id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
1 | For an oscillator with charge $q$, its energy operator without an external field is
\begin{equation*}
H_{0}=\frac{p^{2}}{2 m}+\frac{1}{2} m \omega^{2} x^{2}
\end{equation*}
If a uniform electric field $\mathscr{E}$ is applied, causing an additional force on the oscillator $f= q \mathscr{E}$, the total energy operato... | In $H_{0}$ and $H$, $p$ is the momentum operator,
$p=-\mathrm{i} \hbar \frac{\mathrm{~d}}{\mathrm{~d} x}$
The potential energy term in equation (2) can be rewritten as
$\frac{1}{2} m \omega^{2} x^{2}-q \mathscr{E} x=\frac{1}{2} m \omega^{2}[(x-x_{0})^{2}-x_{0}^{2}]$
where
\begin{equation*}
x_{0}=\frac{q \mathscr{... | [
"E_{n} =(n+\\frac{1}{2}) \\hbar \\omega-\\frac{q^{2} \\mathscr{E}^{2}}{2 m \\omega^{2}}"
] | Expression | Theoretical Foundations | $E_n$: 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 |
2 | A particle of mass $m$ is in the ground state of a one-dimensional harmonic oscillator potential
\begin{equation*}
V_{1}(x)=\frac{1}{2} k x^{2}, \quad k>0
\end{equation*}
When the spring constant $k$ suddenly changes to $2k$, the potential then becomes
\begin{equation*}
V_{2}(x)=k x^{2}
\end{equation*}
Immediat... | (a) The wave function of the particle $\psi(x, t)$ should satisfy the time-dependent Schrödinger equation
\begin{equation*}
\mathrm{i} \hbar \frac{\partial}{\partial t} \psi=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi+V \psi \tag{3}
\end{equation*}
When $V$ undergoes a sudden change (from $V_{1} ... | [
"\\frac{2^{5 / 4}}{1+\\sqrt{2}}"
] | Expression | Theoretical Foundations | |
3 | A harmonic oscillator with charge $q$ is in a free vibration state at $t<0$ and $t>\tau$, with the total energy operator given by
\begin{equation*}
H_{0}=\frac{1}{2 m} p^{2}+\frac{1}{2} m \omega^{2} x^{2}
\end{equation*}
The energy eigenstates are denoted by $\psi_{n}$, and the energy levels $E_{n}^{(0)}=(n+\frac{1... | At $t>\tau$, the external electric field has vanished, and the wavefunction satisfies the Schrödinger equation
\begin{equation*}
\mathrm{i} \hbar \frac{\partial}{\partial t} \psi(x, t)=H_{0} \psi(x, t) \tag{3}
\end{equation*}
The general solution is
\begin{equation*}
\psi(x, t)=\sum_{n} f_{n} \psi_{n}(x) \mathrm{e}... | [
"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}}"
] | Expression | Theoretical Foundations | $P_n$: 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... |
4 | Calculate the result of the commutator $[\boldsymbol{p}, \frac{1}{r}]$. | Using the commutator
\begin{equation*}
[\boldsymbol{p}, F(\boldsymbol{r})]=-\mathrm{i} \hbar \nabla F \tag{1}
\end{equation*}
We obtain
\begin{equation*}
[p, \frac{1}{r}]=-\mathrm{i} \hbar(\nabla \frac{1}{r})=\mathrm{i} \hbar \frac{r}{r^{3}} \tag{2}
\end{equation*}
Using the formula (see question 4.2)
$[\boldsym... | [
"\\mathrm{i} \\hbar \\frac{\\boldsymbol{r}}{r^{3}}"
] | Expression | Theoretical Foundations | $\mathrm{i}$: Imaginary unit.
$\hbar$: Reduced Planck's constant.
$\boldsymbol{r}$: Position vector.
$r$: Magnitude of the position vector $\boldsymbol{r}$. |
5 | 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):
\begin{equation*}
\frac{\lambda+1}{n^{2}}\langle r^{\lambda}\rangle-(2 \lambda+1) \frac{... | The spherical coordinate expression of $\psi_{n l m}$ is
\begin{equation*}
\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}
\end{equation*}
The expectation value of $r^{\lambda}$ is
\begin{equation*}
\langle r^{\lambda}\rangle_{n l m}=\int r^{... | [
"\\langle r\\rangle_{n l m}=\\frac{1}{2}[3 n^{2}-l(l+1)] \\frac{a_{0}}{Z}"
] | Expression | Theoretical Foundations | $\langle r\rangle_{nlm}$: 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. |
6 | Three-dimensional isotropic harmonic oscillator, the total energy operator is
\begin{equation*}
H=\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}
\end{equation*}
For the common eigenstates of $(H, l^{2}, l_{z})$
\begin{equation*... | The energy levels of the three-dimensional isotropic harmonic oscillator are
\begin{equation*}
E_{n, l m}=E_{N}=\left(N+\frac{3}{2}\right) \hbar \omega, \quad N=l+2 n_{r} . \tag{3}
\end{equation*}
For $\psi_{n_{r}, l m}, H$ is equivalent to
\footnotetext{
(1) Refer to H. A. Kramers. Quantum Mechanics. Amsterdam: No... | [
"\\langle\\frac{1}{r^{2}}\\rangle_{n, l m}=\\frac{1}{l+\\frac{1}{2}} \\frac{\\mu \\omega}{\\hbar}"
] | Expression | Theoretical Foundations | $\langle\frac{1}{r^{2}}\rangle_{n, l m}$: 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 |
