id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
201 | Try to find the effective mass of electrons at the top of the band. | [] | Expression | Semiconductors | $m_0$: Mass of the electron (rest mass) | |
202 | Given the lattice constant of a two-dimensional square lattice is $a$, if the electron energy can be expressed as
$$
E(k)=\frac{h^{2}(k_{x}^{2}+k_{y}^{2})}{2 m_{\mathrm{n}}^{*}}
$$
Try to find the density of states. | [] | Expression | Semiconductors | $g(E)$: 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 | |
203 | Calculate the number of quantum states per unit volume between the energy $E=E_{\mathrm{c}}$ and $E=E_{\mathrm{c}}+100(\frac{h^{2}}{8 m_{\mathrm{n}}^{*} L^{2}})$. | [] | Expression | Semiconductors | $\pi$: Mathematical constant pi
$L$: Characteristic length, representing the dimension of the system | |
204 | 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. | [] | Expression | Semiconductors | $E_{\mathrm{F} 2}$: Fermi level of the second piece of material
$E_{\mathrm{c}}$: Conduction band edge
$k_0$: Boltzmann constant
$T$: Temperature | |
205 | 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. | [] | Expression | Semiconductors | $p_0$: 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. | |
206 | 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. | [] | Expression | Semiconductors | $n_0$: 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 | |
207 | 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. | [] | Expression | Semiconductors | $p_{0}$: 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 | |
208 | 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. | [] | Expression | Semiconductors | $f$: 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. | |
209 | 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. | [] | Expression | Semiconductors | $\tau_{\max }$: 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. | |
210 | Assume $\tau_{\mathrm{p}}=\tau_{\mathrm{n}}=\tau_{0}$, and based on the small signal lifetime formula
\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}}
discuss 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. | [] | Expression | Semiconductors | $\tau$: 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. | |
211 | 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:
\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}}
where $\tau_{\mathrm{n}}$ 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 $\tau_{\mathrm{n}}^{\prime}$. | [] | Expression | Semiconductors | $\tau_{\mathrm{n}}$: 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 $n_1 = N_{\mathrm{c}} \exp (-(E_{\mathrm{c}}-E_{\mathrm{t}})/(k_{0} T))$. | |
212 | Derive the small-signal lifetime formula. | [] | Expression | Semiconductors | $\tau$: 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. | |
213 | 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 ($t=\Delta t$). | [] | Expression | Semiconductors | $g_{\mathrm{p}}$: Generation rate of nonequilibrium carriers.
$\tau_{\mathrm{p}}$: Lifetime of nonequilibrium holes.
$\mathrm{e}$: Base of the natural logarithm. | |
214 | 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. | [] | Expression | Semiconductors | $D^{*}$: bipolar diffusion coefficient
$D_{\mathrm{n}}$: electron diffusion coefficient
$D_{\mathrm{p}}$: hole diffusion coefficient | |
215 | Determine the expression for the hole diffusion current variation with $x$ in the n region. | [] | Expression | Semiconductors | $I_{\mathrm{pD}}$: Hole diffusion current.
$x$: Position. | |
216 | Determine the expression for the electron diffusion current variation with $x$ in the n-region. | [] | Expression | Semiconductors | $I_{\mathrm{nD}}$: Electron diffusion current
$x$: Position coordinate | |
217 | Determine the expression for the variation of electron drift current with $x$ in the n-region. | [] | Expression | Semiconductors | $I_{\mathrm{nt}}$: Electron drift current
$\mathrm{e}$: Base of the natural logarithm, approximately $2.71828$
$x$: Position coordinate | |
218 | Determine the expression for the total electron current in the n-region as a function of $x$. | [] | Expression | Semiconductors | $I_{\mathrm{n}}$: Total electron current, defined as $I_{\mathrm{n}}=I_{\mathrm{nt}}+I_{\mathrm{nD}}$
$x$: Position coordinate | |
219 | 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. | [] | Expression | Semiconductors | $C$: 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 $N_{\mathrm{D}}$. | |
220 | Find the expression for depletion layer width $X_{\mathrm{d}}$ when $V_{\mathrm{s}}=0.4 \mathrm{~V}$; | [] | Expression | Semiconductors | $X_{\mathrm{d}}$: 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 | |
221 | Find the expression for the maximum depletion layer width $X_{\mathrm{dm}}$ when $V_{\mathrm{s}}=0.4 \mathrm{~V}$; | [] | Expression | Semiconductors | $X_{\mathrm{dm}}$: 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. | |
222 | Given the resistivity of the material is $1.5 \times 10^{-3} \Omega \cdot \mathrm{~cm}$, find the carrier mobility. | [] | Expression | Semiconductors | $\mu_{\mathrm{H}}$: Hall mobility of the carriers, defined as $\mu_{\mathrm{H}}= |R_{\mathrm{H}} \sigma_{0}|=\frac{1}{\rho_{0}}|R_{\mathrm{H}}|$ | |
223 | Assume the relaxation time $\tau$ is constant, and try to calculate the Hall coefficient of an n-type semiconductor. | [] | Expression | Semiconductors | $R$: Hall coefficient.
