair approximation to the Thomas-Fermi function $\\varphi_{0}(x)$ for a neutral atom. The constant $\\alpha$ shall be determined in such a way as to permit exact normalization of $\\tilde{\\varphi}$. Hint: the answer is a number.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 258, + "context": "", + "question": "Calculate the numerical value for the mean lifetime (in seconds) of the $2P$ state in a hydrogen atom, which decays to the $1S$ state by emission of a photon. This requires first determining the total transition probability $P$ for an electron from a higher $P$ state to a lower $S$ state (summed over all photon directions and polarizations), and then specializing this for the hydrogen $2P \\rightarrow 1S$ transition.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 259, + "context": "", + "question": "To compare the intensities of emission of the two first Lyman lines of atomic hydrogen, Ly $\\alpha$ and Ly $\\beta$. Numerically calculate $I_\\alpha/I_\\beta$ You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$I_\\alpha$": "Intensity of Ly $\\alpha$ emission.", + "$I_\\beta$": "Intensity of Ly $\\beta$ emission." + } + }, + { + "id": 260, + "context": "", + "question": "Consider a scalar field theory with interaction Lagrangian $\\mathcal{L}_{\\mathrm{I}}=-\\frac{g}{3!} \\phi^{3}-\\frac{\\lambda}{4!} \\phi^{4}$. Derive the identity that relates the number of loops ($n_L$), internal lines ($n_I$), trivalent vertices ($n_3$), and tetravalent vertices ($n_4$). You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$n_{L}$": "Number of loops", + "$n_{I}$": "Number of internal lines", + "$n_{3}$": "Number of trivalent vertices", + "$n_{4}$": "Number of tetravalent vertices" + } + }, + { + "id": 261, + "context": "", + "question": "For a massless spin-one particle with four-momentum $p$, its physical helicity polarization vectors are denoted by $\\epsilon_{\\pm}^{\\mu}(\\mathbf{p})$. Under a Lorentz transformation $\\Lambda$ (which transforms the momentum $p$ to $\\Lambda p$), these polarization vectors transform according to the following relation derived using little group properties:\n$$[\\Lambda^{-1}]_{\\nu}^{\\mu} \\epsilon_{ \\pm}^{v}(\\Lambda \\mathbf{p}) = X$$ \nWhat is the expression for $X$ in terms of the original polarization vector $\\epsilon_{ \\pm}^{\\mu}(\\mathbf{p})$, the original four-momentum $p^{\\mu}$, a phase factor $e^{\\mp i \\theta}$ (where $\\theta$ is the Wigner rotation angle), and coefficients $\\beta_{ \\pm}$ which depend on the parameters of the Lorentz transformation and the reference momentum?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$e^{\\mp i \\theta}$": "A phase factor.", + "$\\theta$": "Wigner rotation angle.", + "$\\epsilon_{\\pm}^{\\mu}(\\mathbf{p})$": "Physical helicity polarization vectors for a particle with four-momentum $p$.", + "$\\beta_{\\pm}$": "Coefficients depending on the parameters of the Lorentz transformation and the reference momentum, defined as $\\beta_{ \\pm} \\equiv(\\beta_{1} \\pm i \\beta_{2}) /(\\omega \\sqrt{2})$.", + "$p^{\\mu}$": "Four-momentum of a massless spin-one particle." + } + }, + { + "id": 262, + "context": "", + "question": "Give the expression of the one-loop self-energy in a $\\phi^{4}$ theory in the Matsubara formalism. Calculate it in the limit $\\beta \\mathrm{m} \\ll 1$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\lambda$": "Coupling constant in $\\phi^4$ theory.", + "$\\Lambda$": "Momentum cutoff for regularization.", + "$\\pi$": "Mathematical constant pi.", + "$T$": "Temperature." + } + }, + { + "id": 263, + "context": "Consider two Grassmann variables $\\theta_\\pm$.", + "question": "For the operator $\\tau_3 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{-} \\frac{\\partial}{\\partial \\theta_{-}})$, find two linearly independent eigenfunctions corresponding to the eigenvalue $0$. Express them using $1, \\theta_+, \\theta_-, \\theta_+\\theta_-$ and normalize constant terms or leading $\\theta$ terms to 1. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$\\theta_+$": "Grassmann variable", + "$\\theta_-$": "Grassmann variable" + } + }, + { + "id": 264, + "context": "Consider two Grassmann variables $\\theta_\\pm$.", + "question": "Consider the operator $\\tau_1 \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ acting on functions of two Grassmann variables $\\theta_{\\pm}$. Find all distinct eigenvalues of $\\tau_1$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 265, + "context": "Consider two Grassmann variables $\\theta_\\pm$.", + "question": "Consider the operator $\\tau_2 \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$ acting on functions of two Grassmann variables $\\theta_{\\pm}$. Find all distinct eigenvalues of $\\tau_2$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 266, + "context": "", + "question": "Given the operators $\\tau_{1} \\equiv \\frac{1}{2}(\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}}+\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}})$ and $\\tau_{2} \\equiv \\frac{i}{2}(\\theta_{-} \\frac{\\partial}{\\partial \\theta_{+}}-\\theta_{+} \\frac{\\partial}{\\partial \\theta_{-}})$, calculate the action of the operator $(\\tau_1 - i\\tau_2)$ on the Grassmann variable $\\theta_-$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 267, + "context": "", + "question": "Consider a non-Abelian gauge theory with the usual $\\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields in the adjoint representation and $n_{f}$ Dirac fermions in the adjoint representation. Calculate the expression for $\\frac{1}{g_{\\mathrm{r}}^{2}(\\mu)}$ for this theory, analogous to the standard one-loop running coupling constant equation. The constants for fields in the adjoint representation are given as $\\mathrm{c}_{\\mathrm{adj}, 0}=\\frac{\\mathrm{N}}{3(4 \\pi)^{2}}$ for gauge fields/ghosts, $\\mathrm{c}_{\\mathrm{adj}, 1 / 2}=-\\frac{8 \\mathrm{~N}}{3(4 \\pi)^{2}}$ for Dirac fermions, and we can infer the scalar contribution from the context (scalars have bosonic statistics and contribute with an opposite sign to ghosts or fermions regarding their statistical nature's impact on the beta function coefficient).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$g_{\\mathrm{r}}$": "Renormalized coupling constant, dependent on the renormalization scale $\\mu$.", + "$\\mu$": "Renormalization scale.", + "$g_{\\mathrm{b}}$": "Bare coupling constant.", + "$\\mathrm{N}$": "Dimension of the $\\mathfrak{s u}(N)$ gauge group.", + "$n_{\\mathrm{f}}$": "Number of Dirac fermions in the adjoint representation.", + "$\\mathrm{n}_{\\mathrm{s}}$": "Number of complex scalar fields in the adjoint representation.", + "$\\kappa$": "Reference scale." + } + }, + { + "id": 268, + "context": "", + "question": "For a non-Abelian gauge theory with $\\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields and $n_{f}$ Dirac fermions all in the adjoint representation, the one-loop running of the inverse squared coupling is given by $\\frac{1}{g_{\\mathrm{r}}^{2}(\\mu)}=\\frac{1}{\\mathrm{~g}_{\\mathrm{b}}^{2}}+\\frac{\\mathrm{N}}{3(4 \\pi)^{2}}(11-4 n_{\\mathrm{f}}-\\mathrm{n}_{\\mathrm{s}}) \\ln \\frac{\\mu^{2}}{\\kappa^{2}}$. Determine the condition on $n_s$ and $n_f$ for the gauge coupling to not be running at one loop. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$n_f$": "Number of Dirac fermions.", + "$n_s$": "Number of complex scalar fields." + } + }, + { + "id": 269, + "context": "", + "question": "Carry out explicitly the calculation of the functions $A$ and $B$ in \n\n$$B(q^{2})=-i\\mathrm{D}e\\int\\frac{d^{\\mathrm{D}}l}{(2\\pi)^{\\mathrm{D}}}\\int_{0}^{1}dx\\frac{2\\Delta(x)}{(l^{2}+\\Delta(x))^{2}}.$$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$B$": "A function of $q^2$ being calculated.", + "$q^{2}$": "A squared momentum or a related kinematic variable.", + "$e$": "A constant parameter, possibly a coupling constant, appearing in the calculation.", + "$\\pi$": "The mathematical constant pi, approximately 3.14159." + } + }, + { + "id": 270, + "context": "", + "question": "Carry out explicitly the calculation of the functions $A$ and $B$ in \n$$A(q^{2})=-i\\mathrm{D}e\\int\\frac{d^{\\mathrm{D}}l}{(2\\pi)^{D}}\\int_{0}^{1}dx\\frac{\\Delta(x)+(\\frac{2}{\\mathrm{D}}-1)l^{2}}{(l^{2}+\\Delta(x))^{2}}, $$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$A$": "Function $A$ to be calculated, dependent on $q^2$", + "$q^{2}$": "Squared external momentum, an argument of functions $A$ and $B$", + "$e$": "A constant, likely electric charge or coupling constant", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 271, + "context": "", + "question": "For the time independent, $z^{3}$-dependent electrical field $E_{3}(z^{3}) \\equiv \\frac{E}{\\cosh ^{2}(k z^{3})}$ with gauge potential $A^{4}=-i \\frac{E}{k} \\tanh (k z^{3})$ (other $A^i=0$), the equations of motion for the stationary solutions $z^3(u)$ and $z^4(u)$, (assuming $z^1, z^2$ are constant) are given in first-order form. What are these equations, expressed in terms of $v = \\sqrt{(\\dot{z}^3)^2+(\\dot{z}^4)^2}$ and $\\gamma \\equiv mk/(eE)$? You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$\\dot{z}^3$": "Derivative of the coordinate $z^3$ with respect to $u$", + "$\\dot{z}^4$": "Derivative of the coordinate $z^4$ with respect to $u$", + "$v$": "Constant value of the magnitude of the velocity, $v = \\sqrt{(\\dot{z}^3)^2+(\\dot{z}^4)^2}$", + "$\\gamma$": "Dimensionless parameter defined as $\\gamma \\equiv mk/(eE)$", + "$k$": "Constant parameter in the electrical field and gauge potential", + "$z^3$": "A coordinate" + } + }, + { + "id": 272, + "context": "", + "question": "The Polyakov loop is defined as $\\mathrm{L}(x) \\equiv \\mathrm{N}^{-1} \\operatorname{tr} ( P \\exp \\int_{0}^{\\beta} d \\tau A^{0}(\\tau, x) )$. Under a center transformation, where the gauge transformation $\\Omega(\\tau,x)$ obeys $\\Omega(\\beta, x)=\\xi \\Omega(0, x)$ with $\\xi \\in \\mathbb{Z}_{N}$, how does $\\mathrm{L}(x)$ transform? Express the transformed Polyakov loop $\\mathrm{L}'(x)$ in terms of $\\mathrm{L}(x)$ and $\\xi$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Others", + "final_symbol": { + "$\\mathrm{L}'(x)$": "Transformed Polyakov loop", + "$\\xi$": "Center element of the gauge group, a scalar multiple of the identity matrix, $\\xi = c \\cdot \\mathbf{1}_{N \\times N}$", + "$\\mathrm{L}(x)$": "Polyakov loop, defined as $\\mathrm{N}^{-1} \\operatorname{tr} ( P \\exp \\int_{0}^{\\beta} d \\tau A^{0}(\\tau, x) )$" + } + }, + { + "id": 273, + "context": "", + "question": "A particle with mass $m$, constrained to move freely on a ring with radius $R$, with an added perturbation\n\\begin{equation*}\nH^{\\prime}=V(\\varphi)=\\left\\{\\begin{array}{ll}\nV_{1}, & -\\alpha<\\varphi<0 \\\\\nV_{2}, & 0<\\varphi<\\alpha \\\\\n0, & \\text { other angles }\n\\end{array} \\quad(\\alpha<\\pi)\\right.\n\\end{equation*}\n\nFind the first-order perturbation corrections for the three lowest energy levels. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$\\alpha$": "Angular range for the perturbation, $\\alpha < \\pi$", + "$\\pi$": "Mathematical constant pi", + "$V_1$": "Constant potential value for $-\\alpha < \\varphi < 0$", + "$V_2$": "Constant potential value for $0 < \\varphi < \\alpha$" + } + }, + { + "id": 274, + "context": "", + "question": "A small uniformly charged sphere acquires potential energy in an external electrostatic field\n$$\n\\begin{equation*}\nU(\\boldsymbol{r})=V(\\boldsymbol{r})+\\frac{1}{6} r_{0}^{2} \\nabla^{2} V(\\boldsymbol{r})+\\cdots \n\\end{equation*}\n$$\n\nwhere $r_{0}$ is the radius of the sphere, $\\boldsymbol{r}$ is the position of the sphere's center, and $V(\\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is\n$$\n\\begin{equation*}\nV(\\boldsymbol{r})=-\\frac{e^{2}}{r} \n\\end{equation*}\n$$\n\nIf the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 1s energy levels [equivalent to the Lamb shift].", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 275, + "context": "", + "question": "A small uniformly charged sphere acquires potential energy in an external electrostatic field\n$$\n\\begin{equation*}\nU(\\boldsymbol{r})=V(\\boldsymbol{r})+\\frac{1}{6} r_{0}^{2} \\nabla^{2} V(\\boldsymbol{r})+\\cdots \n\\end{equation*}\n$$\n\nwhere $r_{0}$ is the radius of the sphere, $\\boldsymbol{r}$ is the position of the sphere's center, and $V(\\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is\n$$\n\\begin{equation*}\nV(\\boldsymbol{r})=-\\frac{e^{2}}{r} \n\\end{equation*}\n$$\n\nIf the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 2p energy levels [equivalent to the Lamb shift].", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 276, + "context": "", + "question": "Take the ground state wave function as\n$$\n\\psi_{0}(\\boldsymbol{r}_{1}, \\boldsymbol{r}_{2})=\\psi_{0}(r_{1}) \\psi_{0}(r_{2})\n$$\n\nwhere\n$$\n\\begin{equation*}\n\\psi_{0}(r)=(\\frac{\\lambda^{3}}{\\pi a_{0}^{3}})^{\\frac{1}{2}} \\mathrm{e}^{-x^{\\prime} / a_{0}}, \\quad a_{0}=\\frac{\\hbar^{2}}{m_{\\mathrm{e}} e^{2}}.\n\\end{equation*}\n$$\n\nCalculate the ground state magnetic susceptibility of a helium atom, in the unit of $\\mathrm{eV} /(\\mathrm{Gs})^{2}$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 277, + "context": "", + "question": "A hydrogen atom is situated within a certain ionic lattice, where the potential exerted by the surrounding ions on the electron in the hydrogen atom can be approximately represented as\n\n\\begin{equation*}\nH^{\\prime}=V_{0}(x^{4}+y^{4}+z^{4}-\\frac{3}{5} r^{4}) \n\\end{equation*}\n\n$H^{\\prime}$ can be considered a perturbation. If the 3d state wave functions of the hydrogen atom (orthonormalized) are taken as\n\n\\begin{align*}\n& \\psi_{1}=\\frac{1}{2}(y^{2}-z^{2}) f(r) \\\\\n& \\psi_{2}=\\frac{1}{2 \\sqrt{3}}(2 x^{2}-y^{2}-z^{2}) f(r) \\\\\n& \\psi_{3}=y z f(r) \\\\\n& \\psi_{4}=z x f(r) \\\\\n& \\psi_{5}=x y f(r)\n\\end{align*}\n\n\nUnder the influence of $H^{\\prime}$, what is the degeneracy of the first energy level after the 3d energy level splits (corresponding to $\\psi_1$ and $\\psi_2$)?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 278, + "context": "", + "question": "A certain molecule is composed of three identical atoms ($\\alpha, \\beta, \\gamma$), with the three atoms located at the vertices of an equilateral triangle. There is one valence electron that can move among the three atoms. Denote the unperturbed Hamiltonian of this valence electron as $H_0$, and the atomic orbitals of the electron as $|\\alpha\\rangle, |\\beta\\rangle, |\\gamma\\rangle$ (which are mutually orthogonal and normalized). Assume the electron's atomic energy level is $\\langle k|H_0|k\\rangle = E_0$ (for $k=\\alpha,\\beta,\\gamma$). Between any two different atoms, the matrix element of $H_0$ is $\\langle j|H_0|k\\rangle = -a$ (where $j \\neq k, a>0$). Request to solve the molecular energy levels of the unperturbed Hamiltonian $H_0$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$E_0$": "Atomic energy level of the electron for a single atom, defined as $\\langle k|H_0|k\\rangle$.", + "$a$": "Positive constant representing the magnitude of the coupling (hopping) matrix element between two different atomic orbitals, where $\\langle j|H_0|k\\rangle = -a$ for $j \\neq k$." + } + }, + { + "id": 279, + "context": "", + "question": "For the unperturbed system mentioned above (i.e., without an electric field), solve for the normalized ground state wave function $|\\psi_{GS,unperturbed}\\rangle$ corresponding to the lowest energy $E_0 - 2a$, and express it as a linear combination of $|\\alpha\\rangle, |\\beta\\rangle, |\\gamma\\rangle$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\alpha$": "A symbolic representation of the basis state $|\\alpha\\rangle$ within a linear combination expression.", + "$\\beta$": "A symbolic representation of the basis state $|\\beta\\rangle$ within a linear combination expression.", + "$\\gamma$": "A symbolic representation of the basis state $|\\gamma\\rangle$ within a linear combination expression.", + "$|\\psi_{GS,unperturbed}\\rangle$": "Normalized ground state wave function for the unperturbed system.", + "$|\\alpha\\rangle$": "A basis state.", + "$|\\beta\\rangle$": "A basis state.", + "$|\\gamma\\rangle$": "A basis state." + } + }, + { + "id": 280, + "context": "", + "question": "Apply a uniform weak electric field as a perturbation. Due to this electric field, the on-site energy level at atom $\\alpha$ decreases by $b$, becoming $E_0-b$, while the energy levels at atoms $\\beta$ and $\\gamma$ remain $E_0$. Assume $b \\ll a$. The hopping integral between atoms ($-a$) is not affected by the electric field. The perturbation matrix elements between different atomic orbitals are zero (i.e., $\\langle j|H'|k\\rangle = 0$ when $j \\neq k$, and $\\langle\\beta|H'|\\beta\\rangle = \\langle\\gamma|H'|\\gamma\\rangle = 0$, $\\langle\\alpha|H'|\\alpha\\rangle = -b$). Solve for the new molecular energy levels, with results approximated to first order in $b/a$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$E_0$": "Original on-site energy level of the atoms", + "$a$": "Magnitude of the hopping integral between atoms", + "$b$": "Magnitude of the decrease in the on-site energy level at atom $\\alpha$ due to the electric field" + } + }, + { + "id": 281, + "context": "", + "question": "Initially, the electrons are in the ground state $|\\psi_{GS,perturbed}\\rangle$ (corresponding to the situation where an electric field is perturbing atom $\\alpha$). If the field suddenly rotates so that the perturbation now acts on atom $\\beta$ (with the system's new ground state being $|\\psi'_{GS,perturbed}\\rangle$), what is the probability that the electrons are found in this new ground state $|\\psi'_{GS,perturbed}\\rangle$? Approximate the result to zero order of $(b/a)$, meaning terms containing $b$ should disregard terms of order $(b/a)$ or higher.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 282, + "context": "", + "question": "Which powers among the operators $\\hat{s}$ (with any spin value $s$) are linearly independent? You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\hat{s}_z$": "Z-component of the spin operator", + "$s$": "Spin quantum number or spin value" + } + }, + { + "id": 283, + "context": "The equation is\n\n\\begin{align}\\label{eq:57.4}\n(\\hat s_x \\psi)^1 &= \\tfrac12\\,\\psi^2, \n &(\\hat s_y \\psi)^1 &= -\\tfrac12\\,i\\,\\psi^2, \n &(\\hat s_z \\psi)^1 &= \\tfrac12\\,\\psi^1, \\\\\n(\\hat s_x \\psi)^2 &= \\tfrac12\\,\\psi^1, \n &(\\hat s_y \\psi)^2 &= \\tfrac12\\,i\\,\\psi^1, \n &(\\hat s_z \\psi)^2 &= -\\tfrac12\\,\\psi^2.\n\\end{align}", + "question": "Rewrite equation in the context, expressing the operators of spin $1 / 2$ in terms of the spinor components of the vector $\\hat{\\boldsymbol{S}}$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\hat{s}^{\\lambda \\mu}$": "Spinor operator component.", + "$\\psi^{\\nu}$": "Component of the spinor.", + "$i$": "Imaginary unit.", + "$\\psi^{\\lambda}$": "Component of the spinor.", + "$g^{\\mu \\nu}$": "Component of the metric tensor.", + "$\\psi^{\\mu}$": "Component of the spinor." + } + }, + { + "id": 284, + "context": "", + "question": "Determine all possible states of a three-nucleon system, where each nucleon has an angular momentum $j=3 / 2$ (with the same principal quantum number). You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 285, + "context": "", + "question": "Obtain the density matrix $\\rho(q, q^{\\prime})=\\langle q| e^{-\\beta \\hat{H}}|q^{\\prime}\\rangle$ for the harmonic oscillator at finite temperature, $\\beta=1 / T(k_{\\mathrm{B}}=1)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\rho$": "Density matrix, defined as $\\langle q| e^{-\\beta \\hat{H}}|q^{\\prime}\\rangle$.", + "$q$": "Position coordinate.", + "$q^{\\prime}$": "Another position coordinate.", + "$m$": "Mass of the harmonic oscillator.", + "$\\omega$": "Angular frequency of the harmonic oscillator.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\beta$": "Inverse temperature, defined as $\\beta=1 / T$." + } + }, + { + "id": 286, + "context": "", + "question": "A particle is moving in an infinite potential well $(-a0 \\tag{1}\n\\end{equation*}", + "question": "For the particle in the potential $V(r)=kr$ (s-wave), its exact ground state energy is $E_0 = C (\\frac{\\hbar^{2} k^{2}}{m})^{1 / 3}$. What is the value of the constant $C$ (accurate to four decimal places)? \n\nThe trial wave function is:\n\\begin{enumerate}\n \\item $\\psi \\sim e^{-\\lambda r}$.\n\\end{enumerate}", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 288, + "context": "", + "question": "Assume in the deuteron, the potential between the proton and neutron is expressed as\n\n\\begin{equation*}\nV(r)=-V_{0} \\mathrm{e}^{-r / a} \\tag{1}\n\\end{equation*}\n\n\nTake $V_{0}=32.7 \\mathrm{MeV}, a=2.16 \\mathrm{fm}$ (range of force). Use the variational method to find the ground state energy level of the deuteron.\n\nThe trial function is chosen as\n\n\\begin{equation*}\n\\psi(\\lambda, r)=N \\mathrm{e}^{-\\lambda r / 2 a}\n\\end{equation*}", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 289, + "context": "", + "question": "Two conductors with capacitances $C_{1}$ and $C_{2}$ respectively are separated by a distance $r$, where $r$ is greater than the dimensions of the conductors themselves. Determine the coefficient $C_{12}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$C_{12}$": "Mutual capacitance coefficient between conductor 1 and 2", + "$C_1$": "Capacitance of the first conductor", + "$C_2$": "Capacitance of the second conductor", + "$r$": "Distance separating the two conductors" + } + }, + { + "id": 290, + "context": "", + "question": "Given two conductors with capacitances $C_{1}$ and $C_{2}$ separated by a distance $r$, where $r$ is much larger than the dimensions of the conductors. Try to determine the coefficient $C_{22}$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Magnetism", + "final_symbol": { + "$C_{22}$": "Self-capacitance coefficient for conductor 2", + "$C_2$": "Capacitance of the second conductor", + "$C_1$": "Capacitance of the first conductor", + "$r$": "Distance separating the conductors" + } + }, + { + "id": 291, + "context": "", + "question": "Consider a conductor with a sharp conical tip on its surface. \n\nUsing spherical coordinates, place the origin at the vertex of the conical tip, with the cone axis as the polar axis. \n\nLet the cone's opening angle be $2 \\theta_{0} \\ll 1$, and the polar angle range corresponding to the external region of the conductor is $\\theta_{0} \\leqslant \\theta \\leqslant \\pi$. Assume that the potential $\\varphi$ has the form $\\varphi(r, \\theta) = r^{n} f(\\theta)$. \n\nBased on this setup, and using the boundary condition that the potential on the conductor's surface ($\\theta=\\theta_0$) is constant (i.e., $f(\\theta_0)=0$ for the angular dependence part of the potential), and that for small $\\theta_0$ and small $n \\ll 1$, the function $f(\\theta)$ can be approximately expressed as $f(\\theta) = \\mathrm{const} \\cdot (1+2 n \\ln \\sin (\\theta/2) )$, derive the expression for the exponent $n$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$n$": "Exponent in the potential's radial dependence, $\\varphi = r^n f(\\theta)$.", + "$\\theta_0$": "Half of the cone's opening angle, a parameter defining the cone's surface." + } + }, + { + "id": 292, + "context": "", + "question": "Given that the conductor boundary is an infinite plane with a hemispherical protrusion whose radius is $R$, determine the charge distribution on the surface of the conductor at the hemispherical protrusion.\n\nHint: the potential is\n$$\n\\varphi=-4\\pi \\sigma_0 \\cdot z(1-\\frac{R^{3}}{r^{3}})\n$$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Charge distribution/density on the surface of the conductor.", + "$\\sigma_0$": "Charge density far from the protruding part.", + "$z$": "Vertical coordinate.", + "$R$": "Radius of the hemispherical protrusion." + } + }, + { + "id": 293, + "context": "", + "question": "Find the charge distribution on a non-charged conductor disk (with radius $a$) that is parallel to a uniform external electric field ${ }^{(1)}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\sigma$": "Charge density", + "$\\mathfrak{C}$": "A constant factor in the charge density formula", + "$\\rho$": "Radial polar coordinate in the plane of the disk", + "$\\varphi$": "Angular polar coordinate in the plane of the disk", + "$\\pi$": "Mathematical constant pi", + "$a$": "Radius of the non-charged conductor disk", + "$p$": "A parameter in the charge density formula, appearing as a radial distance squared under the square root" + } + }, + { + "id": 294, + "context": "", + "question": "For a conducting sphere placed in a uniform external electric field $\\mathfrak{C}$, determine its relative volume change $\\frac{\\Delta V}{V}$, the bulk modulus of the material is $K$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\Delta V$": "Change in volume", + "$V$": "Original volume", + "$\\mathfrak{C}$": "Uniform external electric field", + "$K$": "Bulk modulus of the material" + } + }, + { + "id": 295, + "context": "", + "question": "For a conducting sphere placed in a uniform external electric field $\\mathfrak{C}$, determine its shape deformation. Specifically, find the expression for the quantity $\\frac{a-b}{R}$ that describes its deformation, where $R$ is the original radius of the sphere, and $a$ and $b$ are the semi-axes of the ellipsoid along and perpendicular to the field direction, respectively. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Magnetism", + "final_symbol": { + "$a$": "Semi-axis of the ellipsoid along the field direction", + "$b$": "Semi-axis of the ellipsoid perpendicular to the field direction", + "$R$": "Original radius of the sphere", + "$\\pi$": "Mathematical constant pi", + "$\\mu$": "Shear modulus of the material", + "$\\mathfrak{C}$": "Uniform external electric field" + } + }, + { + "id": 296, + "context": "", + "question": "Try to determine the volume change of a dielectric ellipsoid in a uniform electric field, assuming the direction of the electric field is parallel to one of the ellipsoid's axes. Specifically, determine $\\frac{V-V_0}{V}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathfrak{C}$": "Electric field strength (typo for $\\mathfrak{E}$)", + "$\\varepsilon$": "Dielectric constant", + "$n$": "Depolarization factor", + "$K$": "Compressibility coefficient of the object", + "$P$": "Pressure" + } + }, + { + "id": 297, + "context": "", + "question": "Determine the electrothermal effect of a dielectric ellipsoid in a uniform electric field, assuming the direction of the field is parallel to one of the axes of the ellipsoid.\n\n\\footnotetext{\n(1) If the object is thermally insulated, the application of an electric field will cause a temperature change of $\\Delta T=-Q / \\mathscr{C}_{P}$, where $\\mathscr{C}_{P}$ is the constant pressure heat capacity of the object.