7 | A particle with mass $\mu$ moves in a "spherical square well" potential, $$V(r)= \begin{cases}0, & r<a \\ V_{0}>0, & r \geqslant a.\end{cases}$$ Consider only the bound state $(0<E<V_{0})$. As $V_{0}$ gradually increases from small to large, find the value of $V_{0} a^{2}$ when the first bound state (with angular qu... | As a central force problem, the bound state wave function can be taken as the common eigenfunction of $(H, l^{2}, l_{z})$, written as
\begin{equation*} \psi=R(r) \mathrm{Y}_{l m}(\theta, \varphi) \tag{2} \end{equation*}
The radial equation is
\begin{equation*} R^{\prime \prime}+\frac{2}{r} R^{\prime}+[\frac{2 \mu}{\... | [
"V_{0} a^{2}=\\frac{\\pi^{2} \\hbar^{2}}{8 \\mu}"
] | Expression | Theoretical Foundations | $V_0$: Height of the potential barrier
$a$: Radius of the spherical square well
$\hbar$: Reduced Planck's constant
$\mu$: Mass of the particle |
8 | 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 $\gamma$. | Using the basic commutation relation $\boldsymbol{l} \times \boldsymbol{l}=\mathrm{i} \hbar \boldsymbol{l}$, we have
\[
\begin{aligned}
& \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 \\
& \mathrm{i} \hbar l_{y} l_{x}=l_{y}(l_{y} l_{z}-l_{z} l_{y})=l_{y}^{2} l_{z}-l_{y... | [
"m \\hbar \\cos \\gamma"
] | Expression | Theoretical Foundations | $m$: 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. |
9 | 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 ... | One Using the eigenfunctions of $\sigma_{n}$ obtained from the previous problem, it is easy to find
(a) In the spin state $\chi_{\frac{1}{2}}=[\begin{array}{l}1 \\ 0\end{array}]$,
the probability of $\sigma_{n}=1$ is
\begin{equation*}
|\langle\phi_{1} \lvert\, \chi_{\frac{1}{2}}\rangle|^{2}=\cos ^{2} \frac{\theta}{2}=... | [
"\\frac{1}{2}(1+n_z)"
] | Expression | Theoretical Foundations | $n_z$: Component of the unit vector $\boldsymbol{n}$ along the $z$-axis. |
10 | 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. | (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... | [
"\\frac{1}{\\sqrt{2}}(1+\\sigma_x)"
] | Expression | Theoretical Foundations | $1$: The $2 \times 2$ identity matrix
$\sigma_x$: Pauli matrix |
11 | 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:
\chi(0)=[\begin{array}{c}
\cos \delt... | Any definite spin state is an eigenstate (with eigenvalue 1) of the projection $\sigma_{n}$ of $\boldsymbol{\sigma}$ in some direction $(\theta, \varphi)$, and
\begin{equation*}
\boldsymbol{P}=\langle\boldsymbol{\sigma}\rangle=\boldsymbol{n} \tag{3}
\end{equation*}
where $\boldsymbol{n}$ is the unit vector in the di... | [
"2 \\delta"
] | Expression | Theoretical Foundations | $\delta$: Angle parameter in the initial spin wave function $\chi(0)$, a positive real number or 0. |
12 | 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 $\hbar=1$ ). | The basic relationships are as follows (the single particle formula is only written for particle 1)
(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.
(b) $s_{1} \times s_{1}=\mathrm{i} s_{1}, \boldsymbol{\sigma}_{1} \times \boldsymbol{\sigma}_{1}=2 \m... | [
"\\mathrm{i} s_{1} \\cdot s_{2}"
] | Expression | Theoretical Foundations | $\mathrm{i}$: Imaginary unit
$s_1$: Spin angular momentum operator for particle 1
$s_2$: Spin angular momentum operator for particle 2 |
13 | Two localized non-identical particles with spin $1/2$ (ignoring orbital motion) have an interaction energy given by (setting $\hbar=1)$
\begin{equation*}
H=A \boldsymbol{s}_{1} \cdot \boldsymbol{s}_{2}
\end{equation*}
At $t=0$, particle 1 has spin 'up' $(s_{1 z}=1 / 2)$, and particle 2 has spin 'down' $(s_{2 z}=-\fr... | Start by finding the spin wave function of the system. Since
\begin{equation*}
H=A s_{1} \cdot s_{2}=\frac{A}{2}(\boldsymbol{S}^{2}-\frac{3}{2}) \tag{$\prime$}
\end{equation*}
It is evident that the total spin $\boldsymbol{S}$ is a conserved quantity, so the stationary wave function can be chosen as a common eigenfu... | [
"\\cos^{2}(\\frac{A t}{2})"
] | Expression | Theoretical Foundations | $A$: Constant in the interaction energy.
$t$: Time. |
14 | Consider a system consisting of three distinguishable particles each with spin $1/2$, with the Hamiltonian given by
\begin{equation*}
H=A(s_{1} \cdot s_{2}+s_{2} \cdot s_{3}+s_{3} \cdot s_{1}) \quad \text { ( } A \text { is real) }
\end{equation*}
Let $S$ denote the total spin quantum number of the system. Determine... | The Hamiltonian $H$ can be written as
\begin{equation*}
H=\frac{A}{2}(\boldsymbol{S}_{123}^{2}-3 \times \frac{3}{4}) \tag{$\prime$}
\end{equation*}
Therefore, the energy levels (taking $\hbar=1$) are
\begin{equation*}
E_{S}=\frac{A}{2}[S(S+1)-\frac{9}{4}] \tag{2}
\end{equation*}
A complete set of conserved quanti... | [
"E_{3/2} = \\frac{3}{4}A"
] | Expression | Theoretical Foundations | $E_{3/2}$: Energy level of the system when the total spin quantum number $S=3/2$.