$n$: Electron concentration (number density).
$e$: Elementary charge of an electron. | |
224 | Find the ratio $f_{\mathrm{c}}=J_{\mathrm{ny}} / J_{x}$. | [] | Expression | Semiconductors | $f_{\mathrm{c}}$: 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, $\sigma_{\mathrm{n}} = n e \mu_{\mathrm{n}}$ | |
225 | If the Hall angle of a sample is measured to be $\theta=0$, find the corresponding electrical conductivity; | [] | Expression | Semiconductors | $\sigma$: 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) | |
226 | Find the ratio of the transverse electron current to the primary current at equilibrium $f_{\mathrm{e}}=\frac{J_{\mathrm{ey}}}{J_{x}}$. | [] | Expression | Semiconductors | $f_{\mathrm{e}}$: 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. | |
227 | Derive the formula for the density of st ates of a two-dimensional free gas. | [] | Expression | Others | $g(E)$: 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 | |
228 | 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? | [] | Expression | Others | $D$: Degeneracy of the Landau levels
$e$: Elementary charge
$B$: Applied magnetic field perpendicular to the square lattice
$\hbar$: Reduced Planck's constant | |
229 | A particle is incident with kinetic energy $E$, subjected to the following double $\delta$ potential barriers:
V(x)=V_{0}[\delta(x)+\delta(x-a)]
Find the expression for the conditions under which complete transmission occurs. You should return your answer as an equation. | [] | Equation | Theoretical Foundations | $\mathrm{e}$: 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 $\theta=\frac{2 a k}{C}=\frac{k \hbar^{2}}{m V_{0}}$. | |
230 | For the energy eigenstate $|n\rangle$ of the harmonic oscillator, calculate the expression for the uncertainty product $\Delta x \cdot \Delta p$. | [] | Expression | Theoretical Foundations | $\Delta x$: Uncertainty in position.
$\Delta p$: Uncertainty in momentum.
$n$: Quantum number representing the energy level of the harmonic oscillator.
$\hbar$: Reduced Planck's constant. | |
231 | In the coherent state $|\alpha\rangle$, calculate the uncertainty product $\Delta x \cdot \Delta p$. | [] | Expression | Theoretical Foundations | $\Delta x \cdot \Delta p$: Uncertainty product of position and momentum.
$\hbar$: Reduced Planck's constant. | |
232 | Define the radial momentum operator
\begin{equation*}
\boldsymbol{p}_{r}=\frac{1}{2}(\frac{\boldsymbol{r}}{r} \cdot \boldsymbol{p}+\boldsymbol{p} \cdot \frac{\boldsymbol{r}}{r}) \tag{1}
\end{equation*}
Find the commutation relation $[r, p_{r}]$. | [] | Expression | Theoretical Foundations | $r$: 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 | |
233 | A particle with mass $\mu$ moves in a central potential field
\begin{equation*}
V(r)=\lambda r^{\nu}, \quad-2<\nu<\infty \tag{1}
\end{equation*}
We 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:
\begin{equation*}
\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}
\end{equation*}
By 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. | [] | Equation | Theoretical Foundations | $w$: 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 | |
234 | For the hydrogen atom's s states $(n l m=n 00)$, calculate the expression for the uncertainty product $\Delta x \cdot \Delta p_{x}$. | [] | Expression | Theoretical Foundations | $\Delta x$: 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. | |
235 | 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
\begin{equation*}
\psi(r, \theta, \varphi, s_{z})=\alpha Y_{l 0}(\theta, \varphi) R(r)
\end{equation*}
find the only possible measurement value of the total angular momentum $j_{z}$ (taking $\hbar=1$). | [] | Numeric | Theoretical Foundations | ||
236 | 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. | [] | Equation | Theoretical Foundations | $\hat{f}(x)$: An operator, a function of position $x$.