\n}", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$Q$": "Heat generated due to the electrothermal effect.", + "$T$": "Absolute temperature.", + "$V$": "Volume of the dielectric ellipsoid.", + "$\\mathfrak{C}$": "Magnitude of the uniform electric field strength.", + "$\\alpha$": "Coefficient of thermal expansion, defined as $\\alpha=\\frac{1}{V}(\\frac{\\partial V}{\\partial T})_{P}$.", + "$\\varepsilon$": "Dielectric constant of the material.", + "$n$": "Depolarization factor, characterizing the shape and orientation of the dielectric object (e.g., $n=1$ for parallel plane plates perpendicular to the electric field, $n=0$ for longitudinal field)." + } + }, + { + "id": 298, + "context": "", + "question": "Assume the parallel plane plates are perpendicular to the electric field. Try to determine the difference between the heat capacity $\\mathscr{C}_{\\varphi}$ when the potential difference between the plates remains constant and the heat capacity $\\mathscr{C}_{D}$ when the electric displacement remains constant, while the external pressure is also maintained constant in both situations.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$T$": "Temperature.", + "$V$": "Volume of the plates.", + "$\\mathfrak{C}$": "External electric field or electric displacement.", + "$\\pi$": "Mathematical constant pi.", + "$\\varepsilon$": "Dielectric constant (relative permittivity) of the material between the plates.", + "$\\alpha$": "Coefficient related to thermal expansion or temperature dependence of material properties." + } + }, + { + "id": 299, + "context": "", + "question": "Under the same conditions as the previous sub-question (the total volume of the parallel plane panel remains constant, $\\mathscr{C}_{\\varphi} \\equiv \\mathscr{C}_{E}$), consider representing the difference $\\mathscr{C}_{\\varphi}-\\mathscr{C}_{D}$ (i.e., $\\mathscr{C}_{E}-\\mathscr{C}_{D}$) using the external field $\\mathfrak{C}$.\nDetermine the corresponding mathematical expression for this difference.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$T$": "Temperature", + "$V$": "Total volume of the parallel plane panel", + "$\\mathfrak{C}$": "External field, defined as $\\mathfrak{C} = \\varepsilon E$", + "$\\pi$": "Mathematical constant pi", + "$\\varepsilon$": "Permittivity of the material" + } + }, + { + "id": 300, + "context": "", + "question": "In an infinite anisotropic medium, there is a spherical cavity, and the uniform electric field far from the cavity within the medium is known to be $E^{(e)}$. Find the $x$ component of the electric field inside the cavity $E_x^{(i)}$, expressed in terms of $E_x^{(e)}$ and the medium and geometric parameters.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E_{x}^{(i)}$": "x component of the electric field inside the cavity.", + "$\\varepsilon^{(x)}$": "Dielectric constant of the anisotropic medium in the x-direction.", + "$n^{(x)}$": "Depolarization coefficient of the transformed ellipsoid in the x-direction.", + "$E_{x}^{(e)}$": "x component of the uniform electric field far from the cavity within the anisotropic medium." + } + }, + { + "id": 301, + "context": "", + "question": "Assume there is a spherical cavity within an infinite anisotropic medium, and the uniform electric field far from the cavity inside the medium is known to be $E^{(e)}$. Try to find the $y$ component of the electric field inside the cavity $E_y^{(i)}$, expressed in terms of $E_y^{(e)}$ and the dielectric and geometric parameters.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E_y^{(i)}$": "y component of the electric field inside the cavity.", + "$\\varepsilon^{(y)}$": "Dielectric constant of the anisotropic medium along the y-direction.", + "$n^{(y)}$": "Depolarization coefficient of the transformed ellipsoid along the y-direction.", + "$E_y^{(e)}$": "y component of the uniform electric field far from the cavity inside the medium." + } + }, + { + "id": 302, + "context": "", + "question": "Consider a spherical cavity in an infinite anisotropic medium, with a known uniform electric field $E^{(e)}$ far away from the cavity. Determine the $z$ component $E_z^{(i)}$ of the electric field inside the cavity, expressed in terms of $E_z^{(e)}$ and the medium and geometric parameters.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E_{z}^{(i)}$": "Z-component of the electric field inside the cavity.", + "$\\varepsilon^{(z)}$": "Dielectric constant of the anisotropic medium in the z-direction.", + "$n^{(z)}$": "Depolarization factor of the transformed ellipsoid in the z-direction.", + "$E_{z}^{(e)}$": "Z-component of the electric field far away from the cavity." + } + }, + { + "id": 303, + "context": "", + "question": "For antiferromagnetic ferrous carbonate (whose structure belongs to the magnetic class $\\boldsymbol{D}_{3 d}$), derive the expression for the x-component of magnetization $M_x$ under applied stress based on its magnetoelastic effect.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$M_x$": "x-component of magnetization", + "$\\lambda_{1}$": "Magnetoelastic coupling constant (first coefficient)", + "$\\sigma_{x x}$": "x-x component of the stress tensor", + "$\\sigma_{y y}$": "y-y component of the stress tensor", + "$\\lambda_{2}$": "Magnetoelastic coupling constant (second coefficient)", + "$\\sigma_{y z}$": "y-z component of the stress tensor" + } + }, + { + "id": 304, + "context": "", + "question": "For the antiferromagnet iron carbonate (whose structure belongs to the magnetic class $\\boldsymbol{D}_{3 d}$), derive the expression for the magnetization component $M_y$ under applied stress based on its piezomagnetic effect. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Magnetism", + "final_symbol": { + "$M_{y}$": "Magnetization component along the $y$-axis.", + "$\\lambda_{1}$": "Piezomagnetic coefficient.", + "$\\sigma_{x y}$": "Component of the stress tensor.", + "$\\lambda_{2}$": "Piezomagnetic coefficient.", + "$\\sigma_{x z}$": "Component of the stress tensor." + } + }, + { + "id": 305, + "context": "", + "question": "For a crystal belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, determine the magnetization $x$ component $M_x$ induced by the magnetoelastic effect.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$M_x$": "x component of the magnetization", + "$\\lambda_{1}$": "First magnetoelastic coupling constant", + "$\\sigma_{y z}$": "yz component of the stress tensor" + } + }, + { + "id": 306, + "context": "", + "question": "For crystal belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, find the $y$ component of the magnetization $M_y$ induced by the piezomagnetic effect.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$M_y$": "y component of the magnetization.", + "$\\lambda_1$": "Piezomagnetic coefficient.", + "$\\sigma_{xz}$": "Component of the stress tensor." + } + }, + { + "id": 307, + "context": "", + "question": "For crystals belonging to the magnetic crystal class $\\boldsymbol{D}_{4 h}(\\boldsymbol{D}_{2 h})$, determine the $z$ component $M_z$ of the magnetization induced by magnetoelastic effects.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$M_z$": "z component of the magnetization", + "$\\lambda_2$": "Magnetoelastic coupling constant", + "$\\sigma_{xy}$": "Component of the stress tensor" + } + }, + { + "id": 308, + "context": "", + "question": "Ttry to find the magnetic susceptibility $\\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic field perpendicular to its axis.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha$": "Magnetic susceptibility of the conductive cylinder.", + "$\\pi$": "The mathematical constant pi.", + "$a$": "Radius of the conductive cylinder.", + "$k$": "A constant related to the wave number, appearing in the Bessel function argument and the wave equation.", + "$\\mathrm{J}_{1}$": "Bessel function of the first kind, order 1.", + "$\\mathrm{J}_{0}$": "Bessel function of the first kind, order 0." + } + }, + { + "id": 309, + "context": "", + "question": "Ttry to find the magnetic susceptibility $\\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic, but the magnetic field is parallel to the cylinder axis.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\alpha$": "Magnetic susceptibility of the conductive cylinder.", + "$k$": "A constant related to the magnetic field's spatial variation inside the cylinder, appearing in the equation $\\Delta f+k^{2} f=0$.", + "$a$": "Radius of the conductive cylinder." + } + }, + { + "id": 310, + "context": "", + "question": "The surface of a uniaxial metallic crystal is cut so that its normal forms an angle $\\theta$ with the crystal's main axis of symmetry. Considering the thermoelectric effect, under isothermal boundary conditions ($\\tau=0$) and assuming $a \\ll 1$, find the $xx$ component of the surface impedance $\\zeta_{x x}^{(\\mathrm{is})}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\zeta_{x x}^{(\\mathrm{is})}$": "xx component of the surface impedance under isothermal boundary conditions", + "$\\zeta_{0}$": "Surface impedance without considering the thermoelectric effect, $\\zeta_{0}=(\\omega \\rho_{y y} / 8 \\pi)^{1 / 2}(1-\\mathrm{i})$", + "$a$": "Dimensionless parameter related to the thermoelectric effect, $a=\\frac{T \\alpha_{x z}^{2}}{\\rho_{x x} \\varkappa_{z z}}$", + "$b$": "Parameter, $b=\\frac{c^{2} C \\rho_{x x}}{4 \\pi \\varkappa_{z z}}$" + } + }, + { + "id": 311, + "context": "", + "question": "Determine the x-component of the magnetic moment $\\mathscr{M}_{x}$ of a conducting sphere ($\\mu=1$), rotating uniformly in a uniform constant magnetic field whose components are $(\\mathfrak{H}_{x}, 0, \\mathfrak{H}_{z})$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{M}_{x}$": "x-component of the magnetic moment of the conducting sphere in the stationary reference frame.", + "V": "Volume of the conducting sphere.", + "$\\alpha^{\\prime}$": "Component of the complex magnetic polarizability, determining the magnetic moment in the plane of vector $\\boldsymbol{\\Omega}$ and $\\mathfrak{H}$.", + "$\\mathfrak{H}_{x}$": "x-component of the uniform constant magnetic field in the stationary reference frame." + } + }, + { + "id": 312, + "context": "", + "question": "Determine the y-component $\\mathscr{M}_{y}$ of the magnetic moment of a conducting sphere ($\\mu=1$) uniformly rotating in a uniform constant magnetic field whose components are $(\\mathfrak{H}_{x}, 0, \\mathfrak{H}_{z})$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{M}_{y}$": "y-component of the magnetic moment of the conducting sphere.", + "$V$": "Volume of the conducting sphere.", + "$\\alpha^{\\prime \\prime}$": "Parameter determining the component of the magnetic moment perpendicular to the plane of the vector $\\boldsymbol{\\Omega}$ and $\\mathfrak{H}$.", + "$\\mathfrak{H}_{x}$": "x-component of the uniform constant magnetic field in the rest frame." + } + }, + { + "id": 313, + "context": "", + "question": "Try to determine the magnetic moment of a non-uniformly rotating sphere (sphere radius is $a$). Assume the rotation speed is very low so that the penetration depth $\\delta \\gg a$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\mathscr{M}$": "Magnetic moment acquired by the sphere.", + "$m$": "Mass of the sphere.", + "$a$": "Radius of the sphere.", + "$\\sigma$": "Electrical conductivity of the sphere.", + "$c$": "Speed of light.", + "$e$": "Elementary charge.", + "$\\Omega$": "Angular velocity of rotation of the sphere.", + "$t$": "Time.", + "$\\mathcal M$": "Magnetic moment acquired by the sphere." + } + }, + { + "id": 314, + "context": "", + "question": "The tangential magnetic field before the shock wave $\\boldsymbol{H}_{t 1}=0$, while it is $\\boldsymbol{H}_{t 2} \\neq 0$ after the shock wave (this kind of shock wave is called a switch-on shock wave) . Determine the range of values $v_{n 1}$ for such a shock wave in a ideal gas with thermodynamic properties $(c_p/c_v = 5/3)$. You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.", + "answer": "", + "final_answer": [], + "answer_type": "Interval", + "topic": "Magnetism", + "final_symbol": { + "$u_{\\mathrm{A} 1}$": "Alfvén velocity in front of the shock wave.", + "$u_{01}$": "A characteristic velocity related to the gas in front of the shock wave, appearing in the boundary conditions for a switch-on shock wave." + } + }, + { + "id": 315, + "context": "", + "question": "Determine the direction of the extraordinary ray when light is refracted from vacuum into the surface of a uniaxial crystal, assuming the crystal surface is perpendicular to its optical axis. Hint: calculate $\\tan(\\vartheta^{\\prime})$ where $\\vartheta^{\\prime}$ is the refraction angle.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\varepsilon_{\\perp}$": "Perpendicular component of the permittivity (or dielectric constant) of the uniaxial crystal", + "$\\vartheta$": "Incidence angle", + "$\\varepsilon_{\\|}$": "Parallel component of the permittivity (or dielectric constant) of the uniaxial crystal" + } + }, + { + "id": 316, + "context": "", + "question": "Determine the direction of the extraordinary ray when vertically incident on the surface of a uniaxial crystal, assuming the optical axis of the uniaxial crystal is in an arbitrary direction. Hint: calculate $\\tan(\\vartheta^{\\prime})$ where $\\vartheta^{\\prime}$ is the refraction angle.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$\\vartheta^{\\prime}$": "Refraction angle.", + "$\\varepsilon_{\\|}$": "Permittivity parallel to the optical axis.", + "$\\varepsilon_{\\perp}$": "Permittivity perpendicular to the optical axis.", + "$\\alpha$": "Angle between the optical axis and the surface normal." + } + }, + { + "id": 317, + "context": "", + "question": "Determine the asymptotic form of the gyration vector's frequency dependence in the high-frequency regime.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$f$": "Function describing the gyration vector's frequency dependence, $f(\\omega)$.", + "$\\omega$": "Angular frequency.", + "$\\pi$": "Mathematical constant pi.", + "$N$": "Number density of electrons.", + "$e$": "Charge of an electron, $e = -|e|$.", + "$c$": "Speed of light.", + "$m$": "Mass of the electron.", + "$|e|$": "Magnitude of the electron charge.", + "$\\varepsilon$": "Function describing the dielectric function (or permittivity), $\\varepsilon(\\omega)$." + } + }, + { + "id": 318, + "context": "", + "question": "Determine the intensity distribution within the diffraction spot around the main maximum when diffraction occurs on a spherical crystal with radius $a$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "{\\mathrm{d}\\sigma}": "Differential scattering cross-section.", + "$\\pi$": "Mathematical constant pi.", + "$e$": "Elementary charge.", + "$m$": "Mass of the electron.", + "$c$": "Speed of light in vacuum.", + "$\\vartheta$": "Scattering angle.", + "$n_b$": "Number density of scattering centers in the crystal.", + "$\\varkappa$": "Magnitude of the wave vector transfer $\\boldsymbol{\\varkappa}$.", + "$a$": "Radius of the spherical crystal.", + "{\\mathrm{d}o'}": "Infinitesimal solid angle element." + } + }, + { + "id": 319, + "context": "", + "question": "Assume energy levels depend only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, where $n, ~ n^{\\prime}, ~ l, ~ m$ are given. Find the branching ratios for transitions to $l^{\\prime}=l+1$, $m^{\\prime}=m+1, m, m-1$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$l$": "Azimuthal or orbital angular momentum quantum number of the initial state", + "$m$": "Magnetic quantum number of the initial state" + } + }, + { + "id": 320, + "context": "", + "question": "Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, with $n, ~ n^{\\prime}, ~ l, ~ m$ all given. Find the branching ratios for transitions to $l^{\\prime}=l-1$, $m^{\\prime}=m+1, m, m-1$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$l$": "Azimuthal quantum number of the initial state.", + "$m$": "Magnetic quantum number of the initial state." + } + }, + { + "id": 321, + "context": "", + "question": "Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\\prime} l^{\\prime} m^{\\prime})$, with $n, ~ n^{\\prime}, ~ l, ~ m$ all given. Calculate the branching ratio for transitions to $l^{\\prime}=l+1$ and $l^{\\prime}=l-1$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$l$": "Azimuthal quantum number of the initial state" + } + }, + { + "id": 322, + "context": "", + "question": "Irradiate atoms with right circularly polarized light propagating along the positive $z$ direction, causing stimulated transitions of electrons in the atom ($E_n < E_{n'}$). Find the selection rules. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\Delta l$": "Change in the orbital angular momentum quantum number during transition.", + "$\\Delta m$": "Change in the magnetic quantum number during transition." + } + }, + { + "id": 323, + "context": "", + "question": "According to experimental measurements, the energy level of the hydrogen atom's $2\\mathrm{s}_{1 / 2}$ is higher than the $2 \\mathrm{p}_{1 / 2}$ level by 1058 MHz (Lamb shift). Find the average lifetime of the electron's spontaneous transition from the $2 \\mathrm{s}_{1 / 2}$ level to the $2 \\mathrm{p}_{1 / 2}$ level, in years.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 324, + "context": "", + "question": "For two electrons at the $n \\mathrm{p}$ energy level $(l=1)$ of an atom, attempt to determine the number of all possible total angular momentum eigenstates using both $L-S$ coupling and $j-j$ coupling.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 325, + "context": "", + "question": "Estimate the mass of a Uranium nucleus in micrograms, knowing that it contains 92 protons and 143 neutrons.\nHint: $m_{p} c^{2} \\simeq m_{n} c^{2} \\simeq 939 \\mathrm{MeV}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 326, + "context": "", + "question": "Calculate the electric field in volt per meter that a muon feels in the $1 s$-state of muonic lead.\nHints Bohr radius $a_{B}=\\hbar c /(Z \\alpha m_{\\mu} c^{2}), m_{\\mu} c^{2}=105.6 \\mathrm{MeV}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 327, + "context": "", + "question": "For the case of decay into two identical particles, when $V\\frac{v_{0}}{\\sqrt{1-v_{0}^{2}}}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion.", + "answer": "", + "final_answer": [], + "answer_type": "Interval", + "topic": "Theoretical Foundations", + "final_symbol": { + "$v_0$": "Speed of each decay particle in the $C$ frame", + "$V$": "Velocity of the $C$ frame relative to the $L$ frame" + } + }, + { + "id": 330, + "context": "", + "question": "Identify appropriately normalized coefficients in the expansion of the fields in terms of plane wave solutions with annihilation and/or creation operators. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$f(\\mathbf{p})$": "The coefficient function for annihilation operators, which is determined to be $\\frac{1}{(2 \\pi)^{3 / 2}}$.", + "$g(\\mathbf{p})$": "The coefficient function for creation operators, which is determined to be $0$.", + "$\\pi$": "Mathematical constant pi." + } + }, + { + "id": 331, + "context": "Although we won't use coherent states much in this course, coherent states do have applications in all sorts of odd corners of physics, and working out their properties is an instructive exercise in manipulating annihilation and creation operators.\n\nIt suffices to study a single harmonic oscillator; the generalization to a free field (= many oscillators) is trivial. Let\n\nH=\\frac{1}{2}(p^{2}+q^{2})\n\nand, as usual, let us define\n\na=\\frac{1}{\\sqrt{2}}(q+i p) \\quad a^{\\dagger}=\\frac{1}{\\sqrt{2}}(q-i p)\n\n\nDefine the coherent state $|z\\rangle$ by\n\n\\begin{equation*}\n|z\\rangle=N e^{z a^{\\dagger}}|0\\rangle \\tag{P4.2}\n\\end{equation*}\n\nwhere $z$ is a complex number and $N$ is a real, positive normalization factor (dependent on $z$ ), chosen such that $\\langle z \\mid z\\rangle=1$.", + "question": "The set of all coherent states for all values of $z$ is obviously complete. Indeed, it is overcomplete: The energy eigenstates can all be constructed by taking successive derivatives at $z=0$, so the coherent states\n\n\\footnotetext{\n${ }^{1}$ [Eds.] Roy J. Glauber, \"Photon correlations\", Phys. Rev. Lett. 10 (1963) 83-86. Glauber won the 2005 Nobel Prize in Physics for research in optical coherence.\n}\nwith $z$ in some small, real interval around the origin are already enough. Show that, despite this, there is an equation that looks something like a completeness relation, namely\n\n\\begin{equation*}\n1=\\alpha \\int d(\\operatorname{Re} z) d(\\operatorname{Im} z) e^{-\\beta z^{*} z}|z\\rangle\\langle z| \\tag{P4.3}\n\\end{equation*}\n\nand find the real constants $\\alpha$ and $\\beta$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\alpha$": "Real constant in the completeness relation $1=\\alpha \\int d(\\operatorname{Re} z) d(\\operatorname{Im} z) e^{-\\beta z^{*} z}|z\\rangle\\langle z|$", + "$\\beta$": "Real constant in the completeness relation $1=\\alpha \\int d(\\operatorname{Re} z) d(\\operatorname{Im} z) e^{-\\beta z^{*} z}|z\\rangle\\langle z|$", + "$\\pi$": "Pi, the mathematical constant" + } + }, + { + "id": 332, + "context": "The Lagrangian of Model 3 is:\n\\begin{align*}\n\\mathcal{L} = \\frac{1}{2}(\\partial^\\mu \\phi)(\\partial_\\mu \\phi) - \\frac{1}{2}\\mu^2\\phi^2 + \\partial^\\mu \\psi^* \\partial_\\mu \\psi - m^2 \\psi^* \\psi - g \\phi \\psi^* \\psi.\n\\end{align*}", + "question": "In Model 3, compute, to lowest non-vanishing order in $g$, the center-of-momentum differential cross-section and the total cross section for \"nucleon\"-\"antinucleon\" elastic scattering.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\sigma$": "Total cross-section.", + "$g$": "Coupling constant for the interaction between $\\phi$ and $\\psi$.", + "$\\pi$": "Mathematical constant pi.", + "$E_T$": "Total energy in the center-of-momentum frame.", + "$\\mu$": "Mass parameter of the scalar field $\\phi$.", + "$|\\mathbf{p}_i|$": "Magnitude of the initial 3-momentum of a particle." + } + }, + { + "id": 333, + "context": "", + "question": "The two-particle density of states factor, $D$, in the center-of-momentum frame, $\\mathbf{P}_{T}=\\mathbf{0}$:\n\n\\begin{equation*}\nD=\\frac{1}{16 \\pi^{2}} \\frac{|\\mathbf{p}_{f}| d \\Omega_{f}}{E_{T}}.\n\\end{equation*}\nwhere we have used the notation: the final particles' momenta $|\\mathbf{p_f|$ in the center-of-momentum frame The factor $d\\Omega_f$ describes the solid angle associated with $d^3\\mathbf{p}_f$.}\n\n\nFind the formula that replaces this one if $\\mathbf{P}_{T} \\neq \\mathbf{0}$. Comment: Although the center-of-momentum frame is certainly the simplest one in which to work, sometimes we want to do calculations in other frames, for example, the \"lab frame\", in which one of the two initial particles is at rest.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E_{k}$": "Energy of the particle with momentum $\\mathbf{k}$", + "$\\phi$": "Azimuthal angle of $\\mathbf{k}$ about the $\\mathbf{P}_{T}$ axis", + "$|\\mathbf{P}_{T}|$": "Magnitude of the total momentum $\\mathbf{P}_{T}$" + } + }, + { + "id": 334, + "context": "", + "question": "When we attempted to quantize the free Dirac theory\n\n\\begin{equation*}\n\\mathscr{L}= \\pm \\bar{\\psi}(i \\partial \\!\\!\\!/-m) \\psi \\tag{P12.1}\n\\end{equation*}\n\nwith canonical commutation relations, we encountered a disastrous contradiction with the positivity of energy. We succeeded when we used canonical anticommutators (if we chose ( $\\pm$ ) to be + ). Much earlier we were able to quantize the free charged scalar field,\n\n\\begin{equation*}\n\\mathscr{L}= \\pm(\\partial_{\\mu} \\phi^{*} \\partial^{\\mu} \\phi-\\mu^{2} \\phi^{*} \\phi) \\tag{P12.2}\n\\end{equation*}\n\nwith canonical commutators (if we chose $( \\pm)$ to be + ). Attempt to quantize the free charged scalar field with (nearly) canonical anticommutators:\n\n\\begin{align*}\n\\{\\phi(\\mathbf{x}, t), \\phi(\\mathbf{y}, t)\\} & =\\{\\dot{\\phi}(\\mathbf{x}, t), \\dot{\\phi}(\\mathbf{y}, t)\\}=0 \\\\\n{\\phi(\\mathbf{x}, t), \\phi^{*}(\\mathbf{y}, t)} & ={\\dot{\\phi}(\\mathbf{x}, t), \\dot{\\phi}^{*}(\\mathbf{y}, t)}=0 \\tag{P12.3}\\\\\n{\\phi(\\mathbf{x}, t), \\dot{\\phi}^{*}(\\mathbf{y}, t)} & =\\lambda \\delta^{(3)}(\\mathbf{x}-\\mathbf{y})\n\\end{align*}\n\nwhere $\\lambda$ is a (possibly complex) constant.\nShow that one reaches a disastrous contradiction with the positivity of the norm in Hilbert space; that is to say:\n\n\\begin{equation*}\n\\langle\\phi|{\\theta, \\theta^{\\dagger}}|\\phi\\rangle \\geq 0 \\tag{21.20}\n\\end{equation*}\n\nfor any operator $\\theta$ and any state $|\\phi\\rangle$.\n\nHints: (1) Canonical anticommutation implies that, even on the classical level, $\\phi$ and $\\phi^{*}$ are Grassmann variables. If you don't take proper account of this (especially in ordering terms when deriving the canonical momenta), you'll get hopelessly confused. (2) Dirac theory is successfully quantized using anticommutators; the sign of the Lagrangian is fixed by appealing to the positivity of the inner product in Hilbert space. If we attempt to quantize the theory using commutators, we get into trouble with the positivity of the energy. The Klein-Gordon theory is successfully quantized using commutators; the sign of the Lagrangian is fixed by appealing to the positivity of energy. So it's to be expected that we'd get into trouble, if we attempted to quantize the Klein-Gordon theory with anticommutators, with the positivity of the inner product. (3) You should do the expansion\n\\begin{align*}\n\\phi(x) = \\int \\frac{d^3 \\mathbf{p}}{(2\\pi)^{3/2}\\sqrt{2\\omega_{\\mathbf{p}}}} [b_{\\mathbf{p}} e^{-i\\mathbf{p}\\cdot x} + c_{\\mathbf{p}}^\\dagger e^{i\\mathbf{p}\\cdot x}],\n\\end{align*}\nand output $\\{b_p,b_p^\\dagger\\}, \\{c_p,c_p^\\dagger\\}$.