$A$: Real constant in the Hamiltonian. |
15 | 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 $L q=4 \pi N$ ($N$... | (a) For a free particle, the common eigenfunction of the energy $(H_{0})$ and momentum $(p)$ satisfying the periodic condition is
\begin{equation*}
\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}
\end{equation*}
The eigenvalue is
\begin{align*}
p_{n} & ... | [
"-\\frac{\\varepsilon^{2}}{32 E_{N}^{(0)}}"
] | Expression | Theoretical Foundations | $\varepsilon$: 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 $L q=4 \pi N$. |
16 | 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. | Take the even parity state as an example. The energy eigenvalue equation can be written as
\begin{equation}
\begin{array}{lll}
\psi^{\prime \prime}+k^{2} \psi=0, & |x| \leqslant a / 2 & \text { (inside the well) } \tag{1}\\
\psi^{\prime \prime}-\beta^{2} \psi=0, & |x| \geqslant a / 2 & \text { (outside the well) }
\... | [
"\\frac{2 \\hbar E_n}{a V_0 \\sqrt{2 m V_0}}"
] | Expression | Theoretical Foundations | $\hbar$: 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. |
17 | A particle moves freely, and the initial wave function at $t=0$ is given as
$$ \psi(x, 0)=(2 \pi a^{2})^{-1 / 4} \exp [\mathrm{i} k_{0}(x-x_{0})-(\frac{x-x_{0}}{2 a})^{2}], \quad a>0 $$
Find the wave function $\varphi(p)$ in the $p$ representation at $t=0$; | First, consider the shape of the wave packet at $t=0$
\begin{equation*}
|\psi(x, 0)|^{2}=(2 \pi a^{2})^{-1 / 2} \mathrm{e}^{-(x-x_{0})^{2} / 2 a^{2}} \tag{1}
\end{equation*}
which is a Gaussian distribution. According to the mean value formula
\begin{equation}
\overline{f(x)}=\int_{-\infty}^{+\infty}|\psi|^{2} f(x)... | [
"\\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}]"
] | Expression | Theoretical Foundations | $\varphi(p)$: 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 als... |
18 | The particle moves freely, with the initial wave function at $t=0$ given as
$$ \psi(x, 0)=(2 \pi a^{2})^{-1 / 4} \exp [\mathrm{i} k_{0}(x-x_{0})-(\frac{x-x_{0}}{2 a})^{2}], \quad a>0 $$
Find the wave function $\psi(x, t)$ for $t>0$ | In equation \begin{equation*}
\psi(x, 0)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} \varphi(k) \mathrm{e}^{\mathrm{i} k x} \mathrm{~d} k, \tag{3}
\end{equation*} let
$$ \mathrm{e}^{\mathrm{i} k x} \rightarrow \mathrm{e}^{\mathrm{i}(k x-\omega t)}, \quad \omega=\frac{\hbar k^{2}}{2 m}=\frac{k^{2}}{2 m} \quad(\hbar... | [
"\\psi(x, t) = \\frac{\\exp [\\mathrm{i} k_{0}(x-x_{0})-\\mathrm{i} t k_{0}^{2} / 2 m]}{(2 \\pi)^{1 / 4}(a+\\mathrm{i} t / 2 m a)^{1 / 2}} \\exp [-\\frac{1}{4}(x-x_{0}-\\frac{k_{0} t}{m})^{2} \\frac{1-\\mathrm{i} t / 2 m a^{2}}{a^{2}+(t / 2 m a)^{2}}]"
] | Expression | Theoretical Foundations | $\psi(x, t)$: Wave function of the particle at time $t$
$\mathrm{i}$: Imaginary unit
$k_0$: Initial wave number
$x$: Position coordinate
$x_0$: Initial position offset
$t$: Time
$m$: Mass of the particle
$\pi$: Mathematical constant pi
$a$: A positive constant related to the width of the initial wave packet |
19 | Using the raising and lowering operators $a^{+}, ~ a$, find the energy eigenfunctions of the harmonic oscillator (in the $x$ representation), and briefly discuss their mathematical properties. | Start with the ground state wave function $\psi_{0}(x)$. We have,
\begin{equation*}
a|0\rangle=0 \tag{1}
\end{equation*}
In the $x$ representation this reads as
\begin{equation*}
(\mathrm{i} \hat{p}+m \omega x) \psi_{0}(x)=(\hbar \frac{\mathrm{d}}{\mathrm{~d} x}+m \omega x) \psi_{0}(x)=0 \tag{2}
\end{equation*}
L... | [
"\\psi_n(x) = N_n H_n(\\alpha x) e^{-\\frac{1}{2}(\\alpha x)^2}"
] | Expression | Theoretical Foundations | $\psi_n(x)$: Energy eigenfunction for the $n$-th state in the x representation.
$N_n$: Normalization constant for the $n$-th energy eigenfunction.
$H_n(\alpha x)$: Hermite polynomial of degree $n$ evaluated at $\alpha x$.
$\alpha$: Dimensionless constant defined as $\sqrt{m \omega / \hbar}$.