$x$: Position coordinate.
$a$: The period of the function $\hat{f}(x)$. | |
237 | 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. | [] | Equation | Theoretical Foundations | $\sigma_r$: 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 $j=l'-1/2$ | |
238 | 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. | [] | Equation | Theoretical Foundations | $\boldsymbol{\sigma}_{1} \cdot \boldsymbol{\sigma}_{2}$: Dot product of the Pauli operators for particle 1 and particle 2, which is the quantity for which an algebraic equation is sought. | |
239 | Let us write a 4 -component Dirac field as
\psi(x)=\binom{\psi_{L}}{\psi_{R}}
and 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:
\psi_{L}(x)=\chi_{1}(x), \quad \psi_{R}(x)=i \sigma^{2} \chi_{2}^{*}(x)
From 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. | [] | Expression | Others | $\mathrm{i}$: 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 $\chi_2$. | |
240 | To begin, normalize the 16 matrices $\Gamma^{A}$ to the convention
$$
\operatorname{tr}[\Gamma^{A} \Gamma^{B}]=4 \delta^{A B} .
$$
This 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. | [] | Tuple | Others | $1$: 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. | |
241 | 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. | [] | Equation | Others | $C$: 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. | |
242 | QED accounts extremely well for the electron's anomalous magnetic moment. If $a=(g-2) / 2$,
$$
|a_{\text {expt. }}-a_{\mathrm{QED}}|<1 \times 10^{-10}
$$
What 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.
Hint: 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. | [] | Equation | Others | $\lambda$: 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 | |
243 | 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. | [] | Tuple | Others | $x_1$: 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, $s = q^2$. | |
244 | Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,
$$
\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^{2}-\frac{\lambda}{4!} \phi^{4}+\bar{\psi}(i \not \partial) \psi-i g \bar{\psi} \gamma^{5} \psi \phi,
$$
compute the Callan-Symanzik $\beta$ function for $g$:
$$
\beta_{g}(\lambda, g),
$$
to leading order in coupling constants, assuming that $\lambda$ and $g^{2}$ are of the same order. | [] | Expression | Others | $\beta_{g}$: Callan-Symanzik beta function for the coupling constant $g$.
$g$: Yukawa coupling constant.
$\pi$: Mathematical constant pi. | |
245 | Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,
$$
\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^{2}-\frac{\lambda}{4!} \phi^{4}+\bar{\psi}(i \not \partial) \psi-i g \bar{\psi} \gamma^{5} \psi \phi,
$$
compute the Callan-Symanzik $\beta$ function for $\lambda$:
\beta_{\lambda}(\lambda, g),
to leading order in coupling constants, assuming that $\lambda$ and $g^{2}$ are of the same order. | [] | Expression | Others | $\beta_{\lambda}$: Callan-Symanzik beta function for the coupling constant $\lambda$.
$\lambda$: Quartic coupling constant for the $\phi^4$ interaction.
$g$: Yukawa coupling constant. | |
246 | For a theory with two scalar fields $\phi_1$ and $\phi_2$ described by the Lagrangian:
$$
\mathcal{L}=\frac{1}{2}((\partial_{\mu} \phi_{1})^{2}+(\partial_{\mu} \phi_{2})^{2})-\frac{\lambda}{4!}(\phi_{1}^{4}+\phi_{2}^{4})-\frac{2\rho}{4!} \phi_{1}^{2} \phi_{2}^{2}
$$
Working in four dimensions, what is the $\beta$ function for the coupling constant $\lambda$, denoted as $\beta_{\lambda}$, to leading order in the coupling constants? | [] | Expression | Others | $\lambda$: Coupling constant.
$\rho$: Coupling constant.