\n\nHint: You can use $a = \\mathrm{Re}\\lambda,b=\\mathrm{Im}\\lambda$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$b$": "Imaginary part of $\\lambda$, $b = \\mathrm{Im}\\lambda$", + "$\\delta^{(3)}$": "Three-dimensional Dirac delta function", + "$\\mathbf{p}$": "Three-momentum vector", + "$\\mathbf{p}^{\\prime}$": "Another three-momentum vector", + "$b_{\\mathbf{p}}$": "Annihilation operator for a particle with momentum $\\mathbf{p}$", + "$b_{\\mathbf{q}}^{\\dagger}$": "Creation operator for a particle with momentum $\\mathbf{q}$", + "$c_{\\mathbf{p}}$": "Annihilation operator for an antiparticle with momentum $\\mathbf{p}$", + "$c_{\\mathbf{q}}^{\\dagger}$": "Creation operator for an antiparticle with momentum $\\mathbf{q}$", + "$\\mathbf{q}$": "Another three-momentum vector" + } + }, + { + "id": 335, + "context": "Let $\\psi_{A}, \\psi_{B}, \\psi_{C}$ and $\\psi_{D}$ be four Dirac spinor fields. These fields interact with each other (and possibly with unspecified scalar and pseudoscalar fields) in some way that is invariant under $P, C$, and $T$, where these operations are defined in the \"standard way\":\n\n\\begin{equation*}\nU_{P}^{\\dagger} \\psi(\\mathbf{x}, t) U_{P}=\\beta \\psi(-\\mathbf{x}, t) \\tag{22.8}\n\\end{equation*}\n\n\nLikewise,\n\n\\begin{equation*}\nU_{C}^{\\dagger} \\psi(x) U_{C}=\\psi'(x) \\tag{22.49}\n\\end{equation*}\n\nin a Majorana basis (one in which $\\gamma^{\\mu}=-\\gamma^{\\mu *}$ ). Finally,\n\n\\begin{equation*}\n\\Omega_{P T}^{-1} \\psi(x) \\Omega_{P T}=i \\gamma_{5} \\psi(-x) \\tag{22.80}\n\\end{equation*}\n\nagain in a Majorana basis. Now let us consider adding a term to the Hamiltonian density,\n\n\\begin{align*}\n\\mathscr{H}^{\\prime}= & g_{1}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\psi_{D})+g_{2}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\gamma_{5} \\psi_{D})+g_{3}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\gamma_{5} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\psi_{D}) \\\\\n& +g_{4}(\\bar{\\psi}_{A} \\gamma^{\\mu} \\gamma_{5} \\psi_{B})(\\bar{\\psi}_{C} \\gamma_{\\mu} \\gamma_{5} \\psi_{D})+\\text { Hermitian conjugate } \\tag{P14.1}\n\\end{align*}\n\nwhere the $g_{i}$ 's are (possibly complex) numbers.", + "question": "Under what conditions on the $g^{\\prime}$ s is $\\mathscr{H}^{\\prime}(0)$ invariant under $P$. Hint: $\\Omega_{P T}$ is anti-unitary. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$g_2$": "Coupling constant for the second interaction term in $\\mathscr{H}'$", + "$g_3$": "Coupling constant for the third interaction term in $\\mathscr{H}'$" + } + }, + { + "id": 336, + "context": "", + "question": "Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:\n\n\\mathscr{L}=\\mathscr{L}_{\\mathrm{em}}-\\frac{1}{2} \\lambda(\\partial_{\\mu} A^{\\mu}+\\sigma A_{\\mu} A^{\\mu})^{2}+\\mathscr{L}_{\\mathrm{ghost}}\n\n\nHere $\\mathscr{L}_{\\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\\lambda$ and $\\sigma$ are arbitrary real numbers.\n\nWhat are the vertices involving ghost fields?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\sigma$": "Arbitrary real number, a parameter in the gauge-fixing term.", + "$k^{\\mu}$": "Contravariant four-momentum." + } + }, + { + "id": 337, + "context": "", + "question": "Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:\n$$\n\\mathscr{L}=\\mathscr{L}_{\\mathrm{em}}-\\frac{1}{2} \\lambda(\\partial_{\\mu} A^{\\mu}+\\sigma A_{\\mu} A^{\\mu})^{2}+\\mathscr{L}_{\\mathrm{ghost}}\n$$\n\nHere $\\mathscr{L}_{\\mathrm{em}}$ is the standard Lagrangian, with neither gauge-fixing nor ghost terms, and $\\lambda$ and $\\sigma$ are arbitrary real numbers.\n\nWhat is $\\mathscr{L}_{\\text {ghost }}$ ?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mathscr{L}_{\\mathrm{ghost}}$": "Ghost Lagrangian.", + "$\\bar{\\eta}$": "Complex anti-ghost field.", + "$\\square^{2}$": "D'Alembert operator, defined as $\\square^{2} = \\partial_{\\mu}\\partial^{\\mu}$.", + "$\\sigma$": "Arbitrary real number, a gauge-fixing parameter.", + "$A_{\\mu}$": "Covariant four-vector potential.", + "$\\partial^{\\mu}$": "Contravariant four-gradient operator.", + "$\\eta$": "Complex ghost field.", + "$\\partial_{\\mu}$": "Covariant four-gradient operator." + } + }, + { + "id": 338, + "context": "", + "question": "$\\mathrm{SU}(3)$ allows only one possible coupling of the electromagnetic current to a quark and an antiquark. Thus (by the same reasoning used for the decuplet in the previous problem), in the limit of perfect $\\mathrm{SU}(3)$ symmetry, if quarks are observable, their magnetic moments would be proportional to their charges. In the non-relativistic limit,\n\n\\begin{equation*}\n\\boldsymbol{\\mu}=\\kappa q \\boldsymbol{\\sigma}, \\tag{P21.2}\n\\end{equation*}\n\nwhere $\\kappa$ is an unknown constant, $q$ is the electric charge of the quark in question, and $\\boldsymbol{\\sigma}$ is the vector of Pauli spin matrices.\n\n\\footnotetext{\n${ }^{1}$ [Eds.] \"An Introduction to Unitary Symmetry\", the Erice notes from the summer of 1966, originally published in Strong and Weak Interactions - Present Problems, Academic Press, 1966, and reprinted in Coleman Aspects.\n}\n\nIn the naive quark model discussed in class, the baryons are considered as non-relativistic three-quark bound states with no spin-dependent interactions. Thus, as in atomic physics, we can compute the baryon moments in terms of the quark moments, that is, in terms of the single unknown constant $\\kappa$, if we know the baryon wave function. For the lightest baryon octet, the one that contains the proton and the neutron, there is no orbital contribution to the magnetic moments because each quark has zero orbital angular momentum. Thus all we need is the spin-flavor-color part of the wave function. Of course, since the assumption of perfect $\\mathrm{SU}(3)$ already gives all the baryon moments in terms of the proton and neutron moments, the only new information we get from this analysis is the ratio of these moments. Compute the ratio and compare it to experiment.\n\nRemark. It's clear from the way the calculation is set up that it's the total moment you will be computing, not the anomalous moment. Be careful that you don't use the anomalous moments when you make the computation.\n\nHint: You will need the spin-flavor part of the wave functions for both the proton and the neutron to do this problem. Here is an easy way to construct them without resorting to tables of $3 j$ symbols. It is trivial to construct the wave function for the $I_{z}=J_{z}=\\frac{3}{2}$ state of the $\\Delta$; it is $|u \\uparrow, u \\uparrow, u \\uparrow\\rangle$, with all three quarks being up quarks, and all three spinning up. If we apply both an isospin lowering operator and a spin lowering operator to this, we obtain the $I_{z}=J_{z}=\\frac{1}{2}$ state of the $\\Delta$. The $J_{z}=\\frac{1}{2}$ state of the proton must be orthogonal to this. The $J_{z}=\\frac{1}{2}$ state of the neutron (up to an irrelevant phase) is obtained from the proton state by exchanging $u$ and $d$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 339, + "context": "", + "question": "The model\n\\begin{equation*}\n\\mathscr{L}=\\frac{1}{2}(\\partial_{\\mu} \\boldsymbol{\\Phi}) \\cdot(\\partial^{\\mu} \\boldsymbol{\\Phi})-U(\\boldsymbol{\\Phi}) \n\\end{equation*}\nwas a theory with spontaneous breakdown of $\\mathrm{U}(1)$ internal symmetry. The particle spectrum of the theory consisted of a massless Goldstone boson and a massive neutral scalar. Furthermore, this term in the Lagrangian\n\n\\begin{equation*}\n\\frac{1}{2} \\rho^{2}(\\partial_{\\mu} \\theta)^{2}=\\frac{1}{2} a^{2}(\\partial_{\\mu} \\theta)^{2}+a \\rho^{\\prime}(\\partial_{\\mu} \\theta)^{2}+\\frac{1}{2} \\rho^{\\prime 2}(\\partial_{\\mu} \\theta)^{2} \\tag{P24.5}\n\\end{equation*}\n\ngives rise to the decay of the massive meson into two Goldstone bosons, with an invariant Feynman amplitude proportional to $a^{-1}$. (This is not a misprint: before reading the decay amplitude from the Lagrangian, we must first rescale $\\theta$ to put the free Lagrangian in standard form.) Now consider the theory minimally coupled instead to a massive photon with mass $\\mu_{0}$ (before symmetry breaking). What is the photon mass after the symmetry breaks? \n\nComment: The Abelian Higgs model is the same theory minimally coupled to a massless photon.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\mu_{0}$": "Mass of the photon before symmetry breaking.", + "$a$": "Vacuum expectation value (VEV) of the scalar field, or a constant related to symmetry breaking.", + "$e$": "Electric charge or gauge coupling constant." + } + }, + { + "id": 340, + "context": "", + "question": "Attempt to derive the elementary excitation spectrum in a nearly ideal Bose gas, where the elementary excitation spectrum is considered as the dispersion relation of the collective wave function fluctuations. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega$": "Angular frequency of the collective wave.", + "$p$": "Momentum, defined as $p=\\boldsymbol{\\hbar} \\boldsymbol{k}$. It is treated as a scalar magnitude in the equations.", + "$m$": "Mass of the particle.", + "$n$": "Constant mean value related to the Bose gas density.", + "$U_0$": "Interaction strength parameter." + } + }, + { + "id": 341, + "context": "", + "question": "Calculate the probability of a quasiparticle with momentum $p$ (close to the threshold $p_{\\mathrm{c}}$) emitting a phonon, when the quasiparticle speed reaches the speed of sound.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$w$": "Total probability of phonon emission", + "$A$": "Constant related to the interaction strength, defined as $A=p_{\\mathrm{c}}+\\frac{\\rho}{u} \\frac{\\partial \\varepsilon}{\\partial \\rho}|_{p=p_{c}}$", + "$p$": "Momentum of a quasiparticle", + "$p_{\\mathrm{o}}$": "Threshold momentum, appears as a typo for $p_{\\mathrm{c}}$ in the final expression for $w$", + "$\\pi$": "Mathematical constant pi", + "$\\rho$": "Density of the medium", + "$\\hbar$": "Reduced Planck's constant" + } + }, + { + "id": 342, + "context": "", + "question": "There is a planar film with thickness $d \\ll \\xi, \\delta$. Find the critical value of the magnetic field parallel to the planar film, which can destroy superconductivity.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$H_{\\mathrm{c}}^{f}$": "Critical magnetic field of the film.", + "$H_{\\mathrm{c}}$": "Critical magnetic field of large-scale superconductors.", + "$\\delta$": "Penetration depth.", + "$d$": "Thickness of the planar film." + } + }, + { + "id": 343, + "context": "", + "question": "If the average magnetic induction intensity of the cylindrical sample cross-section is $\\bar{B}$, and in the mixed state with an external magnetic field $\\mathfrak{S}$, all vortices are distributed at distances $d \\gg \\delta$ from each other, forming an equilateral triangular lattice in the sample cross-section. Try to determine the relationship between the external field $\\mathfrak{S}$, lower critical field $H_{\\mathrm{cl}}$, and the dimensionless vortex spacing $a = d/\\delta$ at thermodynamic equilibrium under the condition $1/a \\ll 1$. (Hint: the thermodynamic potential per unit volume $\\tilde{f}$ reaches its minimum).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathfrak{S}$": "External magnetic field", + "$H_{\\mathrm{c} 1}$": "Lower critical field", + "$\\phi_{0}$": "Magnetic flux quantum", + "$\\delta$": "Penetration depth", + "$a$": "Dimensionless vortex spacing, $a = d/\\delta$" + } + }, + { + "id": 344, + "context": "", + "question": "If the average magnetic induction intensity of the cross-section of a cylindrical sample is $\\bar{B}$, and in the mixed state with an applied magnetic field of $\\mathfrak{S}$, each vortex line is distributed at a distance $d \\gg \\delta$ from each other and forms an equilateral triangular lattice within the sample cross-section. It is known that the average magnetic induction intensity $\\bar{B} = \\nu \\phi_0$ (where $\\nu$ is the number of vortices per unit area and $\\phi_0$ is the magnetic flux quantum) and the dimensionless vortex spacing $a = d/\\delta$ (where $d$ is the vortex spacing, and $\\delta$ is the London penetration depth). Try to derive the relationship between the dimensionless vortex spacing $a$ and the average magnetic induction intensity $\\bar{B}$ under the condition $1/a \\ll 1$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$a$": "Dimensionless vortex spacing, $a = d/\\delta$", + "$\\phi_0$": "Magnetic flux quantum", + "$\\delta$": "London penetration depth", + "$\\bar{B}$": "Average magnetic induction intensity of the cross-section of a cylindrical sample" + } + }, + { + "id": 345, + "context": "", + "question": "Ignoring interactions between spins, calculate the magnetization of a paramagnet when the ratio of $\\beta \\mathscr{G}$ to $T$ is arbitrary.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "M": "Magnetization of the paramagnet.", + "$\\beta$": "A constant that, when multiplied by the magnetic field $\\mathscr{G}$, gives an energy scale.", + "N": "Total number of spins.", + "V": "Volume.", + "S": "Total spin quantum number.", + "$\\mathscr{G}$": "Magnetic field strength.", + "T": "Absolute temperature." + } + }, + { + "id": 346, + "context": "If the interaction energy of a crystal can be expressed as $u(r)=-\\frac{\\alpha}{r^{m}}+\\frac{\\beta}{r^{n}}$", + "question": "Taking $m=2, n=10, r_{0}=0.3 \\mathrm{~nm}, W=4 \\mathrm{eV}$, calculate the value of $\\beta$, unit $\\mathrm{eV} \\cdot \\mathrm{m}^{10}$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\beta$": "Constant coefficient for the repulsive term in the interaction energy" + } + }, + { + "id": 347, + "context": "", + "question": "For $\\mathrm{H}_{2}$, the Lennard-Jones potential parameters obtained from gas measurements are $\\varepsilon=50 \\times 10^{-6} J, \\sigma=2.96 \\stackrel{\\circ}{\\mathrm{~A}}$. Calculate the binding energy of $\\mathrm{H}_{2}$ when it forms a face-centered cubic solid molecular hydrogen (in units of $\\mathbf{K J} / \\mathrm{mol}$), considering each hydrogen molecule as spherical. The experimental value of the binding energy is $0.751 \\mathrm{~kJ} / \\mathrm{mo1}$. Compare it with the calculated value.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 348, + "context": "", + "question": "Consider the lattice vibrations of a diatomic chain where the force constants between nearest neighbor atoms alternate as $c$ and $10c$. The two types of atoms have the same mass, and the nearest neighbor distance is $\\frac{a}{2}$. Find the vibrational frequency $\\omega(k)$ at $k=0$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$c$": "Force constant between nearest neighbor atoms.", + "$M$": "Mass of the atoms in the diatomic chain." + } + }, + { + "id": 349, + "context": "", + "question": "Consider the lattice vibrations of a diatomic chain where the force constants alternate as $c$ and $10c$ between nearest neighbor atoms on the chain. The two kinds of atoms have the same mass and the nearest neighbor distance is $\\frac{a}{2}$. Find the vibration frequency $\\omega(k)$ at $k=\\frac{\\pi}{a}$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$c$": "Force constant between nearest neighbor atoms.", + "$M$": "Mass of the atoms in the diatomic chain." + } + }, + { + "id": 350, + "context": "One-dimensional Compound Lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$),", + "question": "Find the optical frequencies $\\omega_{\\max }^{0}, \\omega_{\\min }^{0}$, and present the answer in tuple form. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\omega_{\\max}^0$": "Maximum optical frequency", + "$\\omega_{\\min}^0$": "Minimum optical frequency", + "$\\beta$": "Spring constant, $\\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$", + "$m$": "Mass of the lighter atom, $m=5 \\times 1.67 \\times 10^{-24} \\mathrm{~g}$", + "$M$": "Mass of the heavier atom, such that $\\frac{M}{m}=4$" + } + }, + { + "id": 351, + "context": "One-dimensional complex lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$),", + "question": "What is the corresponding phonon energy in electron volts for the optical wave $\\omega_{\\min }^{o}$?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 352, + "context": "One-dimensional complex lattice\n$m=5 \\times 1.67 \\times 10^{-24} g, \\frac{M}{m}=4, \\beta=1.5 \\times 10^{1} \\mathrm{~N} / \\mathrm{m}$ (i.e., $1.51 \\times 10^{4} \\mathrm{dyn} / \\mathrm{cm}$),", + "question": "Determine the wavelength band of the electromagnetic wave corresponding to the optical wave $\\omega_{\\max }^{0}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 353, + "context": "", + "question": "For a one-dimensional lattice with a lattice constant of $2.5 A$, estimate the time required for an electron to move from the bottom of the energy band to the top under an external electric field of $10^{7} \\mathrm{~V} / \\mathrm{m}$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 354, + "context": "", + "question": "The spin of $\\mathrm{He}^{3}$ is $1 / 2$, making it a fermion. The density of liquid $\\mathrm{He}^{3}$ near absolute zero is $0.081 \\mathrm{gcm}^{-3}$. Calculate the Fermi temperature $\\mathbf{T}^{\\mathbf{F}}$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 355, + "context": "If silver is considered to be a monovalent metal with a spherical Fermi surface, calculate the following quantities", + "question": "Find the Fermi energy and Fermi temperature, and provide the answer as a tuple You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 356, + "context": "If silver is considered a single-valence metal with a spherical Fermi surface, calculate the following quantities", + "question": "Find the average free path of electrons at room temperature and low temperature, represented as a tuple You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 357, + "context": "InSb effective electron mass $m_{e}=0.015 m$, dielectric constant $\\varepsilon=18$, lattice constant $a=6.49 A$.", + "question": "Find the ground state orbital radius;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Theoretical Foundations", + "final_symbol": {} + }, + { + "id": 358, + "context": "", + "question": "Given the expression for the free energy of an object, how can the average kinetic energy of the object's particles be calculated?", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$m$": "Mass of the object's identical particles.", + "$F$": "Free energy of the object.", + "$r$": "A subscript indicating that a set of parameters, possibly related to position or other extensive variables, are held constant during the partial differentiation.", + "$v$": "A subscript indicating that a set of parameters, possibly related to volume or other extensive variables, are held constant during the partial differentiation." + } + }, + { + "id": 359, + "context": "", + "question": "Find the expression for heat capacity $C_{\\nu}$ when variables are $T, \\mu, V$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$C_v$": "Heat capacity at constant volume", + "$T$": "Absolute temperature", + "$S$": "Entropy", + "$N$": "Number of particles", + "$\\mu$": "Chemical potential" + } + }, + { + "id": 360, + "context": "", + "question": "Find the probability distribution of atomic kinetic energy.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{d} w_{\\varepsilon}$": "Infinitesimal probability of an atom having kinetic energy in the range $\\varepsilon$ to $\\varepsilon + \\mathrm{d}\\varepsilon$", + "$\\varepsilon$": "Atomic kinetic energy", + "$T$": "Temperature", + "$\\pi$": "Mathematical constant pi", + "$\\mathrm{e}$": "Euler's number, the base of the natural logarithm" + } + }, + { + "id": 361, + "context": "", + "question": "Find the probability distribution of molecular rotational angular velocity.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{d} w_{\\Omega}$": "Differential probability distribution of angular velocity.", + "$\\pi$": "Mathematical constant pi.", + "$T$": "Temperature (or $k_B T$ where $k_B$ is Boltzmann constant).", + "$I_1$": "Principal moment of inertia along the first principal axis.", + "$I_2$": "Principal moment of inertia along the second principal axis.", + "$I_3$": "Principal moment of inertia along the third principal axis.", + "$\\Omega_1$": "Projection of angular velocity on the first principal axis of inertia.", + "$\\Omega_2$": "Projection of angular velocity on the second principal axis of inertia.", + "$\\Omega_3$": "Projection of angular velocity on the third principal axis of inertia.", + "$\\mathrm{d}(\\Omega_1)$": "Differential element for the first component of angular velocity.", + "$\\mathrm{d}(\\Omega_2)$": "Differential element for the second component of angular velocity.", + "$\\mathrm{d}(\\Omega_3)$": "Differential element for the third component of angular velocity." + } + }, + { + "id": 362, + "context": "", + "question": "Determine the coordinate density matrix of the harmonic oscillator.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\rho$": "Coordinate density matrix of the harmonic oscillator.", + "$q$": "Oscillator coordinate.", + "$q^{\\prime}$": "A different oscillator coordinate, used in the density matrix $\\rho(q, q^{\\prime})$.", + "$\\omega$": "Angular frequency of the harmonic oscillator.", + "$\\pi$": "Mathematical constant pi.", + "$\\hbar$": "Reduced Planck's constant.", + "$T$": "Temperature of the system, corresponding to statistical equilibrium." + } + }, + { + "id": 363, + "context": "", + "question": "Find the distribution of particles by momentum for a relativistic ideal gas.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{d} N_{p}$": "Differential number of particles with momentum in a specific range", + "$N$": "Total number of particles", + "$V$": "Volume", + "$c$": "Speed of light", + "$m$": "Mass of the particle", + "$p$": "Magnitude of the momentum of the particle", + "$T$": "Temperature", + "$K_{1}$": "Macdonald function (Hankel function of imaginary argument) of order 1", + "$K_{0}$": "Macdonald function (Hankel function of imaginary argument) of order 0", + "$\\mathrm{d}(p_{x})$": "Differential momentum component in x-direction", + "$\\mathrm{d}(p_{y})$": "Differential momentum component in y-direction", + "$\\mathrm{d}(p_{2})$": "Differential momentum component in the third direction (likely a typo for $p_z$)", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 364, + "context": "", + "question": "Find the number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\\theta$ and $\\theta+\\mathrm{d} \\theta$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{d} \\nu_{\\theta}$": "Number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\\theta$ and $\\theta+\\mathrm{d} \\theta$.", + "$N$": "Total number of gas molecules.", + "$V$": "Volume of the vessel.", + "$T$": "Temperature of the gas.", + "$m$": "Mass of a single gas molecule.", + "$\\pi$": "Mathematical constant pi.", + "$\\theta$": "Angle between the velocity direction of a gas molecule and the surface normal of the vessel wall.", + "$\\mathrm{d} \\theta$": "Infinitesimal change in the angle $\\theta$." + } + }, + { + "id": 365, + "context": "", + "question": "Find the number of gas molecules with speeds between $v$ and $v+\\mathrm{d} v$ that collide with a unit area of the wall per unit time.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{d} \\nu_{v}$": "Number of gas molecules with speeds between $v$ and $v+\\mathrm{d} v$ that collide with a unit area of the wall per unit time.", + "$N$": "Total number of gas molecules.", + "$V$": "Volume.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$m$": "Mass of a single gas molecule.", + "$T$": "Temperature of the gas. In the context of the Maxwell-Boltzmann distribution, it is often multiplied by the Boltzmann constant $k$ to form $kT$.", + "$\\mathrm{e}$": "Base of the natural logarithm, approximately 2.71828.", + "$\\nu$": "Speed of gas molecules, likely a typographical error for $v$ in the exponent term.", + "$v$": "Speed of gas molecules.", + "$\\mathrm{d} v$": "Infinitesimal increment in speed." + } + }, + { + "id": 366, + "context": "", + "question": "Find the work and heat obtained in the process of gas under constant pressure (isobaric process), expressed as a tuple. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$R$": "Work obtained in the process.", + "$Q$": "Heat obtained in the process.", + "$N$": "Number of moles or particles of the gas.", + "$T_1$": "Initial temperature of the gas.", + "$T_2$": "Final temperature of the gas.", + "$c_p$": "Molar specific heat at constant pressure." + } + }, + { + "id": 367, + "context": "", + "question": "If a gas is compressed from volume $V_{1}$ to volume $V_{2}$ following the law $P V^{n}=a$ (polytropic process), calculate the work done on it and the heat received by it, expressed as a tuple. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$R$": "Work done on the gas, $R = -\\int_{V_{1}}^{V_{2}} P \\mathrm{~d} V$.", + "$Q$": "Heat received by the gas.", + "$a$": "Polytropic constant, defined by the process law $P V^{n}=a$.", + "$n$": "Polytropic index.", + "$V_2$": "Final volume of the gas.", + "$V_1$": "Initial volume of the gas.", + "$c_v$": "Molar specific heat at constant volume." + } + }, + { + "id": 368, + "context": "", + "question": "Find the average value $\\langle\\exp (\\alpha_{i} x_{i})\\rangle$, where $\\alpha_{i}$ is a constant and $x_{i}$ is a fluctuation following a Gaussian distribution.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\langle\\exp (\\alpha_{i} x_{i})\\rangle$": "Average value of the exponential of $\\alpha_{i} x_{i}$", + "$\\alpha_{i}$": "Constant coefficient", + "$\\alpha_{k}$": "Constant coefficient", + "$\\langle x_{i} x_{k}\\rangle$": "Average value of the product of fluctuations $x_i$ and $x_k$", + "$\\beta_{ik}^{-1}$": "Inverse of coefficient matrix element $\\beta_{ik}$" + } + }, + { + "id": 369, + "context": "This is a thermodynamics problem.", + "question": "Try to find $\\langle(\\Delta W)^{2}\\rangle$ (with $P,V,C_p,T$ and $S$ as variables).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\langle(\\Delta W)^{2}\\rangle$": "Mean square fluctuation of work, defined by $-T V^{2}(\\frac{\\partial P}{\\partial V})+T^{2} C_{p}$", + "$T$": "Temperature", + "$V$": "Volume", + "$P$": "Pressure", + "$C_p$": "Heat capacity at constant pressure" + } + }, + { + "id": 370, + "context": "This is a thermodynamics problem.", + "question": "Try to find $\\langle\\Delta T \\Delta P\\rangle$ (where $V,C_v,P$ and $T$ are variables).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\Delta T$": "Fluctuation in temperature", + "$\\Delta P$": "Fluctuation in pressure", + "$T$": "Temperature", + "$C_v$": "Heat capacity at constant volume", + "$P$": "Pressure", + "$V$": "Volume" + } + }, + { + "id": 371, + "context": "This is a thermodynamics problem.", + "question": "Try to find $\\langle\\Delta S \\Delta V\\rangle$ ( $V$ and $T$ are variables).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$V$": "Volume, specified as a variable", + "$T$": "Temperature, specified as a variable" + } + }, + { + "id": 372, + "context": "The response function is :\n\\begin{align*}\n\\alpha(\\omega) = \\frac{1}{\\hbar} \\sum_m |x_{mn}|^2 \\left[ \\frac{1}{\\omega_{mn} - \\omega - i0} + \\frac{1}{\\omega_{mn} + \\omega + i0} \\right].\n\\end{align*}", + "question": "Find the asymptotic behavior of $\\alpha(\\omega)$ as $\\omega \\rightarrow \\infty$ (assuming $\\alpha(\\infty)=0$).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\mathrm{i}$": "Imaginary unit, $\\sqrt{-1}$.