$x$: Position variable in t... |
20 | The emission of recoil-free $\gamma$ radiation by nuclei bound in a lattice is a necessary condition for the Mössbauer effect. The potential acting on the nuclei in the lattice can be approximated as a harmonic oscillator potential
$$ V(x)=\frac{1}{2} M \omega^{2} x^{2} $$
where $M$ is the mass of the nucleus, $x$ is... | 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:
\begin{equation*}
\psi_{0}(x)=(2 \pi \hbar)^{-1 / 2} \int_{-\infty}^{+\infty} \varphi(p) \mathrm{e}^{\mat... | [
"P=\\exp (-\\frac{E_{\\gamma}^{2}}{2 \\hbar \\omega M c^{2}})"
] | Expression | Theoretical Foundations | $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.... |
21 | Calculate the probability that the nucleus is in various energy eigenstates after $\gamma$ radiation, as in the previous problem. | From the previous problem, the center-of-mass wave function of the nucleus after $\gamma$ radiation is
\begin{equation*}
\psi(x)=\mathrm{e}^{-\mathrm{i} p_{0} x / \hbar} \psi_{0}(x), \quad p_{0}=E_{\gamma} / c \tag{1}
\end{equation*}
The probability of being in the ${ }^{\prime} n$-th vibrational excited state $\psi... | [
"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}})"
] | Expression | Theoretical Foundations | $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 os... |
22 | Suppose the operator $\hat{H}$ has continuous eigenvalues $\omega$, and its eigenfunctions $u_{\omega}(\boldsymbol{x})$ form an orthonormal complete system, i.e.
\begin{gather*}
\hat{H} u_{\omega}(\boldsymbol{x})=\omega u_{\omega}(\boldsymbol{x}) \tag{1}\\
\int u_{\omega^{*}}^{*}(\boldsymbol{x}) u_{\omega}(\boldsymbo... | Since $u_{\omega}(\boldsymbol{x})$ is a complete system, the solution of equation (4) can always be expressed as
\begin{equation*}\nu(x)=\int C_{\omega} u_{\omega}(x) \mathrm{d} \omega \tag{5}
\end{equation*}
Substitute into equation (4), we obtain
\begin{equation*}
\int(\omega-\omega_{0}) C_{\omega} u_{\omega}(\bo... | [
"\\frac{F_{\\omega}}{\\omega-\\omega_{0}} u_{\\omega}(\\boldsymbol{x})"
] | Expression | Theoretical Foundations | $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}$... |
23 | 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． | Known energy levels of a hydrogen-like ion are
\begin{equation*}
E_{n l m}=E_{n}=-\frac{Z^{2} e^{2}}{2 n^{2} a_{0}}, \quad n=n_{r}+l+1 \tag{1}
\end{equation*}
where $a_{0}=\hbar^{2} / \mu e^{2}$ is the Bohr radius. According to the virial theorem,
\begin{equation*}
\langle\frac{p^{2}}{2 \mu}\rangle_{n l m}=\langle\... | [
"\\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}"
] | Expression | Theoretical Foundations | $\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^... |
24 | 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
\begin{equation*}
V(r)=-\frac{e^{2}}{r}-\lambda \frac{e^{2} a_{0}}{r^{2}}, \quad 0<\lambda \ll 1 \tag{1}
\end{equation*}
where $a_{0}$ is t... | Take the complete set of conserved quantities as $(H, l^{2}, l_{z})$, whose common eigenfunctions are
\begin{equation*}
\psi(r, \theta, \varphi)=R(r) \mathrm{Y}_{l m}(\theta, \varphi)=\frac{u(r)}{r} \mathrm{Y}_{l m}(\theta, \varphi) \tag{2}
\end{equation*}
$u(r)$ satisfies the radial equation
\begin{equation*}
-\fra... | [
"E_{n l}=-\\frac{e^{2}}{2(n^{\\prime})^{2} a_{0}}"
] | Expression | Theoretical Foundations | $E_{nl}$: Energy levels of the valence electron.
$e$: Elementary charge.
$n'$: Modified principal quantum number for the valence electron.
$a_0$: Bohr radius. |
25 | For a particle of mass $\mu$ moving in a spherical shell $\delta$ potential well
\begin{equation*}
V(r)=-V_{0} \delta(r-a), \quad V_{0}>0, a>0 \tag{1}
\end{equation*}
find the minimum value of $V_{0}$ required for bound states to exist. | The ground state is an s-state $(l=0)$, and the wave function can be expressed as
\begin{equation*}
\psi(r)=u(r) / r \tag{2}
\end{equation*}
$u(r)$ satisfies the radial equation
\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}
\end{equation*}
As... | [
"V_{0}=\\frac{\\hbar^{2}}{2 \\mu a}"
] | Expression | Theoretical Foundations | $V_0$: Strength of the spherical shell potential well
$\hbar$: Reduced Planck's constant
$\mu$: Mass of the particle
$a$: Radius of the spherical shell |
26 | Simplify $\mathrm{e}^{\mathrm{i} \lambda_{\sigma_{2}}} \sigma_{\alpha} \mathrm{e}^{-\mathrm{i} \lambda \sigma_{z}}, \alpha=x, ~ y, \lambda$ are constants. | Solution 1: Utilize the formula:
\begin{equation*}
\mathrm{e}^{\mathrm{i} \lambda \sigma_{z}}=\cos \lambda+\mathrm{i} \sigma_{z} \sin \lambda \tag{1}
\end{equation*}
We get
$$\mathrm{e}^{\mathrm{i} \lambda \sigma_{z} \sigma_{x} \mathrm{e}^{-\mathrm{i} \lambda \sigma_{z}}=(\cos \lambda+\mathrm{i} \sigma_{z} \sin \la... | [
"\\mathrm{e}^{\\mathrm{i} \\lambda \\sigma_{z}} \\sigma_{x} \\mathrm{e}^{-\\mathrm{i} \\lambda \\sigma_{z}} = \\sigma_{x} \\cos 2 \\lambda-\\sigma_{y} \\sin 2 \\lambda"
] | Expression | Theoretical Foundations | $\mathrm{e}$: Euler's number, the base of the natural logarithm.
$\mathrm{i}$: Imaginary unit, defined as $\sqrt{-1}$.
$\lambda$: A constant parameter.
$\sigma_{z}$: Pauli matrix in the z-direction.
$\sigma_{x}$: Pauli matrix in the x-direction.