$\pi$: Mathematical constant pi. | |
247 | State the leading term in $\gamma(\lambda)$ for $\phi^{4}$ theory. | [] | Expression | Others | $\gamma$: 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 | |
248 | 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
\[\gamma = (N+2)\,\frac{\lambda^{2}}{(4\pi)^{4}}.\] | [] | Numeric | Others | $\gamma$: Anomalous dimension
$\pi$: Mathematical constant pi | |
249 | Write down the dimension $d$ of $S U(N)$ group. You should return your answer as an equation. | [] | Equation | Others | $d$: Dimension of the $SU(N)$ group
$N$: Degree of the Special Unitary group | |
250 | Estimate, in order of magnitude, the value of the proton lifetime if the proton is allowed to decay through this interaction. | [] | Numeric | Others | ||
251 | Assume that the two Higgs fields couple to quarks by the set of fundamental couplings
\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. }
Find 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. | [] | Expression | Others | 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. | |
252 | 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. | [] | Equation | Others | $\Pi_{WW}$: 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 | |
253 | 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. | [] | Equation | Others | $\Pi_{ZZ}$: 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 | |
254 | 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. | [] | Equation | Theoretical Foundations | $\lambda$: 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 $n(r)=\sqrt{1-\frac{V(r)}{E}}$. | |
255 | The tensor force between two particles 1 and 2 of spin $1/2$ is associated with the operator $T_{12}$ given by:
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})
where $\boldsymbol{\sigma}_{1}$ 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.
If $\chi_{0,0}$ 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 $T_{12} \chi_{0,0}$. | [] | Numeric | Theoretical Foundations | ||
256 | 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). | [] | Numeric | Theoretical Foundations | ||
257 | The function
\begin{equation*}
\tilde{\varphi}(x)=\frac{1}{(1+\alpha x)^{2}} \tag{176.1}
\end{equation*}
with 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. | [] | Numeric | Theoretical Foundations | ||
258 | 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. | [] | Numeric | Theoretical Foundations | ||
259 | 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. | [] | Equation | Theoretical Foundations | $I_\alpha$: Intensity of Ly $\alpha$ emission.
$I_\beta$: Intensity of Ly $\beta$ emission. | |
260 | 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. | [] | Equation | Others | $n_{L}$: Number of loops
$n_{I}$: Number of internal lines
$n_{3}$: Number of trivalent vertices
$n_{4}$: Number of tetravalent vertices | |
261 | 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:
$$[\Lambda^{-1}]_{\nu}^{\mu} \epsilon_{ \pm}^{v}(\Lambda \mathbf{p}) = X$$
What 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? | [] | Expression | Others | $e^{\mp i \theta}$: 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. | |
262 | Give the expression of the one-loop self-energy in a $\phi^{4}$ theory in the Matsubara formalism. Calculate it in the limit $\beta \mathrm{m} \ll 1$. | [] | Expression | Others | $\lambda$: Coupling constant in $\phi^4$ theory.
$\Lambda$: Momentum cutoff for regularization.
$\pi$: Mathematical constant pi.
$T$: Temperature. | |
263 | 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. | [] | Tuple | Others | $\theta_+$: Grassmann variable
$\theta_-$: Grassmann variable | |
264 | 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. | [] | Tuple | Others | ||
265 | 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. | [] | Tuple | Others | ||
266 | 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 $\theta_-$. | [] | Numeric | Others | ||
267 | 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). | [] | Expression | Others | $g_{\mathrm{r}}$: 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. | |
268 | 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. | [] | Equation | Others | $n_f$: Number of Dirac fermions.