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega$": "Angular frequency.", + "$\\dot{\\hat{x}}$": "Time derivative of the position operator, representing the velocity operator.", + "$\\hat{x}$": "Position operator." + } + }, + { + "id": 373, + "context": "", + "question": "Consider a system composed of two independent oscillators (i.e., two types of phonons), with $n_{1}, ~ n_{2}$ representing their quantum numbers (phonon numbers), and $a_{1}^{+}, ~ a_{1}, ~ a_{2}^{+}, ~ a_{2}$ representing the quantum number raising and lowering operators (i.e., the creation and annihilation operators of the two types of phonons), $\\hat{n}_{1}=a_{1}^{+} a_{1}$ and $\\hat{n}_{2}=a_{2}^{+} a_{2}$ represent the particle number operators. The normalized eigenstate in the particle number representation is denoted as $|n_{1} n_{2}\\rangle$. Let\n\n\\begin{gathered}\na=\\binom{a_{1}}{a_{2}}, \\quad a^{+}=(a_{1}^{+} a_{2}^{+}) \\\\\nJ=\\frac{1}{2} a^{+} \\sigma a \\quad(\\sigma \\text { is the Pauli matrix })\n\\end{gathered}\n\n\nThat is,\n\n\\begin{align*}\n& J_{x}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{ll}\n0 & 1 \\\\\n1 & 0\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2}(a_{1}^{+} a_{2}+a_{2}^{+} a_{1}) \\\\\n& J_{y}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{cc}\n0 & -\\mathrm{i} \\\\\n\\mathrm{i} & 0\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2 \\mathrm{i}}(a_{1}^{+} a_{2}-a_{2}^{+} a_{1}) \\tag{1}\\\\\n& J_{z}=\\frac{1}{2}(a_{1}^{+} a_{2}^{+})(\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1\n\\end{array})\\binom{a_{1}}{a_{2}}=\\frac{1}{2}(a_{1}^{+} a_{1}-a_{2}^{+} a_{2})=\\frac{1}{2}(\\hat{n}_{1}-\\hat{n}_{2})\n\\end{align*}\n\n\nAlso,\n\n\\begin{align*}\n& J_{+}=J_{x}+\\mathrm{i} J_{y}=a_{1}^{+} a_{2} \\tag{2}\\\\\n& J_{-}=J_{x}-\\mathrm{i} J_{y}=a_{2}^{+} a_{1}=(J_{+})^{+}\n\\end{align*}\n\n\nFind the eigenvalues of $J_z$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 374, + "context": "", + "question": "Denote the creation and annihilation operators of a single-particle state in a fermion system as $a^{+}$ and $a$, respectively, satisfying the fundamental anti-commutation relations\n\n\\begin{align*}\n& {[a, a^{+}]_{+} \\equiv a a^{+}+a^{+} a=1} \\tag{1}\\\\\n& a^{2}=0, \\quad(a^{+})^{2}=0 \\tag{2}\n\\end{align*}\n\n\nLet $\\hat{n}=a^{+} a$ be the particle number operator on this single-particle state. Calculate the commutator $[\\hat{n}, a^{+}]$ and $[\\hat{n}, a]$, represent it as a tuple. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$a^{+}$": "Creation operator of a single-particle state in a fermion system", + "$a$": "Annihilation operator of a single-particle state in a fermion system" + } + }, + { + "id": 375, + "context": "Assume a Fermi particle system moves in a central force field. The single-particle energy level is related to the total angular momentum of the particle, denoted as $\\varepsilon_j$, its degeneracy is $(2j+1)$, corresponding to the single-particle state, $|jm\\rangle = a_{jm}^\\dagger |0\\rangle$, where $m=\\pm j, \\pm(j-1), \\dots, \\pm \\frac{1}{2}$. $|0\\rangle$ represents the vacuum state; $a_{jm}^\\dagger$ is the Fermi particle creation operator for the $|jm\\rangle$ state. Consider the state on the energy level $\\varepsilon_j$ where a pair of Fermi particles have their angular momentum coupled to $0$, denoted as (coupled representation, $J=M=0$) $|jj00\\rangle$.", + "question": "Introduce the total particle number operator on the energy level $\\varepsilon_{j}$\n\n\\begin{align*}\n\\hat{N}_{j} & =\\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j-m}^{+} a_{j-m}) \\\\\n& =\\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j \\bar{m}}^{+} a_{j \\bar{m}}) \\tag{1}\n\\end{align*}\n\nCalculate $[\\hat{N}_{j}, C_{j}^{+}], ~[C_{j}, C_{j}^{+}], ~[C_{j}^{+} C_{j}, C_{j}^{+}]$. You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": { + "$C_j^{+}$": "Creation operator for a pair of fermions with angular momentum coupled to 0 on level $\\varepsilon_{j}$, defined as $C_{j}^{+}=\\frac{1}{\\sqrt{\\Omega_{j}}} \\sum_{m>0}(-1)^{j-m} a_{j m}^{+} a_{j-m}^{+}$.", + "$\\Omega_j$": "Normalization constant for the pair creation/annihilation operators, related to the degeneracy of the level $j$.", + "$\\hat{N}_j$": "Total particle number operator for the energy level $\\varepsilon_{j}$." + } + }, + { + "id": 376, + "context": "", + "question": "Starting from the Dirac equation describing the motion of electrons in an electromagnetic field, derive the electronic current flow density.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\boldsymbol{j}_{e}$": "Electronic current flow density.", + "$e$": "Elementary charge (electron charge).", + "$m$": "Electron mass.", + "$\\varphi^{+}$": "Hermitian conjugate of the large component of the Dirac spinor wavefunction.", + "$\\boldsymbol{p}$": "Momentum operator.", + "$c$": "Speed of light.", + "$\\boldsymbol{A}$": "Magnetic vector potential.", + "$\\varphi$": "Large component of the Dirac spinor wavefunction $\\psi$, a two-component wavefunction.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\nabla$": "Del operator (gradient operator).", + "$\\boldsymbol{\\sigma}$": "Pauli matrices, a 2x2 matrix vector." + } + }, + { + "id": 377, + "context": "", + "question": "Consider the second-order approximation of the stationary Dirac equation under the non-relativistic limit, determining the specific form of the Hamiltonian operator when an electron is moving in a central force field.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$H$": "Hamiltonian operator.", + "$\\boldsymbol{p}^{2}$": "Square of the magnitude of the momentum operator, $(\\boldsymbol{\\sigma} \\cdot \\boldsymbol{p})^{2}$.", + "$m$": "Mass of the electron.", + "$V$": "Central force potential energy, defined as $V(r) = -e\\phi(r)$.", + "$\\boldsymbol{p}^{4}$": "Fourth power of the magnitude of the momentum operator.", + "$c$": "Speed of light.", + "$\\hbar$": "Reduced Planck's constant.", + "$r$": "Radial distance from the central force origin.", + "$\\boldsymbol{\\sigma}$": "Pauli matrices (vector of matrices).", + "$\\boldsymbol{l}$": "Orbital angular momentum operator, defined as $\\boldsymbol{l}=\\boldsymbol{r} \\times \\boldsymbol{p}$.", + "$\\nabla^{2}$": "Laplacian operator." + } + }, + { + "id": 378, + "context": "Consider a two-dimensional square lattice.", + "question": "The maximum energy value is at the corners of the first Brillouin zone.\nTry to find the number of states $N(E) \\mathrm{d} E$ in the unit area crystal within the energy range $E \\sim(E+\\mathrm{d} E)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$N(E)$": "Number of states per unit energy interval in the unit area crystal.", + "$\\mathrm{d} E$": "Infinitesimal energy range.", + "$m_{\\mathrm{p}}$": "Effective mass parameter in the energy dispersion relation near the extremum.", + "$\\hbar$": "Reduced Planck's constant." + } + }, + { + "id": 379, + "context": "", + "question": "The energy $E$ near the valence band top of a certain semiconductor crystal can be expressed as: $E(k)=E_{\\text {max }}-10^{26} k^{2}(\\mathrm{erg})$. Now, removing an electron with wave vector $k=10^{7} \\mathrm{i} / \\mathrm{cm}$, calculate the effective mass of the hole left by this electron.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 380, + "context": "", + "question": "In an anisotropic crystal, its energy $E$ can be expressed in terms of the components of wave vector $\\boldsymbol{k}$ as $E(k)=A k_{x}^{2}+B k_{y}^{2}+C k_{z}^{2}$. Try to derive the equation of motion of electrons where the left-hand-side is $\\frac{dv{dt}$.} You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Semiconductors", + "final_symbol": { + "$v$": "Velocity of the electron.", + "$t$": "Time.", + "$F$": "Force acting on the electron.", + "$h$": "Planck's constant.", + "$E$": "Energy of the electron in the anisotropic crystal.", + "$k$": "Wave vector (used as a general variable for differentiation in the context of $E(k)$)." + } + }, + { + "id": 381, + "context": "", + "question": "For a one-dimensional lattice with a lattice constant of $2.5 \\AA$, when an external electric field of $10^{2} \\mathrm{~V} / \\mathrm{m}$ is applied, calculate the time required for an electron to move from the bottom of the energy band to the top. $(1 \\AA=10 \\mathrm{~nm}=10^{-10} \\mathrm{~m})$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 382, + "context": "", + "question": "A one-dimensional lattice with a lattice constant of $2.5 \\AA$, calculate the time required for an electron to move from the bottom to the top of the energy band when an external electric field of $10^{7} \\mathrm{~V} / \\mathrm{m}$ is applied. $(1 \\AA=10 \\mathrm{~nm}=10^{-10} \\mathrm{~m})$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 383, + "context": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, using the hydrogen-like model estimate:", + "question": "Donor ionization energy;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 384, + "context": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$ respectively. Using a hydrogen-like model estimate:", + "question": "Acceptor ionization energy;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 385, + "context": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, using a hydrogen-like model to estimate:", + "question": "Ground state electron orbital radius $r_{1}$; You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 386, + "context": "The dielectric constant of semiconductor silicon single crystal $\\varepsilon_{\\mathrm{r}}=11.8$, and the effective masses of electrons and holes are $m_{\\mathrm{n} 1}=$ $0.97 m_{0}, m_{\\mathrm{nt}}=0.19 m_{0}$ and $m_{\\mathrm{pl}}=0.16 m_{0}, m_{\\mathrm{ph}}=0.53 m_{0}$, respectively. Using the hydrogen-like model to estimate:", + "question": "What is the acceptor concentration when the electron orbitals of adjacent impurity atoms overlap significantly?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 387, + "context": "", + "question": "For silicon material doped with n-type impurity phosphorus, try to calculate the concentration of phosphorus when weak degeneracy occurs at room temperature.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 388, + "context": "", + "question": "For the germanium material doped with n-type impurity phosphorus, try to calculate the numerical value of its phosphorus doping concentration at room temperature when weak degeneracy occurs.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 389, + "context": "", + "question": "For silicon materials doped with n-type impurity phosphorus, try to calculate the dopant concentration when weak degeneracy occurs at room temperature.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 390, + "context": "", + "question": "For a semiconductor silicon sample with a donor impurity concentration of $10^{12} \\mathrm{~cm}^{-3}$, calculate the equation that the temperature value (in K) satisfies when its intrinsic carrier concentration $n_i$ equals the donor impurity concentration $N_d$. Assume $E_{\\mathrm{g}}=1 \\mathrm{eV}, m_{\\mathrm{n}}^{*}=m_{\\mathrm{p}}^{*}=0.2 m_{0}$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Semiconductors", + "final_symbol": { + "$T$": "Temperature." + } + }, + { + "id": 391, + "context": "", + "question": "Given a silicon sample with a donor concentration $N_{\\mathrm{D}}=2 \\times 10^{14} \\mathrm{~cm}^{-3}$ and an acceptor concentration $N_{\\mathrm{A}}=10^{14} \\mathrm{~cm}^{-3}$, where the donor ionization energy $\\Delta E_{\\mathrm{D}}=E_{\\mathrm{c}}-E_{\\mathrm{D}}=0.05 \\mathrm{eV}$, find the equation that the temperature value (in K) satisfies when $99\\%$ of the donor impurities are ionized. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Semiconductors", + "final_symbol": { + "$T$": "Temperature in Kelvin." + } + }, + { + "id": 392, + "context": "", + "question": "In a boron-doped non-degenerate p-type silicon, containing a certain concentration of indium, the hole concentration at room temperature is measured as $p_{0}=1.1 \\times 10^{16} \\mathrm{~cm}^{-3}$. Given that the boron doping concentration $N_{\\mathrm{A} 1}=10^{16} \\mathrm{~cm}^{-3}$ and its ionization energy $\\Delta E_{\\mathrm{A} 1}=E_{\\mathrm{A} 1}-E_{\\mathrm{v}}=0.046 \\mathrm{eV}$, and indium's ionization energy $\\Delta E_{\\mathrm{A} 2}=E_{\\mathrm{A} 2}-E_{\\mathrm{v}}=0.16 \\mathrm{eV}$, determine the concentration of indium in this semiconductor. At room temperature, silicon's $N_{\\mathrm{v}}=1.04 \\times 10^{19}$ $\\mathrm{cm}^{-3}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 393, + "context": "", + "question": "At room temperature, the resistivity of intrinsic germanium is $47 \\Omega \\cdot \\mathrm{~cm}$. If antimony impurities are added so that there is one impurity atom per $10^{6}$ germanium atoms. Assume all impurities are ionized. The concentration of germanium atoms is $4.4 \\times 10^{22} / \\mathrm{cm}^{3}$. Calculate the resistivity of this doped germanium material. Assume $\\mu_{\\mathrm{n}}=3600 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), ~ \\mu_{\\mathrm{p}}$ $=1700 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and are unchanged by doping. $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 394, + "context": "", + "question": "When $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}, \\mu_{\\mathrm{p}}=1900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{n}}=3800 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, try to find the intrinsic conductivity of germanium.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 395, + "context": "", + "question": "When $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}, \\mu_{\\mathrm{p}}=1900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{n}}=3800 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, calculate the minimum conductivity of germanium.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 396, + "context": "At room temperature, the electron mobility of high-purity germanium $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Given the effective mass of electrons $m_{\\mathrm{n}}=0.3 m \\approx 3 \\times 10^{-28} \\mathrm{~g}$, try to calculate:", + "question": "Mean free time $\\tau$;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 397, + "context": "At room temperature, the electron mobility of high-purity germanium is $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. Given that the effective mass of the electron $m_{\\mathrm{n}}=0.3 m \\approx 3 \\times 10^{-28} \\mathrm{~g}$, try to calculate:", + "question": "Average free path $l$;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 398, + "context": "At room temperature, we have a a p-type silicon wafer, and want to convert a p-type silicon wafer with a resistivity of $0.2 \\Omega \\cdot \\mathrm{~cm}$.", + "question": "What should be the density of impurities to achieve a resistivity of $0.2 \\Omega \\cdot \\mathrm{~cm}$ for n-type silicon?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 399, + "context": "", + "question": "A boron-doped non-degenerate p-type silicon sample contains a certain concentration of indium, with a measured resistivity at room temperature (300K) of $\\rho=2.84 \\Omega \\cdot \\mathrm{~cm}^{2}$. Given that the doped boron concentration is $N_{\\mathrm{a} 1}=10^{16} / \\mathrm{cm}^{3}$, with boron ionization energy $E_{\\mathrm{a} 1}-E_{\\mathrm{v}}=0.045 \\mathrm{eV}$ and indium ionization energy $E_{\\mathrm{a} 2}-E_{\\mathrm{v}}=0.16 \\mathrm{eV}$, determine the concentration of indium $N_{\\mathrm{a} 2}$ in the sample [At room temperature, $N_{\\mathrm{v}}=$ $1.04 \\times 10^{19} / \\mathrm{cm}^{3}, ~ \\mu_{\\mathrm{p}}=200 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})]$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 400, + "context": "", + "question": "Given that the conductivity of intrinsic germanium is $3.56 \\times 10^{-2} \\mathrm{~S} / \\mathrm{cm}$ at 310 K and $0.42 \\times$ $10^{-2} \\mathrm{~S} / \\mathrm{cm}_{\\circ}$ at 273 K, an n-type germanium sample has a donor impurity concentration of $N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$ at these two temperatures. Calculate the conductivity of the doped germanium at the above temperatures. [Assume $\\mu_{\\mathrm{n}}=3600 \\mathrm{~cm} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=1700 \\mathrm{~cm} /(\\mathrm{V} \\cdot \\mathrm{s})$]", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 401, + "context": "", + "question": "A thermistor made of intrinsic silicon material has a resistance value of $500 \\Omega$ at 290 K. Assuming the band gap of silicon $E_{\\mathrm{q}}=1.12 \\mathrm{eV}$ and does not change with temperature, if we assume the carrier mobility remains unchanged, try to estimate the approximate value of the thermistor at 325 K.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 402, + "context": "", + "question": "The resistivity of intrinsic germanium material with temperature $T$ can be tabulated as follows:\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$T(\\mathrm{~K})$ & 385 & 458 & 556 & 714 \\\\\n\\hline$\\rho(\\Omega \\cdot \\mathrm{~cm})$ & 0.028 & 0.0061 & 0.0013 & 0.00027 \\\\\n\\hline\n\\end{tabular}\n\nAssume $E_{\\mathrm{g}}$ is independent of temperature $T$, and the mobilities of electrons and holes $\\mu_{\\mathrm{n}} \\mu_{\\mathrm{p}}$ both vary as $T^{-\\frac{3}{2}}$. Find the band gap $E_{\\mathrm{g}}$ of germanium.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 403, + "context": "", + "question": "Calculate the resistivity of intrinsic silicon at room temperature (unit $\\Omega \\cdot \\mathrm{~cm}$). It is known that the electron mobility of intrinsic silicon at room temperature is $1350 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, the hole mobility is $500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the intrinsic carrier concentration is $n_{\\mathrm{i}}=1.5 \\times 10^{10} / \\mathrm{cm}^{3}$, elementary charge $q=1.6 \\times 10^{-19} \\mathrm{C}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 404, + "context": "", + "question": "For a certain n-type semiconductor silicon with a doping concentration $N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and a minority carrier lifetime $\\tau_{\\mathrm{p}}=5 \\mu \\mathrm{~s}$, if due to external influences all minority carriers are removed (such as near a reverse-biased pn junction), what is the electron-hole generation rate at this time (let $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$)?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 405, + "context": "The concentration of copper in a copper-doped germanium sample is $10^{15} \\mathrm{~cm}^{-3}$, and the concentration of antimony is $10^{17} \\mathrm{~cm}^{-3}$. Its minority carrier lifetime measured under small injection conditions is $10^{-7} \\mathrm{~s}^{-1}$. Given that $N_{\\mathrm{c}}=1.04 \\times 10^{19} \\mathrm{~cm}^{-1}$.", + "question": "If the effective mass of holes in germanium $m_{\\mathrm{p}}^{*}=0.30 m_{0}(m_{0}$ is the free electron mass), find the hole capture cross-section?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 406, + "context": "", + "question": "In a piece of p-type semiconductor, there exists a recombination-generation center. When slightly doped, the electrons captured by these centers are reemitted to the conduction band with the same probability as their recombination with holes. Try to find the energy level position of this recombination-generation center. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Semiconductors", + "final_symbol": { + "$E_{\\mathrm{t}}$": "Energy level position of the recombination-generation center (trap level)", + "$E_{\\mathrm{i}}$": "Intrinsic Fermi energy level, $E_{\\mathrm{i}}=\\frac{1}{2}(E_{\\mathrm{c}}+E_{\\mathrm{v}}-k_{0} T \\ln \\frac{N_{\\mathrm{c}}}{N_{\\mathrm{v}}})$", + "$E_{\\mathrm{F}}$": "Fermi energy level" + } + }, + { + "id": 407, + "context": "Illuminating an n-type silicon sample with a resistivity of $1 \\Omega \\cdot \\mathrm{~cm}$, non-equilibrium carriers are uniformly generated, with the generation rate of electron-hole pairs being $10^{17} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. Assume the lifetime of the sample is $10 \\mu \\mathrm{~s}$, and the surface recombination velocity is $100 \\mathrm{~cm} / \\mathrm{s}$. Calculate:", + "question": "The number of holes recombined at the surface per unit time per unit surface area;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 408, + "context": "Illuminate a $1 \\Omega \\cdot \\mathrm{~cm}$ n-type silicon sample, uniformly generating non-equilibrium carriers, with an electron-hole pair generation rate of $10^{17} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. Assume the sample's lifetime is $10 \\mu \\mathrm{~s}$, and the surface recombination velocity is $100 \\mathrm{~cm} / \\mathrm{s}$. Calculate:", + "question": "The number of holes recombined within three diffusion lengths from the surface, per unit time and per unit surface area.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 409, + "context": "A silicon wafer with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$.", + "question": "Calculate the bulk lifetime;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 410, + "context": "A silicon wafer with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$.", + "question": "Calculate the surface recombination velocity;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 411, + "context": "A silicon wafer with a dopant concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is gold-doped to saturation concentration at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of the silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$.", + "question": "If the silicon wafer is uniformly illuminated, and the generation rate of electron-hole pairs is $10^{11} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$, what is the hole concentration at the surface?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 412, + "context": "A silicon wafer doped with a donor concentration of $2 \\times 10^{16} \\mathrm{~cm}^{-3}$ is saturated with gold at $920^{\\circ} \\mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \\mathrm{~cm}^{-2}$.", + "question": "If the silicon wafer is illuminated and uniformly absorbed by the sample, the generation rate of electron-hole pairs is $10^{11} \\mathrm{~cm}^{-3} \\cdot \\mathrm{~s}^{-1}$. What is the hole current density flowing towards the surface?", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 413, + "context": "", + "question": "A pn junction composed of p-type germanium with a resistivity of $1 \\Omega \\cdot \\mathrm{~cm}$ and n-type germanium with a resistivity of $0.1 \\Omega \\cdot \\mathrm{~cm}$, calculate the built-in potential difference $V_{\\mathrm{D}}$ at room temperature (300 K). Given that at the above resistivities, the hole mobility in the p-type region $\\mu_{\\mathrm{p}}=1650 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, electron mobility in the n-type region $\\mu_{\\mathrm{n}}=3000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the intrinsic carrier concentration of germanium $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 414, + "context": "", + "question": "A pn junction composed of p-type germanium with resistivity $1 \\Omega \\cdot \\mathrm{~cm}$ and n-type germanium with resistivity $0.1 \\Omega \\cdot \\mathrm{~cm}$, calculate the width of the depletion region at room temperature (300 K). Given that at these resistivities, the hole mobility in the p region is $\\mu_{\\mathrm{p}}=1650 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the electron mobility in the n region is $\\mu_{\\mathrm{n}}=3000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, the intrinsic carrier concentration of germanium $n_{\\mathrm{i}}=2.5 \\times 10^{13} \\mathrm{~cm}^{-3} $.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 415, + "context": "", + "question": "Given a silicon abrupt junction, with resistivities on both sides being $\\rho_{\\mathrm{n}}=10 \\Omega \\cdot \\mathrm{~cm}$ for $\\mathrm{n}-\\mathrm{Si}$ and $\\rho_{\\mathrm{p}}=0.01 \\Omega \\cdot \\mathrm{~cm}$ for $\\mathrm{p}-\\mathrm{Si}$, and mobilities $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2}/(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=300 \\mathrm{~cm}^{2}/(\\mathrm{V} \\cdot \\mathrm{s})$, find the barrier width at room temperature.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 416, + "context": "", + "question": "Assume a silicon abrupt junction with the impurity concentrations on both sides as $N_{\\mathrm{A}}=10^{17} / \\mathrm{cm}^{3}, N_{\\mathrm{D}}=4.5 \\times 10^{15} / \\mathrm{cm}^{3}$. The intrinsic carrier concentration of silicon at room temperature is known as $n_{\\mathrm{i}} = 1.5 \\times 10^{10} / \\mathrm{cm}^{3}$, vacuum permittivity $\\varepsilon_{0} = 8.85 \\times 10^{-14} \\mathrm{F/cm}$, the relative permittivity of silicon $\\varepsilon_{\\mathrm{rs}} = 11.9$, elementary charge $q = 1.6 \\times 10^{-19} \\mathrm{C}$, thermal voltage $kT/q \\approx 0.026 \\mathrm{V}$. Determine the value of total depletion width $x_{\\mathrm{D}}$ at zero applied voltage.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 417, + "context": "", + "question": "Suppose a silicon abrupt junction, with impurity concentrations on either side as $N_{\\mathrm{A}}=10^{17} / \\mathrm{cm}^{3}, N_{\\mathrm{D}}=4.5 \\times 10^{15} / \\mathrm{cm}^{3}$, when a reverse bias voltage of 10 V is applied, find the value of $x_{\\mathrm{D}}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 418, + "context": "Given that the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of the linear graded junction of silicon is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$,", + "question": "Find the value of the barrier width $x_{\\mathrm{D}}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 419, + "context": "Assume the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of silicon's linearly graded junction is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$,", + "question": "Find its maximum electric field,", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 420, + "context": "Assume the impurity concentration gradient $\\alpha_{\\mathrm{j}}$ of a silicon linear gradient junction is $10^{22} / \\mathrm{cm}^{4}, V_{\\mathrm{D}}=0.68 \\mathrm{~V}$,", + "question": "Find the value of $x_{\\mathrm{D}}$ when an external reverse bias of 10 V is applied.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 421, + "context": "", + "question": "For a silicon $n^+p$ junction, given $x_p = 0.2\\ \\mu\\text{m}$, $L_n = 200\\ \\mu\\text{m}$, $N_A = 10^{15}\\ \\text{cm}^{-3}$, $n_i = 1.5 \\times 10^{10}\\ \\text{cm}^{-3}$, at room temperature $T = 300\\ \\text{K}$, find the value of voltage V corresponding to the condition where the barrier recombination current equals the diffusion current.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 422, + "context": "", + "question": "The relationship between the barrier capacitance $C_{\\mathrm{T}}$ of the $\\mathrm{p}^{+} \\mathrm{n}$ junction made from GaP material and the reverse voltage $V_{\\mathrm{R}}$ is measured as follows\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline$V_{\\mathrm{R}}(\\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\\n\\hline$C_{\\mathrm{T}}(\\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\\\\n\\hline\n\\end{tabular}\n\nThe $pn$ junction area $A=4 \\times 10^{-4} \\mathrm{~cm}^{2}$, try to find the built-in potential $V_{\\mathrm{D}}$ of the $\\mathrm{p}^{+} \\mathrm{n}$ junction.