$\sigma_{y}$: Pauli matrix in the y-direction. |
27 | A particle with spin $\hbar / 2$ and magnetic moment $\boldsymbol{\mu}=\mu_{0} \boldsymbol{\sigma}$ is placed in a uniform magnetic field $\boldsymbol{B}$ that points in an arbitrary direction $(\theta, \varphi)$ (where $n_3 = \cos \theta$ is the projection of the direction vector of the magnetic field on the $z$ axis)... | 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:
\begi... | [
"n_{3}^{2}+(1-n_{3}^{2}) \\cos 2 \\omega t"
] | Expression | Theoretical Foundations | $n_3$: 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. |
28 | 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 t... | The Hamiltonian related to the spin motion of the system is
\begin{equation*}
H=-\mu \cdot[\boldsymbol{B}_{0}+\boldsymbol{B}_{1}(t)], \quad t \geqslant 0 \tag{1}
\end{equation*}
In the $s_{z}$ representation, the matrix form of $H$ is
\begin{align}
H & =-\mu_{0} B_{1}(\sigma_{x} \cos 2 \omega_{0} t-\sigma_{y} \sin ... | [
"\\pi \\hbar / 2 \\mu_{0} B_{1}"
] | Expression | Theoretical Foundations | $\pi$: 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. |
29 | The magnetic moment (operator) of an electron is
\begin{equation*}
\boldsymbol{\mu}=\boldsymbol{\mu}_{l}+\boldsymbol{\mu}_{s}=-\frac{e}{2 m_{\mathrm{e}} c}(\boldsymbol{l}+2 \boldsymbol{s}) \tag{1}
\end{equation*}
Try to calculate the expectation value of $\mu_{z}$ for the $|l j m_{j}\rangle$ state. | 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)
\begin{equation*}
\boldsymbol{\mu}=-(\boldsymbol{l}+2 \boldsymbol{s})=-(\boldsymbol{j}+\boldsymbol{s})=-(\boldsymbol{j... | [
"\\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}"
] | Expression | Theoretical Foundations | $l$: 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 |
30 | 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
\begin{equation*}
H=a \sigma_{1 z}+b \sigma_{2 z}+c_{0} \boldsymbol{\sigma}_{1} \cdot \boldsymbol{\sigma}_{2} \tag{1}
\end{... | We will solve using matrix methods in spin state vector space. The basis vectors can be chosen as the common eigenstates of $(\sigma_{1 z}, \sigma_{2 z})$
$$\alpha(1) \alpha(2), \quad \alpha(1) \beta(2), \quad \beta(1) \alpha(2), \quad \beta(1) \beta(2) $$
or as the common eigenstates $\chi_{S M_{s}}$ of the total s... | [
"c_{0} + a + b"
] | Expression | Theoretical Foundations | $c_{0}$: 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 |
31 | Consider a system composed of three non-identical spin $1/2$ particles, with the Hamiltonian given by
\begin{equation*}
H=A \boldsymbol{s}_{1} \cdot \boldsymbol{s}_{2}+B(\boldsymbol{s}_{1}+\boldsymbol{s}_{2}) \cdot \boldsymbol{s}_{3}, \tag{1}
\end{equation*}
where $A, ~ B$ are real constants. Let $\boldsymbol{S}_{12}=... | 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
\begin{equation*}
\boldsymbol{S}_{12}=\boldsymbol{s}_{1}+\boldsymbol{s}_{2}, \quad \boldsymbol{S}_{123}=\boldsymbol{s}_{1}+\boldsymbol{s}_{2}+\boldsymbol{s}_{3}=\boldsymbol{S}_... | [
"H = \\frac{1}{2}(A-B) \\boldsymbol{S}_{12}^{2}+\\frac{B}{2} \\boldsymbol{S}_{123}^{2}-\\frac{3}{8}(2 A+B)"
] | Expression | Theoretical Foundations | $H$: 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. |
32 | Same as the previous question, for any value, find
\begin{equation*}
d_{j m}^{j}(\lambda)=\langle j j| \mathrm{e}^{-\mathrm{i} \lambda J_{y}}|j m\rangle \tag{1}
\end{equation*} | According to the general theory of angular momentum,
\begin{array}{rl}
J_{+}|j m\rangle & =(J_{x}+i J_{y})|j m\rangle \tag{2}\\
J_{-}|j m\rangle & =a_{j m}|j m+1\rangle \\
J_{x}-i J_{y})|j m\rangle & =a_{j,-m}|j m-1\rangle
\end{array}}
where
\begin{equation*}
a_{j m}=\sqrt{(j-m)(j+m+1)} \tag{3}
\end{equation*}
W... | [
"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}"
] | Expression | Theoretical Foundations | $d_{j m}^{j}(\lambda)$: 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 |
33 | 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... | $J_{1}, ~ J_{2}$ satisfy the fundamental commutation relations of angular momentum operators
\begin{equation*}
\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}
\end{equation*}
$\boldsymbol{J}_{1}, ~ J_{... | [
"m \\frac{j(j+1)+j_{1}(j_{1}+1)-j_{2}(j_{2}+1)}{2 j(j+1)}"
] | Expression | Theoretical Foundations | $m$: 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 t... |
34 | 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 s... | 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 $|j j J M\rangle$, ... | [
"1/(2j+1)"
] | Expression | Theoretical Foundations | $j$: Angular momentum quantum number for $\boldsymbol{J}_{1}$ and $\boldsymbol{J}_{2}$. |
35 | 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 $\mathrm{d} \boldsymbol{v} / \mathrm{d} t$. | The Hamiltonian operator can be expressed as
\begin{equation*}
H=\frac{1}{2 \mu}(\boldsymbol{p}-\frac{q}{c} \boldsymbol{A})^{2}=\frac{1}{2} \mu \boldsymbol{v}^{2} \tag{2}
\end{equation*}
Using the commutation relations of $\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that
\begin{equation*}
\frac{\math... | [
"\\frac{q}{2 \\mu c}(\\boldsymbol{v} \\times \\boldsymbol{B}-\\boldsymbol{B} \\times \\boldsymbol{v})"
] | Expression | Theoretical Foundations | $q$: Charge of the particle
$\mu$: Mass of the particle
$c$: Speed of light