$n_s$: Number of complex scalar fields. | |
269 | Carry out explicitly the calculation of the functions $A$ and $B$ in
$$B(q^{2})=-i\mathrm{D}e\int\frac{d^{\mathrm{D}}l}{(2\pi)^{\mathrm{D}}}\int_{0}^{1}dx\frac{2\Delta(x)}{(l^{2}+\Delta(x))^{2}}.$$ | [] | Expression | Others | $B$: 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. | |
270 | Carry out explicitly the calculation of the functions $A$ and $B$ in
$$A(q^{2})=-i\mathrm{D}e\int\frac{d^{\mathrm{D}}l}{(2\pi)^{D}}\int_{0}^{1}dx\frac{\Delta(x)+(\frac{2}{\mathrm{D}}-1)l^{2}}{(l^{2}+\Delta(x))^{2}}, $$ | [] | Expression | Others | $A$: 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 | |
271 | 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. | [] | Tuple | Others | $\dot{z}^3$: 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 | |
272 | 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. | [] | Equation | Others | $\mathrm{L}'(x)$: 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 $\mathrm{N}^{-1} \operatorname{tr} ( P \exp \int_{0}^{\beta} d \tau A^{0}(\tau, x) )$ | |
273 | A particle with mass $m$, constrained to move freely on a ring with radius $R$, with an added perturbation
\begin{equation*}
H^{\prime}=V(\varphi)=\left\{\begin{array}{ll}
V_{1}, & -\alpha<\varphi<0 \\
V_{2}, & 0<\varphi<\alpha \\
0, & \text { other angles }
\end{array} \quad(\alpha<\pi)\right.
\end{equation*}
Find the first-order perturbation corrections for the three lowest energy levels. You should return your answer as a tuple format. | [] | Tuple | Others | $\alpha$: 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 $0 < \varphi < \alpha$ | |
274 | A small uniformly charged sphere acquires potential energy in an external electrostatic field
$$
\begin{equation*}
U(\boldsymbol{r})=V(\boldsymbol{r})+\frac{1}{6} r_{0}^{2} \nabla^{2} V(\boldsymbol{r})+\cdots
\end{equation*}
$$
where $r_{0}$ 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
$$
\begin{equation*}
V(\boldsymbol{r})=-\frac{e^{2}}{r}
\end{equation*}
$$
If 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]. | [] | Numeric | Others | ||
275 | A small uniformly charged sphere acquires potential energy in an external electrostatic field
$$
\begin{equation*}
U(\boldsymbol{r})=V(\boldsymbol{r})+\frac{1}{6} r_{0}^{2} \nabla^{2} V(\boldsymbol{r})+\cdots
\end{equation*}
$$
where $r_{0}$ 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
$$
\begin{equation*}
V(\boldsymbol{r})=-\frac{e^{2}}{r}
\end{equation*}
$$
If 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]. | [] | Numeric | Others | ||
276 | Take the ground state wave function as
$$
\psi_{0}(\boldsymbol{r}_{1}, \boldsymbol{r}_{2})=\psi_{0}(r_{1}) \psi_{0}(r_{2})
$$
where
$$
\begin{equation*}
\psi_{0}(r)=(\frac{\lambda^{3}}{\pi a_{0}^{3}})^{\frac{1}{2}} \mathrm{e}^{-x^{\prime} / a_{0}}, \quad a_{0}=\frac{\hbar^{2}}{m_{\mathrm{e}} e^{2}}.
\end{equation*}
$$
Calculate the ground state magnetic susceptibility of a helium atom, in the unit of $\mathrm{eV} /(\mathrm{Gs})^{2}$ | [] | Numeric | Others | ||
277 | 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
\begin{equation*}
H^{\prime}=V_{0}(x^{4}+y^{4}+z^{4}-\frac{3}{5} r^{4})
\end{equation*}
$H^{\prime}$ can be considered a perturbation. If the 3d state wave functions of the hydrogen atom (orthonormalized) are taken as
\begin{align*}
& \psi_{1}=\frac{1}{2}(y^{2}-z^{2}) f(r) \\
& \psi_{2}=\frac{1}{2 \sqrt{3}}(2 x^{2}-y^{2}-z^{2}) f(r) \\
& \psi_{3}=y z f(r) \\
& \psi_{4}=z x f(r) \\
& \psi_{5}=x y f(r)
\end{align*}
Under 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 $\psi_2$)? | [] | Numeric | Others | ||
278 | 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. | [] | Tuple | Others | $E_0$: 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 $j \neq k$. | |
279 | 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 $|\alpha\rangle, |\beta\rangle, |\gamma\rangle$. | [] | Expression | Others | $\alpha$: 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. | |
280 | 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. | [] | Tuple | Others | $E_0$: 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 | |
281 | 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. | [] | Numeric | Others | ||
282 | Which powers among the operators $\hat{s}$ (with any spin value $s$) are linearly independent? You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | $\hat{s}_z$: Z-component of the spin operator
$s$: Spin quantum number or spin value | |
283 | 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. | [] | Equation | Theoretical Foundations | $\hat{s}^{\lambda \mu}$: 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. | |
284 | 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. | [] | Tuple | Theoretical Foundations | ||
285 | Obtain the density matrix $\rho(q, q^{\prime})=\langle q| e^{-\beta \hat{H}}|q^{\prime}\rangle$ for the harmonic oscillator at finite temperature, $\beta=1 / T(k_{\mathrm{B}}=1)$. | [] | Expression | Others | $\rho$: 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 $\beta=1 / T$. | |
286 | A particle is moving in an infinite potential well $(-a<x<a)$. As an approximation for the ground state wave function, a trial wave function with three terms is used: $\psi(\lambda, x)=N[1+\lambda(\frac{x}{a})^{2}-(1+\lambda)(\frac{x}{a})^{4}] (|x|<a)$, where $N$ is the normalization constant, and $\lambda$ is the variational parameter. Use the variational method to find an approximate value for the ground state energy. | [] | Expression | Theoretical Foundations | $E$: Average value of the Hamiltonian, representing the approximate ground state energy.