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 423, + "context": "", + "question": "The relationship between the barrier capacitance $C_{\\mathrm{T}}$ and reverse voltage $V_{\\mathrm{R}}$ of a $\\mathrm{p}^{+} \\mathrm{n}$ junction made of GaP material is measured as follows\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline$V_{\\mathrm{R}}(\\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\\n\\hline$C_{\\mathrm{T}}(\\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\\\\n\\hline\n\\end{tabular}\n\nThe pn junction area $A=4 \\times 10^{-4} \\mathrm{~cm}^{2}$, try to find the built-in field $N_{\\mathrm{D}}$ of this $\\mathrm{p}^{+} \\mathrm{n}$ junction.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 424, + "context": "", + "question": "A pn junction diode has the following parameters: $N_{\\mathrm{D}}=10^{16} \\mathrm{~cm}^{-3}, ~ N_{\\mathrm{A}}=5 \\times 10^{18} \\mathrm{~cm}^{-3}, ~ \\tau_{\\mathrm{n}}=\\tau_{p}=$ $1 \\mu \\mathrm{~s}, ~ A=0.01 \\mathrm{~cm}^{2}$. Assume that the widths on both sides of the junction are much larger than the diffusion lengths of minority carriers. Find the applied voltage at room temperature (300 K) when the forward current is 1 mA. Assume the electron mobility in the p-type region $\\mu_{\\mathrm{n}}=500 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility in the n-type region $\\mu_{\\mathrm{p}}=180 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 425, + "context": "An n-type single crystal silicon wafer with [100] crystal orientation forms a Schottky diode with a certain metal contact. The parameters are $W_{\\mathrm{m}}=4.7 \\mathrm{eV}, \\chi_{\\mathrm{s}}=4.0 \\mathrm{eV}, N_{\\mathrm{c}}=10^{19} \\mathrm{~cm}^{-3}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and the relative permittivity of semiconductor silicon is $\\varepsilon_{\\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature:", + "question": "Zero-bias depletion width;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 426, + "context": "An n-type monocrystalline silicon wafer with [100] crystal orientation forms a Schottky diode upon contact with a certain metal. Its parameters are $W_{\\mathrm{m}}=4.7 \\mathrm{eV}, \\chi_{\\mathrm{s}}=4.0 \\mathrm{eV}, N_{\\mathrm{c}}=10^{19} \\mathrm{~cm}^{-3}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}$, and the relative dielectric constant of semiconductor silicon is $\\varepsilon_{\\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature:", + "question": "The thermionic emission current when forward biased at 0.2 V. Assume $\\frac{A^{*}}{A}=2.1, A=120 \\mathrm{~A} / \\mathrm{cm}^{2}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 427, + "context": "Consider a metal forming a Schottky diode with (111) crystal plane $\\mathrm{n}-\\mathrm{Si}$. It is known that the barrier height on the semiconductor side after contact is $0.50 \\mathrm{eV}, N_{\\mathrm{D}}=10^{15} \\mathrm{~cm}^{-3}, N_{\\mathrm{c}}=2.8 \\times 10^{19} \\mathrm{~cm}^{-3}$, electron affinity $X=4.05 \\mathrm{eV}, I_{\\mathrm{p}}=10 \\mu \\mathrm{~m}$, $D_{\\mathrm{p}}=15 \\mathrm{~cm}^{2} / \\mathrm{s}, n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}, A^{*}=252 \\mathrm{~A} / \\mathrm{cm}^{2} \\mathrm{~K}^{2}$ (Richardson constant), find:", + "question": "Calculate the minority carrier injection ratio.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 428, + "context": "", + "question": "A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. The barrier height on the semiconductor side is known as $qV_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$. Calculate the maximum electric field in the semiconductor at the interface under a reverse bias of 5 V.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 429, + "context": "", + "question": "A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\\mathrm{D}}=0.6 \\mathrm{eV}, N_{\\mathrm{D}}=5 \\times 10^{16} \\mathrm{~cm}^{-3}$, determine the barrier capacitance per unit area under a reverse bias voltage of 5 V.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 430, + "context": "A metal plate and n-type silicon are separated by $0.4 \\mu \\mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air between them is $\\varepsilon_{\\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state.", + "question": "Ignoring the work function difference between the metal and the semiconductor, what is the voltage $V_{\\mathrm{G}}$ on the metal plate when the depletion layer width just reaches its maximum? ($N_{\\mathrm{D}}=10^{16} \\mathrm{~cm}^{-3}$)", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 431, + "context": "The MOS structure capacitor formed by metal-$\\mathrm{SiO}_{2}-\\mathrm{Si}$ (p-type), with hole concentration $N_{\\mathrm{A}}=1.5 \\times 10^{15}$ $\\mathrm{cm}^{-3}, ~ \\mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \\mu \\mathrm{~m}$, its relative permittivity $\\varepsilon_{\\mathrm{r}_{0}}=3.9$, the relative permittivity of silicon $\\varepsilon_{\\mathrm{rs}}=12, \\varepsilon_{0}=$ $8.85 \\times 10^{-14} \\mathrm{~F} / \\mathrm{cm}$, at room temperature $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$.", + "question": "If there is a fixed positive charge at the SiO$_2$-silicon interface, and the measured $V_{\\mathrm{T}}=2.6 \\mathrm{~V}$, find the amount of fixed positive charge per unit area (neglecting the influence of the work function difference);", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": { + "$N_{\\mathrm{fc}}$": "Number of fixed positive charges per unit area at the Si-SiO$_2$ interface." + } + }, + { + "id": 432, + "context": "The MOS structure capacitor composed of metal-$\\mathrm{SiO}_{2}-\\mathrm{Si}$ (p-type), with hole concentration $N_{\\mathrm{A}}=1.5 \\times 10^{15}$ $\\mathrm{cm}^{-3}, ~ \\mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \\mu \\mathrm{~m}$, relative permittivity of $\\varepsilon_{\\mathrm{r}_{0}}=3.9$, silicon's relative permittivity $\\varepsilon_{\\mathrm{rs}}=12, \\varepsilon_{0}=$ $8.85 \\times 10^{-14} \\mathrm{~F} / \\mathrm{cm}$, intrinsic carrier concentration at room temperature $n_{\\mathrm{i}}=1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$.", + "question": "If the above positive charges are uniformly distributed in $\\mathrm{SiO}_{2}$, what is the measured $V_{\\mathrm{T}}$? (neglecting the effect of work function difference);", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 433, + "context": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, and the overlap area of the gate with the n region is $10^{-3} \\mathrm{cm}^{2}$. The substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide layer thickness is $0.2 \\mu \\mathrm{~m}$, the lifetime $\\tau=1 \\mu \\mathrm{~s}$, the surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, the flat-band voltage is -2 V.", + "question": "Calculate the gate voltage when the substrate surface is intrinsic (when the junction voltage is zero at room temperature).", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 434, + "context": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, the overlap area between the gate and the n region is $10^{-3} \\mathrm{~cm}^{2}$, the substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide thickness is $0.2 \\mu \\mathrm{~m}$, the lifetime $\\tau=1 \\mu \\mathrm{~s}$, the surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, the flat-band voltage is -2 V.", + "question": "For the gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode described in the problem (metallurgical junction area $10^{-3} \\mathrm{~cm}^{2}$, overlap area between the gate and n region $10^{-3} \\mathrm{~cm}^{2}$, substrate impurity concentration $10^{16} \\mathrm{~cm}^{-3}$, oxide thickness $0.2 \\mu \\mathrm{~m}$, minority carrier lifetime $\\tau=1 \\mu \\mathrm{~s}$, flat-band voltage $V_{FB} = -2 \\mathrm{~V}$), when the diode is subjected to a reverse bias voltage of $V_{\\mathrm{R}}=1 \\mathrm{~V}$ at room temperature, calculate the change of the forward current $\\Delta I_p = I_p(V_G=-20V) - I_p(V_G=0V)$ induced by varying the gate voltage $V_G$ from $0 \\mathrm{~V}$ to $-20 \\mathrm{~V}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 435, + "context": "The metallurgical junction area of a gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode is $10^{-3} \\mathrm{~cm}^{2}$, and the overlap area between the gate and n-region is $10^{-3} \\mathrm{~cm}^{2}$. The substrate impurity concentration is $10^{16} \\mathrm{~cm}^{-3}$, the junction depth is $5 \\mu \\mathrm{~m}$, the oxide layer thickness is $0.2 \\mu \\mathrm{~m}$, lifetime $\\tau=1 \\mu \\mathrm{~s}$, surface recombination velocity $s_{0}=5 \\mathrm{~cm} / \\mathrm{s}$, and the flat-band voltage is -2 V. Calculate:", + "question": "For the gate-controlled $\\mathrm{p}^{+} \\mathrm{n}$ diode (with relevant parameters: gate to n-region overlap area $A_s = 10^{-3} \\mathrm{~cm}^{2}$, used to calculate the diffusion current; n-region substrate impurity concentration $N_D = 10^{16} \\mathrm{~cm}^{-3}$; minority carrier hole lifetime $\\tau_p = 1 \\mu \\mathrm{s}$; hole diffusion coefficient $D_p = 13 \\mathrm{~cm}^{2} / \\mathrm{s}$; intrinsic carrier concentration $n_i = 1.5 \\times 10^{10} \\mathrm{~cm}^{-3}$; thermal voltage $k_0T/q = 0.026 \\mathrm{~V}$), when the diode is under forward bias voltage $V_F = 0.4 \\mathrm{~V}$ at room temperature, calculate the diffusion current component $I_D$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 436, + "context": "", + "question": "Estimate the injection ratio between GaAs and $\\mathrm{Al}_{0.3} \\mathrm{Ga}_{0.7} \\mathrm{As}$ at 300 K.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 437, + "context": "", + "question": "Try to derive the relationship between the absorption coefficient $\\alpha$ and the extinction coefficient $\\bar{k}$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Semiconductors", + "final_symbol": { + "$\\alpha$": "Absorption coefficient", + "$\\omega$": "Angular frequency", + "$\\bar{k}$": "Extinction coefficient", + "$c$": "Speed of light in vacuum", + "$\\pi$": "Mathematical constant pi", + "$\\lambda$": "Wavelength in vacuum" + } + }, + { + "id": 438, + "context": "There is an n-type CdS cubic chip, with an edge length of 1 mm and a thickness of 0.1 mm, with a wavelength absorption limit of $5100 \\AA$. Now, a violet light of intensity $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$ $(\\lambda=4096 \\AA)$ is used to illuminate the square surface, with a quantum yield of $\\beta=1$. Assuming all photo-generated holes are trapped and the lifetime of photo-generated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; and assuming the illuminative energy is completely absorbed by the chip.", + "question": "Calculate the number of electron-hole pairs generated per second in the sample;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 439, + "context": "There is an n-type CdS cubic wafer with a side length of 1 mm and a thickness of 0.1 mm, with an absorption edge wavelength of $5100 \\AA$. Now, the square surface is irradiated with purple light $(\\lambda=4096 \\AA)$ at an intensity of $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$, with a quantum yield of $\\beta=1$. Assume all photogenerated holes are trapped, the lifetime of photogenerated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, and the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; further assume all illumination energy is absorbed by the wafer.", + "question": "calculate the increase in the number of electrons in the sample;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 440, + "context": "There is an n-type CdS square crystal with a side length of 1 mm and a thickness of 0.1 mm. Its wavelength absorption limit is $5100 \\AA$. Now, violet light with an intensity of $1 \\mathrm{~mW} / \\mathrm{cm}^{2}$ $(\\lambda=4096 \\AA)$ illuminates the square surface, and the quantum yield is $\\beta=1$. Assume all photogenerated holes are trapped, the lifetime of the photogenerated electrons is $\\tau_{\\mathrm{n}}=10^{-5} \\mathrm{~s}$, and the electron mobility is $\\mu_{\\mathrm{n}}=100 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$; also assume that the illumination energy is completely absorbed by the crystal.", + "question": "When the photocurrent when a 50 V voltage is applied to the sample, calculate photoconductive gain factor.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 441, + "context": "", + "question": "Given a piece of n-type semiconductor material with a room temperature dark conductivity of $100 \\mathrm{~S} / \\mathrm{cm}$, when illuminated with light at an intensity of $I=$ $10^{-6} \\mathrm{~W} / \\mathrm{cm}^{2}$, its absorption coefficient $\\alpha=10^{2} / \\mathrm{cm}$, the measured ratio of steady-state photoconductivity to dark conductivity $\\gamma$ $=10$, and the lifetime $\\tau$ is $10^{-4} \\mathrm{~s}, b=\\mu_{\\mathrm{n}} / \\mu_{\\mathrm{p}}=10, \\mu_{\\mathrm{n}}=10000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, determine the corresponding quantum yield.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 442, + "context": "", + "question": "In a p-type silicon with a hole concentration of $10^{16} \\mathrm{~cm}^{-3}$, a cold end temperature of $0^{\\circ} \\mathrm{C}$, and a hot end temperature of $50^{\\circ} \\mathrm{C}$, assuming long wavelength acoustic wave scattering, calculate the thermoelectric power.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 443, + "context": "", + "question": "For n-type PoTe with a conductivity of $2000 \\mathrm{~S} / \\mathrm{cm}$, an electron mobility of $6000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, determine the thermoelectric power factor at room temperature assuming long-wavelength acoustic phonon scattering.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 444, + "context": "", + "question": "For an n-type PoTe with a conductivity of $2000 \\mathrm{~S} / \\mathrm{cm}$, electron mobility of $6000 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, assuming long-wavelength acoustic scattering, determine the Peltier coefficient at room temperature.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 445, + "context": "", + "question": "The thermal conductivity of bismuth telluride ($\\mathrm{Bi}_{2} \\mathrm{Te}_{3}$) is $2.4[\\mathrm{~W} /(\\mathrm{m} \\cdot \\mathrm{K})$]. Calculate the percentage contribution of carrier to the thermal conductivity for n-type $\\mathrm{Bi}_{2} \\mathrm{Te}_{3}$ at $10^{5} \\mathrm{~s} / \\mathrm{m}$ and $300 \\mathrm{~K}$. (Assume long-wavelength acoustic phonon scattering)", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 446, + "context": "", + "question": "Try to find the Seebeck coefficient of intrinsic silicon at room temperature.\nAssume the effective masses of electrons and holes are equal, the band gap of silicon is 1.12 eV, and the mobilities of electrons and holes are $0.135 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and $0.048 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, respectively.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 447, + "context": "", + "question": "For an indium antimonide sample, the hole concentration at room temperature is 9 times the electron concentration. Calculate the Hall coefficient $R$. Assume at room temperature $b=\\mu_{\\mathrm{n}} / \\mu_{\\mathrm{p}}=100, n_{\\mathrm{i}}=1.1 \\times 10^{16} \\mathrm{~cm}^{-3}$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 448, + "context": "InSb electron mobility is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, with an intrinsic carrier concentration of $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$, at 300 K", + "question": "Hall coefficient of intrinsic material;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 449, + "context": "The electron mobility of InSb is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. The intrinsic carrier concentration is $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$, find at 300 K", + "question": "intrinsic resistivity;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 450, + "context": "The electron mobility of InSb is $7.8 \\mathrm{~m}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the hole mobility is $780 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$. The intrinsic carrier concentration is $1.6 \\times 10^{16} \\mathrm{~cm}^{-3}$ at 300 K.", + "question": "When $B_{z}=0.1 \\mathrm{~Wb} / \\mathrm{m}^{2}$, calculate the resistivity of the material considering the scattering of long acoustic waves.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 451, + "context": "For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is passed along the $x$ direction, and a magnetic field of 0.5 T is applied along the $z$ direction, resulting in a Hall voltage of -0.4 mV.", + "question": "Hall coefficient;", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 452, + "context": "", + "question": "A silicon sample with a conductivity of $0.001 /(\\Omega \\cdot \\mathrm{cm})$ has zero Hall voltage under a weak magnetic field. Assuming the electron mobility $\\mu_{\\mathrm{n}}=1300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$ and the hole mobility $\\mu_{\\mathrm{p}}=300 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$, and the Hall factors for electrons and holes are the same. Try to determine the carrier density of electrons $n$ in the sample.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": { + "$n$": "Carrier density of electrons in the sample" + } + }, + { + "id": 453, + "context": "", + "question": "Given the band gap of InSb $E_{\\mathrm{g}}=0.15 \\mathrm{eV}$, the effective mass of electrons $m_{\\mathrm{e}}=0.014 m_{0}$, and the effective mass of holes $m_{\\mathrm{h}}=0.18 m_{0}$ (with $m_{0}$ as the inertial mass of the electron). If only electrons are the effective carriers, calculate the Hall coefficient of intrinsic InSb at 300 K.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 454, + "context": "Try to demonstrate in the Hall effect under conditions of two types of carriers and a weak magnetic field, the Hall angle $\\theta$ and Hall coefficient $R$ can be expressed as\n\n\\begin{align}\n& \\theta=\\arctan \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}}} B_{z} \\\\\n& R=\\frac{1}{q} \\frac{p \\mu_{\\mathrm{p}}^{2}-n \\mu_{\\mathrm{n}}^{2}}{(p \\mu_{\\mathrm{p}}+n \\mu_{\\mathrm{n}})^{2}}\n\\end{align}", + "question": "If a given germanium sample is placed in a magnetic field of $B=0.1$ T, what is $\\tan \\theta$ when its conductivity is at a minimum? Let $\\mu_{\\mathrm{n}}=3900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s}), \\mu_{\\mathrm{p}}=1$ $900 \\mathrm{~cm}^{2} /(\\mathrm{V} \\cdot \\mathrm{s})$.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 455, + "context": "Briefly answer the following questions:", + "question": "Classified by symmetry type, how many types of point groups are there for Bravais lattices? How many types of space groups are there? How many types of point groups are there for crystal structures? How many types of space groups are there? You should return your answer as a tuple format.", + "answer": "", + "final_answer": [], + "answer_type": "Tuple", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 456, + "context": "A one-dimensional atomic chain consisting of N identical atoms with mass m and spacing a. Each atom has only one valence electron. Using the tight-binding approximation, only nearest-neighbor interactions are considered,", + "question": "Derive the expression for the density of states of the s-band electrons.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$G(E)$": "Density of states of the s-band electrons as a function of energy $E$.", + "$N$": "Number of identical atoms in the chain.", + "$\\pi$": "Mathematical constant pi.", + "$J_1$": "Hopping integral, representing the strength of nearest-neighbor interactions.", + "$E_0$": "On-site energy, representing the energy of an electron in an isolated atom.", + "$E$": "Energy of an electron." + } + }, + { + "id": 457, + "context": "", + "question": "The valence band of a semiconductor material is almost filled with electrons (nearly full band), and the expression for the energy of valence band electrons is $E(k)=-1.016 \\times 10^{-34} k^{2}(J)$, where the energy zero point is taken at the top of the valence band. At this time, if the electron at $k=1 \\times 10^{6} \\mathrm{~cm}^{-1}$ is excited to a higher energy band (conduction band), a hole is generated at this location. Try to find energy of this hole.", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Semiconductors", + "final_symbol": {} + }, + { + "id": 458, + "context": "", + "question": "Calculate the mean free path of electrons at room temperature (T=295K). (The density of silver is $10.5 \\mathrm{~g} / \\mathrm{cm}^{3}$, atomic weight is 107.87, and its resistivity at T=295K is $1.61 \\times 10^{-6} \\Omega \\cdot \\mathrm{~cm}$)", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Others", + "final_symbol": {} + }, + { + "id": 459, + "context": "", + "question": "Phase transition in BCS superconducting systems\n Consider the Hamiltonian with parameter $\\lambda$, $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, and the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$. When $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n \n\n Please:\n Use Feynman's theorem to derive the difference in Gibbs free energy between the normal state and the superconducting phase\n\n Hint: If the answer is an integral with respect to $\\lambda$, you can just output the integrand.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta (\\lambda)$": "Superconducting gap parameter, explicitly dependent on $\\lambda$, defined as $\\Delta (\\lambda) = \\lambda \\sum_k < C_{-k} C_k >_T$.", + "$\\lambda$": "Parameter in the Hamiltonian, interpolating between the normal and superconducting states." + } + }, + { + "id": 460, + "context": "", + "question": "Phase transition in BCS superconducting systems\n Consider a Hamiltonian with a parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, with the corresponding Gibbs free energy $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states on the Fermi surface is $g(0)$.\n\n Please:\n Derive the approximate expression for the difference in free energy near the superconducting critical temperature", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta G$": "Difference in Gibbs free energy between superconducting and normal states.", + "$G_S$": "Gibbs free energy in the superconducting state.", + "$G_N$": "Gibbs free energy in the normal state.", + "$g(0)$": "Density of states on the Fermi surface.", + "$\\pi$": "Mathematical constant pi.", + "$k_B$": "Boltzmann constant.", + "$\\zeta(3)$": "Riemann zeta function evaluated at 3.", + "$T_c$": "Superconducting critical temperature.", + "$T$": "Temperature of the system." + } + }, + { + "id": 461, + "context": "", + "question": "Phase transition in BCS superconducting systems\n Consider a Hamiltonian with parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, and the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$. When $T>0K$, the Feynman theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For BCS superconducting systems, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the change in entropy $\\Delta S = S_\\mathrm{S}-S_\\mathrm{N}$, and analyze the physical significance of the results", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta S$": "Change in entropy, defined as $S_\\mathrm{S}-S_\\mathrm{N}$.", + "$g(0)$": "Density of states at the Fermi surface.", + "$\\pi$": "Mathematical constant pi.", + "$k_B$": "Boltzmann constant.", + "$\\zeta(3)$": "Riemann zeta function evaluated at 3.", + "$T_c$": "Critical temperature for the superconducting transition.", + "$T$": "Temperature of the system.", + "$S_{\\mathrm{S}}$": "Entropy of the superconducting phase.", + "$S_{\\mathrm{N}}$": "Entropy of the normal phase." + } + }, + { + "id": 462, + "context": "", + "question": "Phase transition in BCS superconducting system\n Consider a Hamiltonian with parameter $\\lambda$ given by $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$, the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, the Feynman theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For a BCS superconducting system, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the discontinuity in the electronic specific heat.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta c$": "Discontinuity in the electronic specific heat.", + "$T$": "Temperature of the system.", + "$\\Delta S$": "Change in entropy.", + "$T_c$": "Critical temperature.", + "$g(0)$": "Density of states at the Fermi surface.", + "$k_B$": "Boltzmann constant." + } + }, + { + "id": 463, + "context": "", + "question": "Phase transition in BCS superconducting system\n Consider a Hamiltonian $\\bar{H} = \\bar{H}_0 + \\bar{H}_{\\mathrm{int}}(\\lambda), \\bar{H}_{\\mathrm{int}}(\\lambda) = \\lambda \\bar{H}_{\\mathrm{int}}$ with parameter $\\lambda$, the corresponding Gibbs free energy is $\\Gamma(\\lambda) = -k_B T \\mathrm{Tr}\\exp[-\\beta \\bar{H}_{\\mathrm{int}}(\\lambda)]$, when $T>0K$, Feynman's theorem gives:\n \\begin{equation} \\label{eq:6.5.44}\n \\frac{\\partial G (\\lambda)}{\\partial \\lambda} = \\frac{< \\bar{H}_{\\text{int}} (\\lambda) >_T}{\\lambda}\n \\end{equation}\n For BCS superconducting systems, the Hamiltonian satisfies:\n \\begin{equation}\n \\bar{H}_{\\text{int}} = -\\Delta \\sum_k (C_k^\\dagger C_{-k}^\\dagger + C_{-k} C_k) + (\\Delta^2/V), \\Delta = V \\sum_k < C_{-k} C_k >_T\n \\end{equation}\n where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.\n\n Please:\n Calculate the critical magnetic field $H_c(T)$ with Euler constant", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$H_c (T)$": "Critical magnetic field at temperature $T$", + "$H_c (0)$": "Critical magnetic field at absolute zero temperature", + "$\\gamma$": "Euler-Mascheroni constant", + "$\\zeta (3)$": "Riemann zeta function evaluated at 3", + "$T$": "Temperature", + "$T_c$": "Critical temperature" + } + }, + { + "id": 464, + "context": "", + "question": "London Theory\n\n Superconductors have two properties:\n \\begin{itemize}\n \\item[(i)] The DC resistance disappears when $T < T_c$, and a resistance-free supercurrent exists, which is the ideal conductivity of the superconductor.\n \\item[(ii)] The Meissner effect, a weak magnetic field cannot penetrate the interior of a bulk superconducting sample, exhibiting complete diamagnetism.