$\boldsymbol{v}$: Velocity operator
$\boldsymbol{B}$: Magnetic field, defined as $\boldsymbol{B}=\nabla \times \boldsymbol{A}$ |
36 | 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} \boldsymb... | The Hamiltonian operator can be expressed as
\begin{equation*}
H=\frac{1}{2 \mu}(\boldsymbol{p}-\frac{q}{c} \boldsymbol{A})^{2}=\frac{1}{2} \mu \boldsymbol{v}^{2} \tag{2}
\end{equation*}
Using the commutation relations of $\boldsymbol{v}$ and $v^{2}$, it can be easily demonstrated that
\begin{equation*}
\frac{\math... | [
"\\frac{q}{2 c}[\\boldsymbol{r} \\times(\\boldsymbol{v} \\times \\boldsymbol{B})+(\\boldsymbol{B} \\times \\boldsymbol{v}) \\times \\boldsymbol{r}]"
] | Expression | Theoretical Foundations | $q$: Charge of the particle
$c$: Speed of light
$\boldsymbol{r}$: Position vector
$\boldsymbol{v}$: Velocity operator
$\boldsymbol{B}$: Magnetic field |
37 | 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. | 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
\begin{align*}
H & =\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... | [
"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}}"
] | Expression | Theoretical Foundations | $E_{n_{1} n_{2} m}$: 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.
$n_{2}$: Ra... |
38 | 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... | With the electric field direction as the $x$ axis and the magnetic field direction as the $z$ axis, then
\begin{equation*}
\mathscr{L}=(\mathscr{E}, 0,0), \quad \boldsymbol{B}=(0,0, B) \tag{1}
\end{equation*}
Taking the scalar and vector potentials of the electromagnetic field as
\begin{equation*}
\phi=-\mathscr{E}... | [
"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}"
] | Expression | Theoretical Foundations | $E$: 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 uni... |
39 | A particle moves in one dimension. When the total energy operator is
\begin{equation*}
H_{0}=\frac{p^{2}}{2 m}+V(x) \tag{1}
\end{equation*}
the energy level is $E_{n}^{(0)}$. If the total energy operator becomes
\begin{equation*}
H=H_{0}+\frac{\lambda p}{m} \tag{2}
\end{equation*}
find the energy level $E_{n}$. | First, treat $\lambda$ as a parameter, then
\begin{equation*}
\frac{\partial H}{\partial \lambda}=\frac{p}{m} \tag{3}
\end{equation*}
According to the Hellmann theorem, we have
\begin{equation*}
\frac{\partial E_{n}}{\partial \lambda}=\langle\frac{\partial H}{\partial \lambda}\rangle_{n}=\frac{1}{m}\langle p\rangle... | [
"E_{n}=E_{n}^{(0)}-\\frac{\\lambda^{2}}{2 m}"
] | Expression | Theoretical Foundations | $E_n$: 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. |
40 | A particle with mass $\mu$ moves in a central force field,
\begin{equation*}
V(r)=\lambda r^{\nu}, \quad-2<\nu, \quad \nu / \lambda>0 \tag{1}
\end{equation*}
Use the Hellmann theorem and the virial theorem to analyze the dependence of the energy level structure on $\hbar, ~ \lambda, ~ \mu$. | The energy operator is
\begin{equation*}
H=T+V=-\frac{\hbar^{2}}{2 \mu} \nabla^{2}+\lambda r^{\nu} \tag{2}
\end{equation*}
Let $\beta=\hbar^{2} / 2 \mu, \lambda$ and $\beta$ be independent parameters. It is evident that
\begin{equation*}
\beta \frac{\partial H}{\partial \beta}=T, \quad \lambda \frac{\partial H}{\pa... | [
"E=C \\lambda^{2 /(2+\\nu)}(\\frac{\\hbar^{2}}{2 \\mu})^{\\nu /(2+\\nu)}"
] | Expression | Theoretical Foundations | $E$: 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 ... |
41 | A particle moves in a potential field
\begin{equation*}
V(x)=V_{0}|x / a|^{\nu}, \quad V_{0}, a>0 \tag{1}.
\end{equation*}
Find the dependence of energy levels on parameters as $\nu \rightarrow \infty$. | The total energy operator is
\begin{equation*}
H=T+V=-\frac{\hbar^{2}}{2 \mu} \frac{\mathrm{~d}^{2}}{\mathrm{~d} x^{2}}+V_{0}|x / a|^{\nu} \tag{2}
\end{equation*}
From dimensional analysis, if $x_{0}$ represents the characteristic length, we have
\begin{equation*}
E \sim \frac{\hbar^{2}}{\mu x_{0}^{2}} \sim \frac{V... | [
"E_{n}=\\frac{n^{2} \\pi^{2} \\hbar^{2}}{8 \\mu a^{2}}"
] | Expression | Theoretical Foundations | $E_n$: 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. |
42 | 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. | The total energy operator is
\begin{equation*}
H=T+V=\frac{p^{2}}{2 m}+F x \tag{1}
\end{equation*}
In momentum representation, the operator for $x$ is given by
\begin{equation*}
\hat{x}=\mathrm{i} \hbar \frac{\mathrm{~d}}{\mathrm{~d} p} \tag{2}
\end{equation*}
The operator for $H$ is given by
\begin{equation*}
\h... | [
"E_{1}=1.8558(\\frac{\\hbar^{2} F^{2}}{m})^{1 / 3}"
] | Expression | Theoretical Foundations | $E_1$: ground state energy level
$\hbar$: reduced Planck's constant
$F$: magnitude of the uniform force field
$m$: mass of the particle |
43 | Calculate the transmission probability of a particle (energy $E>0$) through a $\delta$ potential barrier $V(x)=V_{0} \delta(x)$ in the momentum representation. | The stationary Schrödinger equation in the $x$ representation is
\begin{equation*}
\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}
\end{equation*}
Let
\begin{equation*}
\psi(x)=(2 \pi \hbar)^{-1 / 2} \int_{-\infty}^{+\infty} \mathrm{d} p \varphi(p) \... | [
"\\frac{1}{1+(m V_{0} / \\hbar^{2} k)^{2}}"
] | Expression | Theoretical Foundations | $m$: Mass of the particle.