$\lambda_1$: Value of the variational parameter $\lambda$ that minimizes the approximate ground state energy.
$\hbar$: Reduced Planck's constant.
$m$: Mass of the particle.
$a$: Half-width of the infinite potential well.
$\lambda$: Variational parameter. | |
287 | 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)?
The trial wave function is:
\begin{enumerate}
\item $\psi \sim e^{-\lambda r}$.
\end{enumerate} | [] | Numeric | Theoretical Foundations | ||
288 | Assume in the deuteron, the potential between the proton and neutron is expressed as
\begin{equation*}
V(r)=-V_{0} \mathrm{e}^{-r / a} \tag{1}
\end{equation*}
Take $V_{0}=32.7 \mathrm{MeV}, a=2.16 \mathrm{fm}$ (range of force). Use the variational method to find the ground state energy level of the deuteron.
The trial function is chosen as
\begin{equation*}
\psi(\lambda, r)=N \mathrm{e}^{-\lambda r / 2 a}
\end{equation*} | [] | Numeric | Theoretical Foundations | ||
289 | 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 $C_{12}$. | [] | Expression | Magnetism | $C_{12}$: 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 | |
290 | 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. | [] | Equation | Magnetism | $C_{22}$: 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 | |
291 | Consider a conductor with a sharp conical tip on its surface.
Using spherical coordinates, place the origin at the vertex of the conical tip, with the cone axis as the polar axis.
Let 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 $\varphi(r, \theta) = r^{n} f(\theta)$.
Based 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 $n$. | [] | Expression | Magnetism | $n$: 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. | |
292 | 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.
Hint: the potential is
$$
\varphi=-4\pi \sigma_0 \cdot z(1-\frac{R^{3}}{r^{3}})
$$ | [] | Expression | Magnetism | $\sigma$: 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. | |
293 | Find the charge distribution on a non-charged conductor disk (with radius $a$) that is parallel to a uniform external electric field ${ }^{(1)}$. | [] | Expression | Magnetism | $\sigma$: 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 | |
294 | 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 $K$. | [] | Expression | Magnetism | $\Delta V$: Change in volume
$V$: Original volume
$\mathfrak{C}$: Uniform external electric field
$K$: Bulk modulus of the material | |
295 | 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. | [] | Equation | Magnetism | $a$: 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 | |
296 | 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 $\frac{V-V_0}{V}$. | [] | Expression | Magnetism | $\mathfrak{C}$: Electric field strength (typo for $\mathfrak{E}$)
$\varepsilon$: Dielectric constant
$n$: Depolarization factor
$K$: Compressibility coefficient of the object
$P$: Pressure | |
297 | 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.
\footnotetext{
(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.
} | [] | Expression | Magnetism | $Q$: 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). | |
298 | 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. | [] | Expression | Magnetism | $T$: 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. | |
299 | 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 $\mathfrak{C}$.
Determine the corresponding mathematical expression for this difference. | [] | Expression | Magnetism | $T$: 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 | |
300 | 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. | [] | Expression | Magnetism | $E_{x}^{(i)}$: 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. |
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