\n\\end{itemize}\n Now consider the following current\n \\begin{equation}\\label{eq:6.7.1}\n \\mathbf{j}_s = - \\frac{c}{\\Lambda}\\mathbf{A} \\quad \\Lambda = \\frac{m}{n_se^2},\n \\end{equation}\n where at $T=0K$ $n_s=n$ is the density of conduction electrons, choosing the transverse field condition $\\nabla \\cdot \\mathbf{A}=0$, please explain using Maxwell's equations how the above expression encompasses the two main properties of the superconductor.\n\n1. Ideal Conductivity You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\Lambda$": "London penetration depth parameter, defined as $\\Lambda = \\frac{m}{n_se^2}$", + "$\\mathbf{j}_s$": "Supercurrent density, the current density of superconducting electrons", + "$t$": "Time", + "$\\boldsymbol{E}$": "Electric field" + } + }, + { + "id": 465, + "context": "", + "question": "London theory\n\n\n Superconductors have two properties:\n \\begin{itemize}\n \\item[(i)] When $T < T_c$, the DC resistance disappears, and there exists a resistance-free supercurrent, which is the ideal conductivity of a superconductor.\n \\item[(ii)] Meissner effect, a weak magnetic field cannot penetrate inside a bulk superconducting sample, exhibiting perfect diamagnetism.\n\\end{itemize}\n Now consider the following current\n \\begin{equation}\\label{eq:6.7.1}\n \\mathbf{j}_s = - \\frac{c}{\\Lambda}\\mathbf{A} \\quad \\Lambda = \\frac{m}{n_se^2},\n \\end{equation}\n where when $T=0K$, $n_s=n$ is the density of conduction electrons. Choose the transversal gauge condition $\\nabla \\cdot \\mathbf{A}=0$, and using Maxwell's equations, explain how the above expression encompasses the two main properties of a superconductor.\n\n Consider a semi-infinite superconducting sample with $z > 0$, and explain the perfect diamagnetism when the external magnetic field is parallel to the $z=0$ plane.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\boldsymbol{B}(z)$": "Magnetic field as a function of depth $z$", + "$\\boldsymbol{B}(0)$": "Magnetic field at the surface of the superconductor ($z=0$)", + "$z$": "Spatial coordinate representing depth into the superconducting sample", + "$\\lambda_L$": "London penetration depth, defined by $\\lambda_L^2 = \\frac{mc^2}{4\\pi n_s e^2}$" + } + }, + { + "id": 466, + "context": "", + "question": "Pippard Theory\n\n In superconductors, within the coherence length $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$, there exists a correlation of electron motion, hence a perturbing potential acting at one point will inevitably affect the velocity of superconducting electrons and current density within the spatial scale of $\\xi_0$. Conversely, the supercurrent density at a certain point in space must also be influenced by the perturbing potentials within its neighboring scale of $\\xi_0$. Thus, $j_s (\\boldsymbol{r})$ depends not only on $\\boldsymbol{A}(\\boldsymbol{r})$ at the same point but should also include contributions from the vector potential $\\boldsymbol{A}(\\boldsymbol{r}')$ at all points $\\boldsymbol{r}'$ within the range $|\\boldsymbol{r} - \\boldsymbol{r}'| < \\xi_0$. For this reason, one can consider the nonlocal relationship between $j_s$ and $\\boldsymbol{A}$ as (Pippard Theory):\n \\begin{equation*}\n j_s (\\boldsymbol{r}) = - \\frac{3}{4\\pi \\xi_0 \\lambda_c} \\int \\frac{\\boldsymbol{R} [\\boldsymbol{R} \\cdot \\boldsymbol{A}(\\boldsymbol{r}')]}{R^4} e^{-R/\\xi_0} d\\boldsymbol{r}' \\quad (\\boldsymbol{R} = \\boldsymbol{r} - \\boldsymbol{r}')\n \\end{equation*}\n Complete the following question:\n Analyze the approximate value of $j_s$ under the condition $q\\xi_0 \\ll 1$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathbf{j}(\\mathbf{q})$": "Supercurrent density in momentum space", + "$c$": "Speed of light", + "$\\lambda_L^2(0)$": "London penetration depth squared at zero temperature", + "$\\mathbf{A}(\\mathbf{q})$": "Vector potential in momentum space" + } + }, + { + "id": 467, + "context": "", + "question": "Pippard Theory\n In superconductors, there is a correlation of electron motion within the coherence length $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$, thus a perturbative potential acting at one point will inevitably affect the velocity and current density of superconducting electrons within the spatial scale of $\\xi_0$. Conversely, the supercurrent density at a certain point in space will inevitably be influenced by the perturbative potential within the scale of $\\xi_0$ around it. Therefore, $j_s (\\boldsymbol{r})$ is not only determined by $\\boldsymbol{A}(\\boldsymbol{r})$ at the same point, but should also include the contribution of vector potentials $\\boldsymbol{A}(\\boldsymbol{r}')$ at various points within the range $|\\boldsymbol{r} - \\boldsymbol{r}'| < \\xi_0$. To this end, one can consider the nonlocal relationship between $j_s$ and $\\boldsymbol{A}$ as follows (Pippard theory):\n \\begin{equation*}\n j_s (\\boldsymbol{r}) = - \\frac{3}{4\\pi \\xi_0 \\lambda_c} \\int \\frac{\\boldsymbol{R} [\\boldsymbol{R} \\cdot \\boldsymbol{A}(\\boldsymbol{r}')]}{R^4} e^{-R/\\xi_0} d\\boldsymbol{r}' \\quad (\\boldsymbol{R} = \\boldsymbol{r} - \\boldsymbol{r}')\n \\end{equation*}\n Please complete the following question:\n Analyze the approximate value of $j_s$ in the case of $q\\xi_0 \\gg 1$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathbf{j}(\\mathbf{q})$": "Supercurrent density in momentum space", + "$c$": "Speed of light", + "$K(0)$": "Kernel function $K(q)$ evaluated at $q=0$", + "$q$": "Wave vector or momentum variable", + "$\\xi_0$": "Coherence length, defined as $\\frac{\\hbar v_F}{\\pi \\Delta(0)}$", + "$\\mathbf{A}(\\mathbf{q})$": "Vector potential in momentum space", + "$\\lambda_L^2(0)$": "Square of the London penetration depth at zero temperature", + "$[j_s]_{\\mu}(\\bq)$": "$\\mu$-th component of the supercurrent density in momentum space", + "$A_{\\mu}(\\bq)$": "$\\mu$-th component of the vector potential in momentum space" + } + }, + { + "id": 468, + "context": "", + "question": "The Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is given by:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n In terms of notation, we set $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian of the system interaction $H_1$ can be expressed in terms of quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n Where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n Where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n1. Under the influence of a weak magnetic field, calculate the current $j_2(r)$ at zero temperature, and express the result using the London penetration depth $\\lambda_L = \\left( \\frac{mc^2}{4\\pi ne^2} \\right)^{1/2}$;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\mathbf{j}_2(\\mathbf{r})$": "Current density $\\mathbf{j}_2(\\mathbf{r})$.", + "$n$": "Electron density.", + "$e$": "Elementary charge.", + "$m$": "Mass of the charge carrier.", + "$c$": "Speed of light.", + "$\\mathbf{A}(\\mathbf{r})$": "Magnetic vector potential in real space.", + "$\\pi$": "Mathematical constant pi.", + "$\\lambda_L(0)$": "London penetration depth at zero temperature." + } + }, + { + "id": 469, + "context": "", + "question": "Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n Symbolically, we note $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian of the system interaction $H_1$ can be expressed using the quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n Under a weak magnetic field, the system is in the superconducting ground state. Considering the first-order approximation, calculate the current $j_1$. Hint: if the answer exists in an integral, then find the integrand.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$c$": "Speed of light.", + "$\\pi$": "Mathematical constant pi.", + "$\\lambda_L(0)$": "London penetration depth at zero temperature.", + "$\\mathbf{A}(\\mathbf{q})$": "Fourier transform of the vector potential.", + "$\\epsilon_+$": "Shorthand for single-particle energy $\\epsilon_{\\mathbf{k}+\\mathbf{q}/2}$.", + "$\\epsilon_-$": "Shorthand for single-particle energy $\\epsilon_{\\mathbf{k}-\\mathbf{q}/2}$.", + "$\\Delta$": "Superconducting energy gap.", + "$\\xi_+$": "Shorthand for quasiparticle energy $\\xi_{\\mathbf{k}+\\mathbf{q}/2}$.", + "$\\xi_-$": "Shorthand for quasiparticle energy $\\xi_{\\mathbf{k}-\\mathbf{q}/2}$." + } + }, + { + "id": 470, + "context": "", + "question": "Current in Superconductors\n According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation, it is expressed as:\n \\begin{align}\n \\hat{\\mathbf{j}}(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] - \\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &\\equiv \\hat{\\mathbf{j}}_1(\\mathbf{r}) + \\hat{\\mathbf{j}}_2(\\mathbf{r}) \\label{eq:6.8.9}\n \\end{align}\n Where\n \\begin{align}\n \\hat{\\mathbf{j}}_1(\\mathbf{r}) &= \\frac{e\\hbar}{2m_1} [\\Psi^{\\dagger} (\\nabla \\Psi) - (\\nabla \\Psi^{\\dagger}) \\Psi] \\\\\n &= \\frac{e\\hbar}{2m} \\sum_{\\mathbf{k},\\mathbf{q}} (2\\mathbf{k} + \\mathbf{q}) e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} - C_{-\\mathbf{k}\\downarrow}^{\\dagger} C_{-\\mathbf{k}\\downarrow}) \\label{eq:6.8.10}\n \\end{align}\n \\begin{align}\n \\hat{\\mathbf{j}}_2(\\mathbf{r}) &= -\\frac{e^2}{mc} \\Psi^{\\dagger} \\mathbf{A} \\Psi \\\\\n &= -\\frac{e^2}{mc} \\mathbf{A}(\\mathbf{r}) \\sum_{\\mathbf{k},\\mathbf{q}} e^{-i\\mathbf{q}\\cdot\\mathbf{r}} (C_{\\mathbf{k}+\\mathbf{q}}^{\\dagger} C_{\\mathbf{k}} + C_{-\\mathbf{k}-\\mathbf{q}}^{\\dagger} C_{-\\mathbf{k}-\\mathbf{q}}) \\label{eq:6.8.11}.\n \\end{align}\n In terms of notation, we consider $(\\mathbf{k},\\uparrow) = k,(-\\mathbf{k},\\downarrow)=-k$.\n\n The Hamiltonian $H_1$ of the system interaction can be expressed using the quasiparticle operators $\\alpha,\\alpha^+$ as:\n \\begin{align}\n H_1 = &-\\frac{e\\hbar}{mc} \\sum_{\\mathbf{k},\\mathbf{q}} [\\mathbf{k} \\cdot \\mathbf{A}(\\mathbf{q})] [(u_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}} - u_{\\mathbf{k}}v_{\\mathbf{k}+\\mathbf{q}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger} + \\alpha_{-\\mathbf{k}\\downarrow}\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}) + \\nonumber \\\\\n & (u_{\\mathbf{k}+\\mathbf{q}}u_{\\mathbf{k}} + v_{\\mathbf{k}+\\mathbf{q}}v_{\\mathbf{k}}) (\\alpha_{\\mathbf{k}+\\mathbf{q}\\uparrow}^{\\dagger}\\alpha_{\\mathbf{k}\\uparrow} - \\alpha_{-\\mathbf{k}\\downarrow}^{\\dagger}\\alpha_{-\\mathbf{k}\\downarrow})] \\label{eq:6:8.8},\n \\end{align}\n where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:\n \\begin{equation}\n u_k^2 = \\frac{1}{2} \\left(1 + \\frac{\\epsilon_k}{\\xi_k}\\right), \\quad v_k^2 = \\frac{1}{2} \\left(1 - \\frac{\\epsilon_k}{\\xi_k}\\right),\n \\end{equation}\n where $\\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2}$.\n\n Based on the above information, please complete the following calculation:\n The total current can be written as $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$, find the expression for $K(q)$.\n If the answer exists in an integral, then find the integrand.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\lambda_L^2(0)$": "Square of the London penetration depth at zero temperature.", + "$\\xi_+$": "Quasiparticle energy $\\xi_{\\mathbf{k}+\\mathbf{q}/2} = \\sqrt{\\epsilon_{+}^2 + \\Delta^2}$.", + "$\\xi_-$": "Quasiparticle energy $\\xi_{\\mathbf{k}-\\mathbf{q}/2} = \\sqrt{\\epsilon_{-}^2 + \\Delta^2}$.", + "$\\epsilon_+$": "Single-particle energy $\\epsilon_{\\mathbf{k}+\\mathbf{q}/2}$.", + "$\\epsilon_-$": "Single-particle energy $\\epsilon_{\\mathbf{k}-\\mathbf{q}/2}$.", + "$\\Delta$": "Superconducting energy gap." + } + }, + { + "id": 471, + "context": "", + "question": "Meissner Effect in Superconductors\n Assume the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$. If $K(0)=0$, it indicates the absence of the Meissner effect in the superconductor; otherwise, it exists.\n \n At the microscopic level, the current in a BCS superconductor can be written as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n Here, $K(q)$ is isotropic, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n The coherence length of the superconductor is $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n According to the information, perform the calculation under the assumption $q\\ll k_F$:\n1. For a normal conductor, calculate $K(q)$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$K_n(q)$": "Kernel function for a normal conductor." + } + }, + { + "id": 472, + "context": "", + "question": "Meissner Effect in Superconductors\n Assume the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$. If $K(0)=0$, it can be shown that there is no Meissner effect in the superconductor, otherwise it exists.\n \n Microscopically, the BCS superconducting current can be written as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n where $K(q)$ is orientation-independent, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n The coherence length of the superconductor is $\\xi_0 =\\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n Based on the information, please complete the calculation under the assumption $q\\ll k_F$:\n If $q\\xi_0 \\ll 1$, calculate $K(q)$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$K(q)$": "Kernel function relating current density to vector potential, orientation-independent", + "$\\lambda_L(0)$": "London penetration depth at zero temperature" + } + }, + { + "id": 473, + "context": "", + "question": "Meissner effect in superconductors\n Assuming the current in a superconductor follows $\\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})$, if $K(0)=0$, it can be stated that the superconductor does not exhibit the Meissner effect, otherwise it does.\n \n Microscopically, BCS superconductor current can be expressed as\n \\begin{equation*}\n \\mathbf{j}(\\mathbf{q}) = -\\frac{c}{4\\pi} K(q) \\mathbf{A}(\\mathbf{q})\n \\end{equation*}\n \\begin{equation}\n K(q) = \\frac{1}{\\lambda_L^2(0)} \\left\\{1 - \\frac{3}{4} \\int_{-1}^{+1} (1-Z^2) dZ \\times \\int_{-\\infty}^{\\infty} d\\epsilon \\frac{1}{2} \\frac{\\xi_+ \\xi_- - \\epsilon_+ \\epsilon_- - \\Delta^2}{\\xi_+ \\xi_- (\\xi_+ + \\xi_-)} \\right\\} \\label{eq:6.8.24}\n \\end{equation}\n Where $K(q)$ is orientation-independent, and\n \\begin{align}\n \\epsilon_{\\pm} &= \\epsilon_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad \\xi_{\\pm} = \\xi_{\\mathbf{k}\\pm\\mathbf{q}/2} = \\sqrt{\\epsilon_{\\pm}^2 + \\Delta^2} \\\\\n u_{\\pm} &= u_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad v_{\\pm} = v_{\\mathbf{k}\\pm\\mathbf{q}/2}, \\quad Z = \\cos\\theta \n \\end{align}\n Coherence length of the superconductor $\\xi_0 =\\frac{\\hbar v_F}{\\pi \\Delta(0)}$\n Based on the information, under the assumption $q\\ll k_F$, complete the calculations:\n If $q\\xi_0 \\gg 1$, calculate $K(q)$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$K(q)$": "Orientation-independent kernel function describing the current response in a superconductor.", + "$\\lambda_L(0)$": "London penetration depth at zero temperature.", + "$q$": "Magnitude of the wave vector.", + "$\\xi_0$": "Coherence length of the superconductor, defined as $\\xi_0 = \\frac{\\hbar v_F}{\\pi \\Delta(0)}$.", + "$K(0)$": "Value of the kernel function $K(q)$ at zero wave vector, used to determine the presence of the Meissner effect." + } + }, + { + "id": 474, + "context": "", + "question": "London penetration depth at finite temperatures\n The London equation is a significant equation describing superconductors, reflecting the perfect diamagnetism of superconductors and can be written as\n \\begin{equation}\n \\mathbf{j}_s(\\mathbf{r}) = - \\frac{c}{4\\pi}\\frac{1}{\\lambda_L^2}\\mathbf{A}(\\mathbf{r}),\n \\end{equation}\n where $\\lambda_L$ is called the London penetration depth, depicting the depth to which a magnetic field penetrates into a superconducting sample, and at zero temperature can be expressed as\n \\begin{equation}\n \\lambda_L^2(0) = \\left( \\frac{mc^2}{4\\pi ne^2} \\right)^{1/2}.\n \\end{equation}\n Based on the context, complete the calculation:\n\n Calculate the linear response coefficient of the diamagnetic current under first order approximation. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$K_2(q, T)$": "Linear response coefficient of the diamagnetic current", + "$q$": "Wave number", + "$T$": "Temperature", + "$\\pi$": "Mathematical constant pi", + "$n$": "Number density of superconducting electrons", + "$e$": "Elementary charge", + "$m$": "Mass of the charge carrier", + "$c$": "Speed of light" + } + }, + { + "id": 475, + "context": "", + "question": "Ginzberg Landau Theory\n Ginzburg and Landau proposed using a complex quantity $\\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$, and the corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The system's free energy is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\nThe condition for thermodynamic equilibrium (consider variational performance with respect to the order parameter and vector potential) to derive the equation satisfied by $\\psi$, namely the Ginzburg-Landau equation. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$m$": "Mass, appearing in the Ginzburg-Landau equation (corresponds to $m^*$ in the problem statement).", + "$i$": "Imaginary unit.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\nabla$": "Nabla operator.", + "$e$": "Charge, appearing in the Ginzburg-Landau equation (corresponds to $e^*$ in the problem statement).", + "$c$": "Speed of light.", + "$\\mathbf{A}$": "Vector potential.", + "$\\psi$": "Complex quantity describing the 'effective wave function' of superconducting electrons, also known as the order parameter.", + "$\\alpha$": "Coefficient in the Ginzburg-Landau equation (corresponds to $\\alpha(T)$ in the problem statement).", + "$\\beta$": "Coefficient in the Ginzburg-Landau equation (corresponds to $\\beta(T)$ in the problem statement)." + } + }, + { + "id": 476, + "context": "", + "question": "Ginzburg-Landau Theory\n Ginzburg and Landau proposed using a complex variable $\\psi(r)$ to describe the \"effective wave function\" of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The free energy of the system is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n \n Solve the Ginzburg-Landau equation under $A=0$;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\psi_0$": "spatially uniform solution for the effective wave function, representing a superconducting state", + "$\\alpha(T)$": "temperature-dependent coefficient in the Ginzburg-Landau free energy density", + "$\\beta(T)$": "temperature-dependent coefficient in the Ginzburg-Landau free energy density", + "$T$": "temperature", + "$T_c$": "critical temperature, above which superconductivity is lost" + } + }, + { + "id": 477, + "context": "", + "question": "Ginzburg Landau Theory\n Ginzburg and Landau propose using a complex quantity $\\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The system's free energy is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\n The penetration depth $\\lambda(T)$ in a weak magnetic field (expressed in terms of the result from question two);", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\lambda(T)$": "Penetration depth, temperature-dependent.", + "$m$": "Mass of superconducting electrons (used in one form of the final answer, corresponding to $m^*$ in the problem statement).", + "$c$": "Speed of light in vacuum.", + "$\\pi$": "Mathematical constant pi, approximately 3.14159.", + "$e$": "Charge of superconducting electrons (used in one form of the final answer, corresponding to $e^*$ in the problem statement).", + "$\\psi_0$": "Field-free order parameter, magnitude of the effective wave function in the absence of a magnetic field.", + "$m^*$": "Mass of superconducting electrons.", + "$e^*$": "Charge of superconducting electrons.", + "$|\\psi_0|$": "Magnitude of the field-free order parameter $\\psi_0$." + } + }, + { + "id": 478, + "context": "", + "question": "Ginzburg-Landau Theory\n Ginzburg and Landau proposed using a complex quantity $\\psi(r)$ to describe the \"effective wave function\" of superconducting electrons, with charge $e^*$ and mass $m^*$; the corresponding system free energy density and free energy are:\n \\begin{equation}\n f_s = f_n + \\alpha(T) |\\psi(\\mathbf{r})|^2 + \\frac{1}{2} \\beta(T) |\\psi(\\mathbf{r})|^4 + \\frac{1}{8\\pi} (\\nabla \\times \\mathbf{A}) \\cdot (\\nabla \\times \\mathbf{A}) + \\frac{1}{2m^*} \\left| \\left(-i\\hbar \\nabla - \\frac{e^*}{c}\\mathbf{A}\\right) \\psi(\\mathbf{r}) \\right|^2 \\label{eq:6.10.6},\n \\end{equation}\n The free energy of the system is\n \\begin{equation}\n F_s = \\int f_s dr \\label{eq:6.10.7}\n \\end{equation}\n Please calculate:\n\n Calculate the magnetic flux and thereby demonstrate the quantization of magnetic flux.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Phi$": "Magnetic flux, $\\Phi = \\int_S \\mathbf{B} \\cdot d\\mathbf{S}$", + "$n$": "Integer multiple, quantum number", + "$\\Phi_0$": "Magnetic flux quantum, $\\Phi_0 = \\frac{2\\pi\\hbar c}{e^*}$", + "$\\pi$": "Mathematical constant pi", + "$\\hbar$": "Reduced Planck's constant", + "$c$": "Speed of light", + "$e^*$": "Charge of superconducting electrons", + "$\\mathbf{B}$": "Magnetic field, $\\mathbf{B} = \\nabla \\times \\mathbf{A}$", + "$S$": "Area enclosed by the integration loop $C$", + "$\\mathbb{Z}$": "Set of integers" + } + }, + { + "id": 479, + "context": "", + "question": "Hubbard Model in Narrow Bandwidth\n Consider the following single band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, $n$ is the particle number operator\n\n Please complete the following calculation:\n\n Calculate the off-diagonal elements of the single-particle Green's function obtained in the Bloch representation You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$G_{kk'}^\\sigma(\\omega)$": "Off-diagonal elements of the single-particle Green's function in Bloch representation." + } + }, + { + "id": 480, + "context": "", + "question": "Hubbard Model under Narrow Bandwidth — Green's Function Analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c, c^\\dagger$ are electron annihilation and creation operators, $n$ is the particle number operator.\n\n In the case of bandwidth $\\Delta \\neq 0$, the single-particle green function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - (n_{\\bar{\\sigma}}))}{(\\omega - E_k)(\\omega - T_0 - U) + (n_{\\bar{\\sigma}}) U(T_0 - E_k)}\n \\end{equation}\n\n Please solve the following problem:\n\n Calculate the energy spectrum of the system;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$E_{k\\sigma}^{(1,2)}$": "Combined notation for the two branches of the energy spectrum (poles of the Green's function) for momentum $k$ and spin $\\sigma$.", + "$E_k$": "Single-particle energy for momentum $k$ in the non-interacting system.", + "$U$": "On-site Coulomb repulsion strength.", + "$T_0$": "A parameter in the Green's function, representing an effective on-site energy or average hopping term.", + "$\\langle n_{\\bar\\sigma}\\rangle$": "Average number of particles with opposite spin $\\bar{\\sigma}$." + } + }, + { + "id": 481, + "context": "", + "question": "Hubbard Model in Narrow Band - Green's Function Analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators of electrons, and $n$ is the particle number operator.\n\n For non-zero bandwidth $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - < n_{\\bar{\\sigma}} >)}{(\\omega - E_k)(\\omega - T_0 - U) + < n_{\\bar{\\sigma}} > U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n\n Calculate the Green's function for $U=0$ You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$G_{k}^{\\sigma}(\\omega)$": "Single-particle Green's function for momentum $k$ and spin $\\sigma$ at energy $\\omega$.", + "$U$": "On-site Coulomb repulsion energy.", + "$\\omega$": "Energy (or frequency).", + "$E_k$": "Energy of the band electron with momentum $k$." + } + }, + { + "id": 482, + "context": "", + "question": "Hubbard Model in the Narrow Band Limit—Green's Function Analysis\n Consider the single-band Hubbard model as follows,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.\n\n In the case where bandwidth $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - < n_{\\bar{\\sigma}} >)}{(\\omega - E_k)(\\omega - T_0 - U) + < n_{\\bar{\\sigma}} > U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n3. Calculate the density of states for the case of $U=0$ You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\rho_\\sigma (\\omega)$": "Density of states for spin $\\sigma$ at energy $\\omega$", + "$U$": "On-site Coulomb repulsion energy (Hubbard U)", + "$\\omega$": "Energy or frequency", + "$D(\\omega)$": "Density of states of a single band $E_k$ at energy $\\omega$", + "$\\sigma$": "Spin index" + } + }, + { + "id": 483, + "context": "", + "question": "Narrow-band Hubbard model—Green's function analysis\n Consider the following single-band Hubbard model,\n \\begin{equation}\n H = \\sum_{i,j,\\sigma} T_{ij} c_{i\\sigma}^{\\dagger} c_{j\\sigma} + \\frac{U}{2} \\sum_{i,\\sigma} n_{i\\sigma} n_{i\\bar{\\sigma}} \\label{eq:11.1.13}\n \\end{equation}\n where $c,c^\\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.\n\n When the band width $\\Delta \\neq 0$, the single-particle Green's function approximately satisfies\n \\begin{equation}\n G_k^\\sigma(\\omega) = \\frac{\\omega - T_0 - U(1 - (n_{\\bar{\\sigma}}))}{(\\omega - E_k)(\\omega - T_0 - U) + (n_{\\bar{\\sigma}}) U(T_0 - E_k)}\n \\end{equation}\n\n Please complete the following problem:\n\n Calculate the Green's function in the case $U\\geq \\Delta$ You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$G_k^\\sigma(\\omega)$": "Single-particle Green's function for momentum $k$ and spin $\\sigma$ at frequency $\\omega$.", + "$n_{\\bar{\\sigma}}$": "Particle number operator for an electron at site $i$ with spin $\\bar{\\sigma}$ (opposite spin). In the context of the Green's function, it often represents the expectation value $\\langle n_{\\bar{\\sigma}} \\rangle$.", + "$\\omega$": "Frequency.", + "$T_0$": "A constant energy term, possibly related to the average hopping energy or a chemical potential.", + "$E_k$": "Single-particle energy dispersion for momentum $k$.", + "$U$": "On-site Coulomb repulsion (Hubbard U)." + } + }, + { + "id": 484, + "context": "", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that to discuss the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, two factors must be considered: First, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant impact on the formation of local magnetic moments in the crystal; second, since the free atomic d-orbital states $\\phi_d(\\mathbf{r})$ of impurities in crystals are no longer purely eigenstates, especially due to the tendency for electron delocalization in metal crystals into Bloch states (s orbitals), there exists an electron transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson called s-d mixing. Therefore, he pointed out that the system's Hamiltonian $H$ should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and suppose the Lande factor of electrons and impurities $g_0 = g_i = 2$, which is the non-degenerate orbital Anderson s-d mixing model.