$V_0$: Strength of the delta potential barrier.
$\hbar$: Reduced Planck's constant.
$k$: Wave number, defined as $k=\sqrt{2 m E} / \hbar$. |
44 | The energy operator of a one-dimensional harmonic oscillator is
\begin{equation*}
H=\frac{p_{x}^{2}}{2 \mu}+\frac{1}{2} \mu \omega^{2} x^{2} \tag{1}
\end{equation*}
Try to derive its energy level expression using the Heisenberg equation of motion for operators and the fundamental commutation relation. | Using the Heisenberg equation of motion, we obtain
\begin{equation*}
\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}
\end{equation*}
Take the matrix element in the energy representation, yielding
\begin{equation*}
\mat... | [
"E_{n}=(n+\\frac{1}{2}) \\hbar \\omega"
] | Expression | Theoretical Foundations | $E_n$: 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 |
45 | 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. | It is easy to obtain
\begin{align*}
{[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}) \\
& =\frac{i \hbar}{m \omega} \sin \omega(t_{2}-t_{1}). \tag{1}
\end{align*}
Similarly, it can be obtained
\begin{align*}
& {[p(t_{1}), p(t_{2})]=\mathrm{... | [
"\\mathrm{i} \\hbar \\cos \\omega(t_{2}-t_{1})"
] | Expression | Theoretical Foundations | $\mathrm{i}$: Imaginary unit
$\hbar$: Reduced Planck's constant
$\omega$: Angular frequency of the harmonic oscillator
$t_2$: Second time variable
$t_1$: First time variable |
46 | Two localized non-identical particles with spin $1 / 2$ (ignoring orbital motion) have an interaction energy given by
$$H=A \boldsymbol{s}_{1} \cdot \boldsymbol{s}_{2} $$
where $\boldsymbol{s}_{1}, \boldsymbol{s}_{2}$ are spin operators (the eigenvalues of their $z$ components $s_{iz}$ being $\pm 1/2$), and $A$ is a ... | 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 c... | [
"w(t)=\\cos ^{2} \\frac{A t}{2}"
] | Expression | Theoretical Foundations | $w(t)$: Probability that particle 1 has spin 'up' at time $t$
$A$: A constant related to energy
$t$: Time |
47 | 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-var... | According to the Heisenberg equation of motion.
$$\frac{\mathrm{d}}{\mathrm{~d} t} \sigma_{z}=\frac{1}{\mathrm{i} \hbar}[\sigma_{z}, H]=0,$$
Therefore,
\begin{equation*}
\langle\sigma_{z}\rangle_{t}=\langle\sigma_{z}\rangle_{t=0}=\cos \theta_{0}=\cos 2 \delta \tag{1}
\end{equation*}
Moreover,
\begin{align*}
& \fra... | [
"\\langle\\sigma_{x}\\rangle_{t}=\\sin 2 \\delta \\cos \\varphi(t)"
] | Expression | Theoretical Foundations | $\langle\sigma_{x}\rangle_{t}$: 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}... |
48 | Evaluate the function
$$ \langle 0| \phi(x) \phi(y)|0\rangle=D(x-y)=\int \frac{d^{3} p}{(2 \pi)^{3}} \frac{1}{2 E_{\mathbf{p}}} e^{-i p \cdot(x-y)} $$
for $(x-y)$ spacelike so that $(x-y)^{2}=-r^{2}$, explicitly in terms of Bessel functions.