\n\n In dealing with the $s-d$ exchange model, the Green's function equation of motion method is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n By utilizing a technique, differentiate the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$, yielding the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Perform a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n Obtaining two forms of the general equation of motion for Green’s function\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n\n Calculate the Green's function equation of motion for $\\ll C_{k\\sigma} | C_{k'\\sigma}^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{kk'\\sigma}$ symbolize $\\ll C_{k\\sigma} | C_{k'\\sigma}^+ \\gg_\\omega$, and $b_{k'\\sigma}$ symbolize $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\\\omega$": "Frequency.", + "$E_{k\\\\sigma}$": "Energy of an electron in state $k$ with spin $\\sigma$, $E_{k\\\\sigma} = E_k + \\\\sigma \\\\mu_B h$.", + "$a_{kk'\\\\sigma}$": "Shorthand notation for the Green's function $\\\\ll C_{k\\\\sigma} | C_{k'\\\\sigma}^\\\\dagger \\\\gg_\\\\omega$.", + "$\\\\delta_{k, k'}$": "Kronecker delta function.", + "$V_{kd}$": "Electron transfer matrix element (s-d mixing matrix element) between state $k$ and d-orbital.", + "$b_{k'\\\\sigma}$": "Shorthand notation for the Green's function $\\\\ll d_\\\\sigma | C_{k'\\\\sigma}^\\\\dagger \\\\gg_\\\\omega$.", + "$\\\\langle\\\\langle C_{k\\\\sigma} | C_{k'\\\\sigma}^\\\\dagger \\\\rangle\\\\rangle_\\\\omega$": "Green's function $\\\\ll C_{k\\\\sigma} | C_{k'\\\\sigma}^\\\\dagger \\\\gg_\\\\omega$.", + "$\\\\langle\\\\langle d_\\\\sigma | C_{k'\\\\sigma}^\\\\dagger \\\\rangle\\\\rangle_\\\\omega$": "Green's function $\\\\ll d_\\\\sigma | C_{k'\\\\sigma}^\\\\dagger \\\\gg_\\\\omega$." + } + }, + { + "id": 485, + "context": "", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that when discussing the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, it is essential to consider two factors: Firstly, similar to the inherent magnetic moment formed by a free atom, the Coulomb interaction among d-shell electrons in impurity atoms significantly affects the formation of local magnetic moments in the crystal. Secondly, since the d-orbital state of impurity atoms, $\\phi_d(\\mathbf{r})$, in a crystal are no longer rigorous eigenstates, especially due to the tendency of electrons becoming delocalized into Bloch states (s orbitals) in the metal crystal, there exists an electron transfer between $\\phi_d$ and $\\phi_k$ states, a phenomenon that Anderson termed as s-d mixing. Therefore, he stated that the Hamiltonian $H$ of the system should be composed of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and assuming the Lande's g-factor of electron and impurity $g_0 = g_i = 2$, this constitutes the non-orbitally degenerate Anderson s-d mixing model.\n\n When dealing with the $s-d$ exchange model, the following Green's function equation of motion is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n a technique is used to take derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ separately, yielding the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n By performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n two representations of the general form of the Green's function equation of motion are obtained:\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question\n\n Calculate the Green's function equation of motion related to $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{\\sigma}$ syimbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$, $b_\\sigma$ symbolize $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$, and $c_{\\sigma}$ symbolize $\\ll n_{d\\bar{\\sigma}} d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\omega$": "Frequency.", + "$E_{d\\sigma}$": "Energy of a d-orbital electron with spin $\\sigma$, defined as $E_{d\\sigma} = E_d + \\sigma \\mu_B h$.", + "$a_\\sigma$": "Symbol for the Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$.", + "$U$": "Coulomb interaction strength among d-shell electrons.", + "$c_\\sigma$": "Symbol for $\\ll n_{d\\bar{\\sigma}} d_\\sigma | d_\\sigma^+ \\gg_\\omega$.", + "$b_\\sigma$": "Symbol for $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$." + } + }, + { + "id": 486, + "context": "", + "question": "Anderson s-d exchange model and Green's function equations of motion\n In 1961, Anderson proposed the s-d mixing model, arguing that to discuss the formation of localized magnetic moments by transition metal impurity atoms in a non-magnetic metallic matrix, two factors must be considered simultaneously: First, similar to the intrinsic magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms significantly influences the formation of localized magnetic moments in the crystal; Second, because the d orbital states $\\phi_d(\\mathbf{r})$ of the impurity in the crystal are no longer eigenstates, particularly due to the tendency of electron delocalization into Bloch states in the metallic crystal (s orbital), there is mutual electron transfer between the states $\\phi_d$ and $\\phi_k$, which Anderson termed the s-d mixing interaction. Therefore, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and the Landé factor for the electron and impurity is set as $g_0 = g_i = 2$, which defines the non-degenerate orbital Anderson s-d mixing model.\n\n In dealing with the $s-d$ exchange model, the following Green's function equation of motion method is often employed:\n\n Starting from the two-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n By employing a technique, the differential of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ can be taken respectively, resulting in the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n achieves the two representations of the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n1. Derive the equation of motion for the s-d exchange model concerning the mixed Green's function $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$. Hint: Let $a_{k\\sigma}$ symbolize $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$, and $b_{\\sigma}$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\omega$": "Frequency.", + "$E_{k\\sigma}$": "Energy of an s-electron with momentum $k$ and spin $\\sigma$.", + "$a_{k\\sigma}$": "Symbol representing the mixed Green's function $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$.", + "$V_{kd}$": "s-d mixing interaction strength, representing the matrix element for electron transfer between s and d states.", + "$b_{\\sigma}$": "Symbol representing the Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$." + } + }, + { + "id": 487, + "context": "", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model, suggesting that discussing the formation of local magnetic moments of transition-metal impurity atoms in a non-magnetic metal matrix must simultaneously consider two factors: First, similar to the formation of inherent magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant effect on the formation of local magnetic moments in crystals. Second, since the free atomic d-orbital state $\\phi_d(\\mathbf{r})$ of impurities in crystals is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch orbital states (s-orbital) in metal crystals, there is electronic transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson termed s-d mixing. To this end, he pointed out that the system's Hamiltonian $H$ should be composed of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n here $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, assuming the Lande factor of electrons and impurities $g_0 = g_i = 2$, this is the non-degenerate orbital Anderson s-d mixing model.\n\n In handling the $s-d$ exchange model, the Green's function equation of motion method is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n Using a technique, derive the derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ separately to obtain the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n leads to two expressions for the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the problem\n\n Calculate the equation of motion for the mixed Green's function $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$ in the s-d exchange model. Hint: Let $a_{k'\\sigma}$ symbolize $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$, and $b_\\sigma$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\omega$": "Energy or frequency in Fourier space.", + "$E_{k'\\sigma}$": "Energy of an s-orbital electron with wave vector $k'$ and spin $\\sigma$, following the general definition $E_{k\\sigma} = E_k + \\sigma \\mu_B h$.", + "$a_{k'\\sigma}$": "Symbol representing the mixed Green's function $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$.", + "$V_{k'd}$": "s-d mixing matrix element, representing the hopping amplitude between s-orbital and d-orbital states, specific to $k'$.", + "$b_\\sigma$": "Symbol representing the Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$." + } + }, + { + "id": 488, + "context": "", + "question": "In 1961, Anderson proposed the s-d mixing model, where he considered that the formation of localized magnetic moments by transition metal impurity atoms in non-magnetic metal matrices must take into account two factors: first, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant influence on forming localized magnetic moments in crystals; second, due to the fact that the d-orbital state $\\phi_d(\\mathbf{r})$ of free atoms in the crystal is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch states (s orbitals) in metal crystals, there exists electron transfer between $\\phi_d$ and $\\phi_k$ states, which Anderson referred to as s-d hybridization. Consequently, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, and the Landé factor for electrons and impurities is set as $g_0 = g_i = 2$, which is the non-degenerate s-d mixing model of Anderson.\n\n In dealing with the $s-d$ exchange model, the following method of the equation of motion for Green's functions is often employed:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n Utilizing a technique, the derivatives of the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$ respectively yield the following two equations of motion:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing the Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n leads to two representations of the general form of the equation of motion for Green's functions\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question:\n\n Truncate the higher order terms in the equations to write out the approximate equations.\n Hint: Let $a_\\sigma$ symbolize $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$, $b_\\sigma$ symbolize $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$. Anderson s-d exchange model and the equation of motion for Green's functions You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\omega$": "Frequency in the Fourier transform.", + "$E_{d\\sigma}$": "Energy of an electron in d-orbital state with spin $\\sigma$.", + "$U$": "Coulomb interaction strength between d-shell electrons.", + "$n_{d\\bar{\\sigma}}$": "Number operator for electrons in d-orbital state with spin $\\bar{\\sigma}$. In the final approximate equation, it represents its expectation value $\\langle n_{d\\bar{\\sigma}}\\rangle$.", + "$a_\\sigma$": "Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$.", + "$b_\\sigma$": "Sum of Green's functions $\\sum_k V_{kd} \\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$." + } + }, + { + "id": 489, + "context": "", + "question": "Anderson s-d exchange model and Green's function equation of motion\n In 1961, Anderson proposed the s-d mixing model. He considered that the discussion of the formation of a localized magnetic moment by transition metal impurity atoms in non-magnetic metallic matrices must account for two factors: Firstly, similar to the inherent magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms has a significant impact on the formation of localized magnetic moments in crystals. Secondly, due to the tendency of electron delocalization to Bloch orbital states (s orbitals) in metallic crystals, there is an exchange between states $\\phi_d(\\mathbf{r})$ and $\\phi_k$, which Anderson termed s-d mixing. Thus, he pointed out that the system's Hamiltonian $H$ should consist of the following four components:\n \\begin{equation}\n H = \\sum_{k,\\sigma} E_{k\\sigma} n_{k\\sigma} + \\sum_{\\sigma} E_{d\\sigma} n_{d\\sigma} + \\frac{U}{2}\\sum_{\\sigma} n_{d\\sigma} n_{d\\bar{\\sigma}} + \\sum_{k,\\sigma} V_{kd} (C_{k\\sigma}^\\dagger d_\\sigma + d_\\sigma^\\dagger C_{k\\sigma})\n \\label{eq:11.2.5}\n \\end{equation}\n where\n \\begin{equation}\n E_{k\\sigma} = E_k + \\sigma \\mu_B h, \\quad E_{d\\sigma} = E_d + \\sigma \\mu_B h\n \\label{eq:11.2.6}\n \\end{equation}\n Here, $\\mu_B = \\left( \\left| \\frac{e}{2mc} \\right| \\hbar \\right)$ is the Bohr magneton, with a Landé factor of $g_0 = g_i = 2$ for both electrons and impurities. This is the non-degenerate orbital Anderson s-d mixing model.\n\n When handling the s-d exchange model, the following Green's function equation of motion is often used:\n\n Starting from the double-time Green's function\n \\begin{equation*}\n \\ll A(t); B(t') \\gg = -i \\theta(t-t') <[A(t), B(t')]_+>\n \\end{equation*}\n by employing a technique to differentiate the function $\\ll A(t); B(t') \\gg$ with respect to $t$ and $t'$, the following two equations of motion can be obtained:\n \\begin{equation}\n i \\frac{d}{dt} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll [A, H]; B(t') \\gg\n \\label{eq:11.2.7}\n \\end{equation}\n \\begin{equation}\n -i \\frac{d}{dt'} \\ll A(t); B(t') \\gg = \\delta(t-t') <[A, B]_+> + \\ll A(t); [B, H] \\gg\n \\label{eq:11.2.8}\n \\end{equation}\n Performing a Fourier transform\n \\begin{equation}\n \\ll A | B \\gg_\\omega = \\int dte^{i\\omega(t-t')} \\ll A(t); B(t') \\gg\n \\label{eq:11.2.9}\n \\end{equation}\n yields the two forms of the Green's function equation of motion\n \\begin{align}\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> + \\ll [A, H] | B \\gg_\\omega\n \\label{eq:11.2.10} \\\\\n \\omega \\ll A | B \\gg_\\omega &= <[A, B]_+> - \\ll A | [B, H] \\gg_\\omega\n \\label{eq:11.2.11}\n \\end{align}\n Please complete the question\n Solve the Green's function using the truncation approximation", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$\\delta_{kk'}$": "Kronecker delta, equal to 1 if $k=k'$ and 0 otherwise.", + "$\\omega$": "Frequency in the Fourier transformed Green's function.", + "$E_{k\\sigma}$": "Energy of an s-orbital electron with wave vector $k$ and spin $\\sigma$, $E_{k\\sigma} = E_k + \\sigma \\mu_B h$.", + "$V_{kd}$": "s-d mixing matrix element (hybridization strength) between s-orbital state $k$ and d-orbital state.", + "$V_{k'd}$": "s-d mixing matrix element (hybridization strength) between s-orbital state $k'$ and d-orbital state.", + "$E_{k'\\sigma}$": "Energy of an s-orbital electron with wave vector $k'$ and spin $\\sigma$.", + "$E_{d\\sigma}$": "Energy of a d-orbital electron with spin $\\sigma$, $E_{d\\sigma} = E_d + \\sigma \\mu_B h$.", + "$U$": "Coulomb interaction strength between d-shell electrons.", + "$n_{d\\bar{\\sigma}}$": "Number operator for d-orbital electrons with opposite spin $\\bar{\\sigma}$ (used here as its expectation value).", + "$i$": "Imaginary unit.", + "$\\Gamma$": "Constant representing the hybridization width, $\\Gamma \\equiv \\pi V_{kd}^2 \\rho_F^{(0)}$." + } + }, + { + "id": 490, + "context": "The known Hamiltonian for electron-phonon interaction:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now use plane wave instead of Bloch wave function.", + "question": "Please analyze the conservation laws during the electron-phonon interaction process. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\hbar$": "Reduced Planck's constant", + "$\\mathbf{k}'$": "Final wavevector of an electron", + "$\\mathbf{k}$": "Initial wavevector of an electron", + "$\\mathbf{q}$": "Wavevector of a phonon", + "$\\mathbf{K}_n$": "Reciprocal lattice vector" + } + }, + { + "id": 491, + "context": "The known Hamiltonian for electron-phonon interaction:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now replace Bloch wave functions with plane waves", + "question": "Please provide the expression for the transition probability of the system from the initial to the final state during electron-phonon interaction (considering the long-time limit). You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$W(i \\rightarrow f)$": "Transition probability from initial state $i$ to final state $f$", + "$\\hbar$": "Reduced Planck's constant", + "$|f\\rangle$": "Final state of the system", + "$H_{ep}$": "Hamiltonian for electron-phonon interaction", + "$|i\\rangle$": "Initial state of the system", + "$\\delta$": "Dirac delta function", + "$E_f$": "Energy of the final state", + "$E_i$": "Energy of the initial state" + } + }, + { + "id": 492, + "context": "The known electron-phonon interaction Hamiltonian:\n \\begin{equation}\n H_{ep} = -i \\sum_{\\mathbf{k}, \\mathbf{k}'} \\sum_{\\mathbf{q}, s} \\sum_{\\mathbf{p}} \n \\left( \\frac{N \\hbar}{2 M \\omega_{\\mathbf{q}s}} \\right)^{1/2} \n (\\mathbf{e}_{\\mathbf{q}s} \\cdot \\mathbf{p}) \n \\left\\{ \\frac{1}{N} \\sum_{\\mathbf{l}} e^{i(\\mathbf{k} + \\mathbf{q} - \\mathbf{k}')\\cdot \\mathbf{l}} \\right\\} \\times\n V_{\\mathbf{p}} \\langle \\mathbf{k}' | e^{i\\mathbf{p}\\cdot\\mathbf{r}} | \\mathbf{k} \\rangle \n (a_{\\mathbf{q}s} + a_{-\\mathbf{q}s}^\\dagger) \n C_{\\mathbf{k}}^\\dagger C_{\\mathbf{k}'}\n \\end{equation}\n Now replace Bloch functions with plane waves", + "question": "Please provide the energy conservation relation in the electron-phonon interaction process and explain the specific relationship between electron energy and phonon energy before and after scattering. We only consider the case that the electron absorbs a phonon. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\varepsilon_{\\mathbf{k} + \\mathbf{q}}$": "Energy of the electron after absorbing a phonon, with its wave vector changed from $\\mathbf{k}$ to $\\mathbf{k} + \\mathbf{q}$.", + "$\\varepsilon_{\\mathbf{k}}$": "Energy of the electron before scattering.", + "$\\hbar$": "Reduced Planck's constant.", + "$\\omega_{\\mathbf{q}}$": "Frequency of the absorbed phonon with wave vector $\\mathbf{q}$." + } + }, + { + "id": 493, + "context": "The known collective coordinate expression of the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal's simple lattice composed of $N$ ions immersed in a uniform electron gas. The frequency of perturbed LA phonons at this moment can be expressed using the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal occupies unit volume, $\\Omega$ is the primitive cell volume, $N=\\Omega^{-1}$, corresponding to the LA phonon's Hamiltonian:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n Among them, $H_e$ is the free electron approximation Hamiltonian, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electron normal coordinates $Q$.\n\n Under the long-wavelength approximation, complete the following calculation", + "question": "Find the equation of motion for $Q_q$, and discuss how to derive the phonon frequency correction You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\ddot{Q}_{\\mathbf{q}}$": "Second time derivative of the normal coordinate $Q_{\\mathbf{q}}$.", + "$\\Omega_{\\mathbf{q}}$": "Frequency of perturbed LA phonons for wave vector $\\mathbf{q}$.", + "$Q_{\\mathbf{q}}$": "Normal coordinate of phonons for wave vector $\\mathbf{q}$.", + "$M_{-\\mathbf{q}}$": "Coupling coefficient for electron-phonon interaction for wave vector $-\\mathbf{q}$, $M_{-\\mathbf{q}} = -i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})$.", + "$\\rho_{\\mathbf{q}}$": "Fourier component of electron density operator for wave vector $\\mathbf{q}$." + } + }, + { + "id": 494, + "context": "The known collective coordinate expression of the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Considering a monovalent metal, where a simple lattice consisting of $N$ ions is immersed in a uniform electron gas, the perturbated LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here, the crystal is taken with unit volume, $\\Omega$ is the volume of the primitive cell, $N=\\Omega^{-1}$, corresponding to the LA phonon Hamiltonian:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons approximation, $H_{ee}$ represents the Coulomb interaction, both of which are independent of the normal coordinate $Q$.\n\n Complete the following calculation under the long-wavelength approximation", + "question": "Calculate the ionic density fluctuations $\\rho^i_q$ produced by lattice vibrations", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\rho_{\\mathbf{q}}^i$": "Fourier component of ionic density fluctuation with wave vector $\\mathbf{q}$", + "$i$": "Imaginary unit", + "$N$": "Number density of ions (number of ions per unit volume)", + "$M$": "Mass of a single ion", + "$\\mathbf{e}_{\\mathbf{q}}$": "Polarization vector of phonons with wave vector $\\mathbf{q}$", + "$\\mathbf{q}$": "Wave vector", + "$Q_{\\mathbf{q}}$": "Normal coordinate of phonons with wave vector $\\mathbf{q}$" + } + }, + { + "id": 495, + "context": "The known collective coordinate expression for the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal where a simple lattice composed of $N$ ions is immersed in a uniform electron gas. In this case, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here, the crystal is taken to be of unit volume, $\\Omega$ is the unit cell volume of the primitive lattice, and $N=\\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons approximately, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electronic normal coordinate $Q$.\n\n Under the long-wavelength approximation, complete the following calculation", + "question": "Using linear response theory, calculate the relationship between $\\rho_q$ and $\\rho^i_q$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\rho_{\\mathbf{q}}$": "Fourier component of the electron charge density at wave vector $\\mathbf{q}$", + "$\\epsilon(\\mathbf{q})$": "Static dielectric function at wave vector $\\mathbf{q}$", + "$\\rho_{\\mathbf{q}}^i$": "Fourier component of the ionic charge density at wave vector $\\mathbf{q}$" + } + }, + { + "id": 496, + "context": "The collective coordinate expression for the known electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal immersed in a uniform electron gas, forming a simple lattice composed of $N$ ions. The perturbed LA phonon frequency can be expressed in terms of the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal is taken with unit volume, $\\Omega$ is the unit cell volume of the Bravais lattice, and $N=\\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian of free electrons in approximation, and $H_{ee}$ represents Coulomb interaction, both of which are independent of the normal coordinate $Q$ of phonons.\n\n Under the long-wavelength approximation, complete the following calculations.", + "question": "Under the long-wavelength approximation, considering the electron screening effect, derive the dispersion relation between the LA phonon angular frequency $\\omega_{\\mathbf{q}}$ and the wave vector $q$, and specify its form $\\omega_{\\mathbf{q}}$ with respect to $q$. Hint: You can use the Thomas-Fermi dielectric function. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$\\omega_{\\mathbf{q}}$": "LA phonon angular frequency for wave vector $\\mathbf{q}$.", + "$c_L$": "Longitudinal sound velocity.", + "$q$": "Magnitude of the wave vector $\\mathbf{q}$." + } + }, + { + "id": 497, + "context": "The known collective coordinate expression for the electron-phonon Hamiltonian is:\n \\begin{equation}\n H_{ep} = \\sum_{\\mathbf{q}} M_{\\mathbf{q}} Q_{\\mathbf{q}} \\rho_{-\\mathbf{q}}, \\quad M_{\\mathbf{q}} = i \\left( \\frac{N}{M} \\right)^{1/2} \\frac{4\\pi e^2}{q^2} (\\mathbf{e}_{\\mathbf{q}} \\cdot \\mathbf{q})\n \\end{equation}\n Consider a monovalent metal, a simple lattice consisting of $N$ ions submerged in a uniform electron gas. At this time, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:\n \\begin{equation}\n \\Omega_q^2 + \\frac{4\\pi Ne^2}{M},\n \\end{equation}\n Here the crystal takes unit volume, $\\Omega$ is the positive lattice unit cell volume, $N=\\Omega^{-1}$, corresponding to the Hamiltonian of the LA phonon:\n \\begin{equation}\n P_{-q} = \\dot{Q}_q,\n \\end{equation}\n The electron-phonon interaction Hamiltonian is:\n \\begin{equation}\n H_{\\textrm{ep}} = \\sum_q M_q Q_q \\rho_{-q},\n \\end{equation}\n The total Hamiltonian is:\n \\begin{equation}\n H = H_p + H_{ep} + H_e + H_{ee},\n \\end{equation}\n where $H_e$ is the Hamiltonian for free electrons, and $H_{ee}$ represents the Coulomb interaction, both terms are independent of the electron normal coordinate $Q$.\n\n Under the long wavelength approximation, complete the following calculations", + "question": "With the known dispersion relation of LA phonons in the form $\\omega_{\\mathbf{q}} = c_L q$, please provide the specific expression for the speed of sound of LA phonons $c_L$ (i.e., the Bohm-Staver speed of sound formula), and explain each physical quantity. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Superconductivity", + "final_symbol": { + "$c_L$": "Speed of sound of LA phonons", + "$m$": "Electron mass", + "$M$": "Ion mass", + "$v_F$": "Fermi velocity" + } + }, + { + "id": 498, + "context": "The longitudinal optical (LO) vibration mode in ionic crystals generates a polarization field, which strongly couples with conduction electrons in ionic crystals. This interaction is much stronger than the effect of longitudinal acoustic (LA) phonons (which involves center of mass motion and does not generate a polarization field) on conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier properties in ionic crystals.", + "question": "When electrons move within an ionic crystal, they cause relative displacements between positive and negative ions, forming a local polarization field. This polarization, accompanying the electron motion, excites LO phonons, leading to the renormalization of the electron ground state energy and effective mass, forming a quasi-particle coupled with phonons—polaron. Please discuss the classification of polarons. Then use the perturbation method starting from the Hamiltonian to calculate the effective mass of the polaron in the case of slow phonons $(\\mathbf{k}\\rightarrow 0)$.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$m^*$": "Effective mass of the polaron.", + "$m$": "Effective mass of the band electron.", + "$\\alpha$": "Dimensionless electron-LO phonon coupling constant, defined as $\\alpha = \\frac{e^2}{2\\hbar\\omega_L} \\left( \\frac{2m\\omega_L}{\\hbar} \\right)^{1/2} \\left( \\frac{1}{\\epsilon_\\infty} - \\frac{1}{\\epsilon_0} \\right)$." + } + }, + { + "id": 499, + "context": "The longitudinal optical (LO) phonon mode in an ionic crystal generates a polarization field, which strongly couples with the conduction electrons in the ionic crystal. This coupling is much stronger than the effect of longitudinal acoustic (LA) phonons (which represent center of mass motion and do not generate a polarization field) on the conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier characteristics in ionic crystals.", + "question": "Please calculate the average number of virtual phonons excited around the electron.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\langle N_{\\text{ph}} \\rangle$": "Average number of virtual phonons excited around the electron", + "$\\alpha$": "Fröhlich coupling constant" + } + }, + { + "id": 500, + "context": "Effective interaction of current electronic exchange virtual phonons\n \\begin{equation*}\n H_{\\text{eff}} = \\frac{1}{2} \\sum_{\\substack{\\mathbf{k}_1, \\mathbf{k}_2 \\\\ q_1, q_2}} V_{\\mathbf{k}_1, \\mathbf{q}} C^\\dagger_{\\mathbf{k}_1 + \\mathbf{q}, q_1} C^\\dagger_{\\mathbf{k}_2 - \\mathbf{q}, q_2} C_{\\mathbf{k}_2, q_2} C_{\\mathbf{k}_1, q_1} c\n \\end{equation*}\n \n The interaction coefficient is:\n \\begin{equation*}\n V_{\\mathbf{k}_1, \\mathbf{q}} = |D_{\\mathbf{q}}|^2 \\frac{2\\hbar\\omega_{\\mathbf{q}}}{(E_{\\mathbf{k}_1 + \\mathbf{q}} - E_{\\mathbf{k}_1})^2 - (\\hbar\\omega_{\\mathbf{q}})^2} \n \\end{equation*}", + "question": "Analyze the situation near the Fermi surface and describe the interaction when the attractive potential is greater than the screened Coulomb potential, and elaborate on the approximation method of BCS theory.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$V_{\\text{net}}$": "Net potential, sum of the attractive electron-phonon mediated potential and the repulsive screened Coulomb potential.", + "$V_{\\mathbf{k}_1,\\mathbf{q}}$": "Interaction coefficient/potential for electron-phonon mediated interaction.", + "$e$": "Elementary charge of an electron.", + "$q$": "Magnitude of the momentum transfer wave vector $\\mathbf{q}$.", + "$\\lambda$": "Screening length parameter in the Coulomb potential." + } + }, + { + "id": 501, + "context": "Please discuss the interaction problem of adding two electrons outside a filled Fermi sea at $T=0K$. Approximation: You can assume that the electrons inside the Fermi sea are free electrons.", + "question": "Please calculate the interaction between two electrons in the weak coupling case.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$E$": "Energy of the two additional electrons", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega_D$": "Debye frequency", + "$g(0)$": "Density of states at the Fermi surface for one spin orientation", + "$V$": "Interaction strength (attractive interaction potential between electrons)" + } + }, + { + "id": 502, + "context": "It is known that the BCS superconducting Hamiltonian can be written as:\n \\begin{equation} \\label{eq:6.4.23_again}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the superconducting energy gap, and $E_s(0)$ denotes the ground state energy, given by the expression:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n with:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please solve the following problem based on the Hamiltonian:", + "question": "Calculate the superconducting gap $\\Delta$ at zero temperature;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta$": "Superconducting energy gap", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega_D$": "Debye frequency", + "$g(0)$": "Density of states for one spin orientation at the Fermi surface", + "$V$": "Interaction strength (attractive interaction between electrons)" + } + }, + { + "id": 503, + "context": "The known Hamiltonian of BCS superconductors can be expressed as:\n \\begin{equation} \\label{eq:6.4.23_again}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n Here, $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, and $E_s(0)$ is the ground state energy, expressed as:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please solve the following problem starting from the Hamiltonian:", + "question": "Please calculate the superconducting critical temperature $T_c$ based on the gap equation.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$k_B$": "Boltzmann constant", + "$T_c$": "Superconducting critical temperature", + "$\\mathrm{e}$": "Base of natural logarithm", + "$\\gamma$": "Euler's constant, $\\gamma \\approx 0.5772$", + "$\\pi$": "Pi constant", + "$\\hbar$": "Reduced Planck's constant", + "$\\omega_D$": "Debye frequency", + "$g(0)$": "Density of states at the Fermi level", + "$V$": "Interaction strength in the gap equation" + } + }, + { + "id": 504, + "context": "It is known that BCS superconducting Hamiltonian can be written as: \\begin{equation} \\label{eq:6.4.23_context} \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}), \\end{equation} where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ denotes the ground state energy, given by: \\begin{equation} E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V}, \\end{equation} where: \\begin{equation*} \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F), \\end{equation*} Please solve the following problem starting from the Hamiltonian:", + "question": "At $T \\ll T_c$, calculate the approximate expression for $\\Delta(T)$ according to the previous results", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta(T)$": "Temperature-dependent energy gap", + "$\\Delta(0)$": "Energy gap at zero temperature", + "$\\pi$": "Mathematical constant pi", + "$k_B$": "Boltzmann constant", + "$T$": "Temperature" + } + }, + { + "id": 505, + "context": "The Hamiltonian of known BCS superconductivity can be written as:\n \\begin{equation} \\label{eq:6.4.23_context}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ represents the ground state energy, expressed as:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please complete the following problem starting from the Hamiltonian:", + "question": "Calculate the approximate expression for $\\Delta(T)$ as $T \\rightarrow T_c$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$\\Delta(T)$": "Temperature-dependent energy gap", + "$k_B$": "Boltzmann constant", + "$T_c$": "Critical temperature for superconductivity", + "$T$": "Temperature", + "$\\pi$": "Mathematical constant pi", + "$\\zeta(3)$": "Value of the Riemann zeta function at 3" + } + }, + { + "id": 506, + "context": "It is known that the Hamiltonian for BCS superconductors is \n\n \\begin{equation}\n \\bar{H} = E_s(0) + \\sum_k \\sqrt{\\epsilon_k^2 + \\Delta^2} (\\alpha_k^\\dagger \\alpha_k + \\alpha_{-k}^\\dagger \\alpha_{-k}),\n \\end{equation}\n where $\\alpha$ is the quasiparticle operator, $\\Delta$ indicates the quasiparticle energy gap, $E_s(0)$ represents the ground state energy, and the expression is:\n \\begin{equation}\n E_s(0) = 2 \\sum_k \\epsilon_k v_k^2 - 2\\Delta \\sum_k u_k v_k + \\frac{\\Delta^2}{V},\n \\end{equation}\n where:\n \\begin{equation*}\n \\xi_k = \\sqrt{\\epsilon_k^2 + \\Delta^2} \\quad (\\epsilon_k = E_k - E_F),\n \\end{equation*}\n\n Please calculate:", + "question": "In the case $T \\ll T_c$, the low-temperature approximation of the electronic specific heat", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Superconductivity", + "final_symbol": { + "$c_{es}$": "Electronic specific heat of superconductors", + "$g(0)$": "Density of states at the Fermi level (or at $\\epsilon=0$)", + "$\\Delta(0)$": "Quasiparticle energy gap at zero temperature", + "$T$": "Temperature", + "$k_B$": "Boltzmann constant" + } + }, + { + "id": 507, + "context": "Given the ferromagnetic Heisenberg model $H = -J \\sum_{l,\\delta}S_l \\cdot S_{l+\\delta}$, where $\\delta$ represents the difference in position between neighboring lattice sites, discuss the case of $J>0$. Please perform the following calculations:", + "question": "Solve for the ground state energy of the Heisenberg model.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Magnetism", + "final_symbol": { + "$E_0$": "Ground state energy eigenvalue of the Hamiltonian $H$.", + "$J$": "Exchange coupling constant.", + "$N$": "Total number of lattice sites.", + "$Z$": "Coordination number of the lattice.", + "$S$": "Total spin quantum number (ion spin) for a lattice site." + } + }, + { + "id": 508, + "context": "Consider the action of the Hubbard model: \\begin{align}\n S_{\\text{loc}}[\\Phi^\\dagger, \\Phi] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger(\\partial_\\tau - \\mu - i\\Delta_c - \\Delta\\sigma^z)\\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_1[\\Phi^\\dagger, \\Phi, \\Omega] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger \\dot{R}_{\\mathbf{r}} \\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_2[\\Phi^\\dagger, \\Phi, \\Omega] &= - \\int_0^\\beta d\\tau \\sum_{\\mathbf{r},\\mathbf{r}'} t_{\\mathbf{r}\\mathbf{r}'} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}'} \\Phi_{\\mathbf{r}'}.\n \\label{eq:6.141}\n \\end{align}\nWhere: \\begin{equation} R_{\\mathbf{r}} = e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}\\sigma^z} e^{-\\frac{i}{2}\\theta_{\\mathbf{r}}\\sigma^y} e^{-\\frac{i}{2}\\psi_{\\mathbf{r}}\\sigma^z} = \\begin{pmatrix} \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} & -\\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\\\ \\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} & \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\end{pmatrix}, \\end{equation} \nWe consider the fields $\\delta_c$ and $\\delta$ at the saddle-point approximation: $\\delta_c = i(U/2)\\langle\\phi_{\\mathbf{r}}^\\dagger\\phi_{\\mathbf{r}}\\rangle$ and $\\delta = (U/2)\\langle\\phi_{\\mathbf{r}}^\\dagger\\sigma^z\\phi_{\\mathbf{r}}\\rangle$, for the half-filled case, $\\mu + i\\delta_c = 0$. Please calculate", + "question": "First-order moment $\\langle S_2 \\rangle$", + "answer": "", + "final_answer": [], + "answer_type": "Numeric", + "topic": "Strongly Correlated Systems", + "final_symbol": {} + }, + { + "id": 509, + "context": "The first-order perturbation calculation for the Hamiltonian of interacting electron systems \\begin{equation}H = H_0 + H' = \\sum_{\\mathbf{k},\\sigma} \\frac{\\hbar^2 k^2}{2m} C_{\\mathbf{k}\\sigma}^{\\dagger} C_{\\mathbf{k}\\sigma} + \\frac{1}{2V} \\sum_{\\mathbf{q}} v(q) (\\rho_{\\mathbf{q}}^{\\dagger} \\rho_{\\mathbf{q}} - N) \\label{eq:4.9.1}\\end{equation} is called the Hartree-Fock approximation. Therefore, within the Hartree-Fock approximation, one only needs to take the diagonal average of $H$ with the ground state of non-interacting electrons (Fermi surface state) \\begin{equation}|0\\rangle_0 = \\prod_{k \\le k_F, \\sigma} C_{\\mathbf{k}\\sigma}^{\\dagger} |Vac\\rangle \\label{eq:4.9.2}\\end{equation}. Here, $|Vac\\rangle$ represents the state where all $\\mathbf{k}$ spaces are unoccupied, i.e., the true vacuum state.", + "question": "The average ground state energy per electron under the Hartree-Fock approximation (expressed in terms of Fermi wave vector);", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\hbar$": "Reduced Planck's constant", + "$k_F$": "Fermi wave vector", + "$m$": "Mass of the electron", + "$e$": "Elementary charge of an electron", + "$\\pi$": "Mathematical constant pi" + } + }, + { + "id": 510, + "context": "In ionic crystals, long-wavelength optical modes represent the opposite motion of positive and negative ions within a unit cell, accompanied by polarization and interacting strongly with electromagnetic waves, thus having an important impact on the electrical and optical properties of ionic crystals. For simplicity, assume each unit cell contains only two ions with equal charge magnitude but opposite sign, still confined to an isotropic continuum model. Since, in the long-wavelength limit, the relative displacement $(u_+ - u_-)$ of the positive and negative ions within each unit cell is almost the same, a vector $\\mathbf{W}$ can be used to describe the optical branch vibration \\begin{equation} \\mathbf{W} \\equiv \\rho^{1/2} (u_+ - u_-) \\label{eq:2.6.1} \\end{equation} here $\\rho$ represents the reduced mass density \\begin{equation} \\rho = \\frac{M}{\\omega}, \\quad M = \\frac{M_+ M_-}{M_+ + M_-} \\label{eq:2.6.2} \\end{equation} $M_\\pm$ are the masses of positive and negative ions, $M$ is the reduced mass. The vector $\\mathbf{W}$ can be called the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, and satisfy $P = \\gamma_{12} W + \\gamma_{22}E$, with elastic energy as $\\frac{1}{2}\\gamma_{11}\\mathbf{W}\\cdot \\mathbf{W}$ Please complete the following question:", + "question": "Calculate the Hamiltonian density", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\dot{\\mathbf{W}}$": "Time derivative of the reduced displacement vector $\\mathbf{W}$", + "$\\gamma_{11}$": "Coefficient for the elastic energy density, and appearing in the total potential energy density", + "$\\mathbf{W}$": "Vector describing the optical branch vibration, also called the reduced displacement, defined as $\\mathbf{W} \\equiv \\rho^{1/2} (u_+ - u_-)$", + "$\\gamma_{12}$": "Coefficient relating polarization intensity to reduced displacement and electric field, and appearing in the potential energy density", + "$\\mathbf{E}$": "Macroscopic electric field", + "$\\gamma_{22}$": "Coefficient relating polarization intensity to the electric field, and appearing in the potential energy density" + } + }, + { + "id": 511, + "context": "In an ionic crystal, long-wavelength optical modes represent the opposite movement of positive and negative ions within the unit cell, accompanied by polarization and strong interaction with electromagnetic waves, thus having an important impact on the electrical and optical properties of the ionic crystal. For simplicity, assume that each unit cell contains only two ions with equal and opposite charges, still restricted to the isotropic continuous model. Because the relative displacement of positive and negative ions $(u_+ - u_-)$ in each unit cell at the long-wavelength limit is nearly the same, a vector $\\mathbf{W}$ can describe the optical branch vibration \\begin{equation} \\mathbf{W} \\equiv \\rho^{1/2} (u_+ - u_-) \\label{eq:2.6.1} \\end{equation} where $\\rho$ represents the density of the reduced mass \\begin{equation} \\rho = \\frac{M}{\\Omega}, \\quad M = \\frac{M_+ M_-}{M_+ + M_-} \\label{eq:2.6.2} \\end{equation} $M_\\pm$ are the masses of the positive and negative ions, and $M$ is the reduced mass. The vector $\\mathbf{W}$ can be termed as the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, satisfying $P = \\gamma_{12} W + \\gamma_{22}E$, and the elastic energy is $\\frac{1}{2}\\gamma_{11}\\mathbf{W}\\cdot \\mathbf{W}$. Please complete the following question:", + "question": "Calculate the squared transverse vibration frequency $\\omega^2_L$", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Others", + "final_symbol": { + "$\\gamma_{11}$": "Coefficient for the elastic energy, elastic energy is $\\frac{1}{2}\\gamma_{11}\\mathbf{W}\\cdot \\mathbf{W}$.", + "$\\gamma_{12}$": "Coefficient relating polarization intensity to reduced displacement and macroscopic field, $P = \\gamma_{12} W + \\gamma_{22}E$.", + "$\\gamma_{22}$": "Coefficient relating polarization intensity to macroscopic field, $P = \\gamma_{12} W + \\gamma_{22}E$." + } + }, + { + "id": 512, + "context": "The problem of motion in the Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi = $ constant and $\\eta = $ constant are rotational paraboloids around the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following question:", + "question": "Write the line element in parabolic coordinates", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\xi$": "Parabolic coordinate, ranging from 0 to $\\infty$", + "$\\eta$": "Parabolic coordinate, ranging from 0 to $\\infty$", + "$\\varphi$": "Parabolic coordinate, azimuthal angle, ranging from 0 to $2\\pi$", + "$d\\xi$": "Differential change in the parabolic coordinate $\\xi$", + "$d\\eta$": "Differential change in the parabolic coordinate $\\eta$", + "$d\\varphi$": "Differential change in the parabolic coordinate $\\varphi$" + } + }, + { + "id": 513, + "context": "The problem of motion in a Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following equations: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi =$ constant and $\\eta =$ constant are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem:", + "question": "Write out the Laplacian in parabolic coordinates", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\xi$": "Parabolic coordinate, defined as $\\xi = r + z$", + "$\\eta$": "Parabolic coordinate, defined as $\\eta = r - z$", + "$\\varphi$": "Parabolic coordinate, azimuthal angle, defined as $\\varphi = \\arctan\\left(\\frac{y}{x}\\right)$" + } + }, + { + "id": 514, + "context": "The problem of motion in a Coulomb field can be well handled in parabolic coordinates, where the parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. Surfaces of constant $\\xi$ and constant $\\eta$ are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem:", + "question": "Write down the single-particle Schrödinger equation You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\xi$": "One of the parabolic coordinates, defined as $\\xi = r + z$.", + "$\\eta$": "One of the parabolic coordinates, defined as $\\eta = r - z$.", + "$\\psi$": "Wave function of the single particle.", + "$\\varphi$": "Azimuthal angle, one of the parabolic coordinates, defined as \\varphi = \\arctan(y/x).", + "$E$": "Total energy of the single particle." + } + }, + { + "id": 515, + "context": "The problem of motion in a Coulomb field can be well-handled in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined as follows: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} or conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi =$ constant and $\\eta =$ constant are rotational paraboloids around the $z$-axis, with the origin as the focus. This is an orthogonal coordinate system. Please complete the following task:", + "question": "Use the method of separation of variables to solve the Schrödinger equation corresponding to the discrete spectrum;", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$n$": "Principal quantum number, $n = \\frac{1}{\\sqrt{-2E}}$", + "$\\xi$": "Parabolic coordinate, $\\xi = r + z$", + "$\\eta$": "Parabolic coordinate, $\\eta = r - z$", + "$m$": "Magnetic quantum number", + "$L_{n_1}^{|m|}$": "Associated Laguerre polynomial", + "$L_{n_2}^{|m|}$": "Associated Laguerre polynomial", + "$n_1$": "Parabolic quantum number, $n_1 = \\frac{|m|+1}{2} + n\\beta_1$", + "$n_2$": "Parabolic quantum number, $n_2 = \\frac{|m|+1}{2} + n\\beta_2$", + "$\\varphi$": "Azimuthal angle in parabolic coordinates, $\\varphi = \\arctan\\left(\\frac{y}{x}\\right)$" + } + }, + { + "id": 516, + "context": "The problem of motion in a Coulomb field can be well treated in parabolic coordinates. The parabolic coordinate system $\\xi, \\eta, \\varphi$ is defined by the following: \\begin{align} x &= \\sqrt{\\xi \\eta} \\cos \\varphi, \\quad y = \\sqrt{\\xi \\eta} \\sin \\varphi, \\quad z = \\frac{1}{2}(\\xi - \\eta), \\ r &= \\sqrt{x^2 + y^2 + z^2} = \\frac{1}{2}(\\xi + \\eta). \\label{eq:37.1} \\end{align} Conversely: \\begin{equation} \\xi = r + z, \\quad \\eta = r - z, \\quad \\varphi = \\arctan\\left(\\frac{y}{x}\\right); \\label{eq:37.2} \\end{equation} The values of $\\xi$ and $\\eta$ can range from 0 to $\\infty$, and $\\varphi$ ranges from 0 to $2\\pi$. The surfaces $\\xi = $ constant and $\\eta = $ constant are rotational paraboloids about the $z$-axis centered at the origin. This forms an orthogonal coordinate system. Please complete the following problem:", + "question": "Assuming a hydrogen atom in a uniform electric field, solve for the energy level correction to second order approximation (Stark effect).", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$E$": "Energy of the system.", + "$n$": "Principal quantum number, related to $n_1, n_2, m$ by $n = n_1 + n_2 + |m| + 1$ for the hydrogen atom.", + "$\\mathscr{E}$": "Strength of the uniform electric field.", + "$n_1$": "Parabolic quantum number associated with the $\\xi$ coordinate.", + "$n_2$": "Parabolic quantum number associated with the $\\eta$ coordinate.", + "$m$": "Magnetic quantum number." + } + }, + { + "id": 517, + "context": "We discuss the propagation of light in conductive media, considering a uniform isotropic medium whose dielectric constant is $\\varepsilon$, magnetic permeability is $\\mu$, and conductivity is $\\sigma_0$. Utilizing the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, the Maxwell equations take the following form: \\begin{align}\n\\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\label{eq:1} \\\\\n\\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\label{eq:2} \\\\\n\\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\label{eq:3} \\\\\n\\text{div } \\mathbf{H} &= 0. \\label{eq:4}\n\\end{align} Please calculate:", + "question": "Charge density within a metal", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$\\rho_0$": "Initial charge density at time $t=0$.", + "$t$": "Time.", + "$\\varepsilon$": "Dielectric constant of the uniform isotropic medium.", + "$\\sigma$": "Conductivity of the uniform isotropic medium." + } + }, + { + "id": 518, + "context": "We discuss the propagation of light in conductive media, considering a homogeneous isotropic medium with a dielectric constant $\\varepsilon$, magnetic permeability $\\mu$, and conductivity $\\sigma_0$. Using the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, Maxwell's equations take the following form:\n\n\\begin{align}\n\\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\label{eq:1} \\\\\n\\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\label{eq:2} \\\\\n\\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\label{eq:3} \\\\\n\\text{div } \\mathbf{H} &= 0. \\label{eq:4}\n\\end{align}", + "question": "Introduce the complex dielectric constant $\\hat{\\epsilon} = \\epsilon + i \\frac{4\\pi \\sigma}{\\omega}$, the complex phase velocity $\\hat{v} = \\frac{c}{\\sqrt{\\mu\\hat{\\epsilon}}}$, and the complex refractive index $\\hat{n} = \\frac{c}{\\hat{v}} = \\frac{c}{\\omega}k$, let $\\hat{n}=n(1+i\\kappa)$, where $\\kappa$ is the attenuation coefficient. Please express the refractive index $n$ using the material constants $\\epsilon,\\mu,\\sigma$. You should return your answer as an equation.", + "answer": "", + "final_answer": [], + "answer_type": "Equation", + "topic": "Theoretical Foundations", + "final_symbol": { + "$n$": "Refractive index (real part of complex refractive index).", + "$\\mu$": "Magnetic permeability of the medium.", + "$\\varepsilon$": "Dielectric constant of the medium.", + "$\\sigma$": "Conductivity of the medium.", + "$v$": "Frequency, related to angular frequency by $v = \\omega / (2\\pi)$." + } + }, + { + "id": 519, + "context": "We discuss the propagation of light in conducting media, considering a homogeneous isotropic medium with dielectric constant $\\varepsilon$, magnetic permeability $\\mu$, and conductivity $\\sigma_0$. Using the material equations $\\mathbf{j} = \\sigma \\mathbf{E}$, $\\mathbf{D} = \\varepsilon \\mathbf{E}$, $\\mathbf{B} = \\mu \\mathbf{H}$, the Maxwell equations take the following form:\n\\begin{align} \\text{curl } \\mathbf{H} - \\frac{\\varepsilon}{c} \\dot{\\mathbf{E}} &= \\frac{4\\pi}{c} \\sigma \\mathbf{E}, \\\\ \\text{curl } \\mathbf{E} + \\frac{\\mu}{c} \\dot{\\mathbf{H}} &= 0, \\\\ \\text{div } \\mathbf{E} &= \\frac{4\\pi}{\\varepsilon} \\rho, \\\\ \\text{div } \\mathbf{H} &= 0. \\end{align} \n\nPlease answer the following question.", + "question": "If the electric field is a plane wave, calculate the energy density of the wave.", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Theoretical Foundations", + "final_symbol": { + "$w_0$": "Initial energy density of the wave (amplitude of the energy density at $\\mathbf{r}=0$).", + "$\\chi$": "Attenuation coefficient, defined as $\\chi = \\frac{2\\omega}{c} n\\kappa$.", + "$|\\mathbf{r}|$": "Magnitude of the position vector." + } + }, + { + "id": 520, + "context": "", + "question": "Consider the action of the Hubbard model:\n\\begin{align}\n S_{\\text{loc}}[\\Phi^\\dagger, \\Phi] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger(\\partial_\\tau - \\mu - i\\Delta_c - \\Delta\\sigma^z)\\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_1[\\Phi^\\dagger, \\Phi, \\Omega] &= \\int_0^\\beta d\\tau \\sum_{\\mathbf{r}} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger \\dot{R}_{\\mathbf{r}} \\Phi_{\\mathbf{r}}, \\nonumber \\\\\n S_2[\\Phi^\\dagger, \\Phi, \\Omega] &= - \\int_0^\\beta d\\tau \\sum_{\\mathbf{r},\\mathbf{r}'} t_{\\mathbf{r}\\mathbf{r}'} \\Phi_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}}^\\dagger R_{\\mathbf{r}'} \\Phi_{\\mathbf{r}'}.\n \\label{eq:6.141}\n\\end{align}\nwhere:\n\\begin{equation}\n R_{\\mathbf{r}} = e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}\\sigma^z} e^{-\\frac{i}{2}\\theta_{\\mathbf{r}}\\sigma^y} e^{-\\frac{i}{2}\\psi_{\\mathbf{r}}\\sigma^z} = \\begin{pmatrix} \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} & -\\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{-\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\\\ \\sin\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} & \\cos\\left(\\frac{\\theta_{\\mathbf{r}}}{2}\\right)e^{\\frac{i}{2}\\varphi_{\\mathbf{r}}} \\end{pmatrix},\n\\end{equation}\nWe consider the fields $\\Delta_c$ and $\\Delta$ at the saddle-point approximation level: $\\Delta_c = i(U/2)\\langle\\Phi_{\\mathbf{r}}^\\dagger\\Phi_{\\mathbf{r}}\\rangle$ and $\\Delta = (U/2)\\langle\\Phi_{\\mathbf{r}}^\\dagger\\sigma^z\\Phi_{\\mathbf{r}}\\rangle$. At half-filling, we have $\\mu + i\\Delta_c = 0$.\n\nPlease answer the problem\n\\begin{enumerate}\n \\item the effective action $S[\\Omega]$ accurate to the first order in $\\partial_\\tau$ and the second order in $t$:\n \\begin{equation}\n S[\\Omega] = \\langle S_1 + S_2 \\rangle - \\frac{1}{2}\\langle S_2^2 \\rangle_c,\n \\label{eq:6.142}\n \\end{equation}\n where the expectation value $\\langle \\cdots \\rangle$ is taken with respect to the local action $S_{\\text{loc}}$.\n\n This coincides with the action of the spin- $1/2$ Heisenberg model, what is its exchange coupling $J$?\n\\end{enumerate}", + "answer": "", + "final_answer": [], + "answer_type": "Expression", + "topic": "Strongly Correlated Systems", + "final_symbol": { + "$J$": "Exchange coupling", + "$t$": "Hopping parameter (simplified notation for $t_{\\mathbf{r}\\mathbf{r}'}$ for nearest neighbors)", + "$U$": "On-site Coulomb interaction strength" + } + } +] \ No newline at end of file