\footnotetext{
$\ddagger$ With some additional work you can show that there... | We evaluate the correlation function of a scalar field at two points,
\begin{equation*}
D(x-y)=\langle 0| \phi(x) \phi(y)|0\rangle \tag{2.28}
\end{equation*}
with $x-y$ 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:
\begin{equatio... | [
"D(x-y) = \\frac{m}{4 \\pi^{2} r} K_{1}(m r)"
] | Expression | Others | $D(x-y)$: 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)^... |
49 | 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, \math... | Now we work out the $C, P$ and $T$ transformation properties of a scalar field $\phi$. Our starting point is
P 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}}
Then, for a complex scalar field
\begin{equation*}
\phi(x)=\int \frac{\mathrm{d}^{3} k}{(... | [
"P J^{\\mu}(t, \\mathbf{x}) P =(-1)^{s(\\mu)} J^{\\mu}(t,-\\mathbf{x})"
] | Expression | Others | $P$: 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. |
50 | Compute the probability that the source creates one particle of momentum $k$ to all orders in the source $j$ by summing the perturbation series. | The probability that the source creates one particle with momentum $\mathbf{k}$ is given by,
$$P(\mathbf{k})=|\langle\mathbf{k}| T \exp\{i \int \mathrm{~d}^{4} x j(x) \phi_{I}(x)\}| 0\rangle|^{2}. $$
Expanding the amplitude to the first order in \( j \), we get:
\begin{equation*}
\begin{split}
P(k) &= \left| \langle \m... | [
"P(k) = \\frac{1}{2 E_{\\mathbf{k}}} |\\tilde{\\jmath}(k)|^{2} e^{-\\langle N \\rangle}"
] | Expression | Others | $P(k)$: Probability that the source creates one particle with momentum $k$
$\tilde{j}(k)$: Fourier transform of the source $j$ evaluated at momentum $k$ |
51 | Decay of a scalar particle. Consider the following Lagrangian, involving two real scalar fields $\Phi$ and $\phi$ :
$$\mathcal{L}=\frac{1}{2}(\partial_{\mu} \Phi)^{2}-\frac{1}{2} M^{2} \Phi^{2}+\frac{1}{2}(\partial_{\mu} \phi)^{2}-\frac{1}{2} m^{2} \phi^{2}-\mu \Phi \phi \phi .$$
The last term is an interaction that... | This problem is based on the following Lagrangian,
\begin{equation*}
\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}
\end{equation*}
When $M>2 m$, a $\Phi$ particle can decay into two $\phi$ pa... | [
"\\tau = \\frac{8\\pi M}{\\mu^2} (1 - \\frac{4m^2}{M^2})^{-1/2}"
] | Expression | Others | $\tau$: 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 |
52 | 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... | This problem is based on the following Lagrangian,
\begin{equation*}
\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 .
\end{equation*}
When $M>2 m$, a $\Phi$ particle can decay into two $\phi$ particles. W... | [
"\\frac{\\sqrt{5}}{24\\pi}"
] | Expression | Others | $\pi$: Mathematical constant pi. |
53 | 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... | It is easy to find that
\epsilon_{1}^{\mu}=N(0, p^{\prime} \cos \theta-p, 0,-p^{\prime} \sin \theta), \quad \epsilon_{2}^{\mu}=(0,0,1,0),
where $N=(E^{2}+E^{\prime 2}-2 E E^{\prime} \cos \theta)^{-1 / 2}$ is the normalization constant. Then, for the righthanded electron with spinor $u_{+}(p)=\sqrt{2 E}(0,0,1,0)^{T}$ ... | [
"-\\sqrt{E E^{\\prime}} \\frac{E+E^{\\prime}}{|E-E^{\\prime}|} \\theta"
] | Expression | Others | $E$: Initial energy of the electron
$E^{\prime}$: Final energy of the electron
$\theta$: Scattering angle |
54 | The unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to
$$ H_{\mathrm{int}}=\int d^{3} x \frac{\lambda}{\sqrt{2}} h \bar{\psi} \psi $$
Compute the contribution of a virtual Higgs boson to the electron $(g-2)$, in te... | The 1-loop vertex correction from Higgs boson is,
\begin{align*}
\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}}{\... | [
"\\frac{\\lambda^{2} m^2}{8 \\pi^{2} m_h^2}[\\log (\\frac{m_h^2}{m^2})-\\frac{7}{6}]"
] | Expression | Others | $\lambda$: Higgs-electron coupling constant.
$m$: Mass of the electron.
$\pi$: Mathematical constant pi.
$m_h$: Mass of the Higgs boson. |
55 | Some more complex versions of this theory contain a pseudoscalar particle called the axion, which couples to the electron according to
$$ H_{\mathrm{int}}=\int d^{3} x \frac{i \lambda}{\sqrt{2}} a \bar{\psi} \gamma^{5} \psi $$
The axion may be as light as the electron, or lighter, and may couple more strongly than t... | The 1-loop correction from the axion is given by,
\begin{align*}
\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{... | [
"-\\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)}"
] | Expression | Others | $\lambda$: Coupling constant between the axion and the electron.
$m_{a}$: Mass of the axion.
$m$: Mass of the electron. |
56 | 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 conve... | 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
\begin{equation*}
\mathrm{i} \mathcal{M}=Q_{f}(-\mathrm{i} e)^{2}(-\mathrm{i} g) \epsilon_{\nu}^{*}(k_{3}) \bar{u}(k_{1})[\gamma^{\nu} \frac{\mathr... | [
"\\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})}"
] | Expression | Others | $\sigma$: 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, sq... |
57 | 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 expli... | Now we reevaluate the averaged squared amplitude, with $\mu$ kept nonzero in the formula
\begin{align*}
\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}} \\
&+(\fra... | [
"\\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}]"
] | Expression | Others | $\pi$: 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 $\alpha_{g}=\frac{g^{2}}{4 \pi}$.
$... |
58 | 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 corre... | Now we calculate the spin-averaged differential cross section for the process $e^{+} e^{-} \rightarrow$ $\phi^{*} \phi$. The scattering amplitude is given by
\begin{equation*}
\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}
\end{equat... | [
"(\\frac{d\\sigma}{d\\Omega})_{CM} = \\frac{\\alpha^2}{8s} (1 - \\frac{m^2}{E^2})^{3/2} \\sin^2{\\theta}"
] | Expression | Others | $(\frac{d\sigma}{d\Omega})_{\mathrm{CM}}$: 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 particl... |
59 | 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 $M$. | Now we calculate the 1-loop effective potential of the model. We know that 1-loop correction of the effective Lagrangian is given by,
\begin{equation*}
\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... | [
"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... | Expression | Others | $V_{\mathrm{eff}}$: 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 e... |
60 | 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 $e^{2}$... | 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 ... | [
"m_{\\sigma}/m_A = \\frac{\\sqrt{6}e}{4\\pi}"
] | Expression | Others | $m_{\sigma}$: Mass of the scalar $\sigma$ particle.
$m_A$: Mass of the gauge boson.
$e$: Electric charge coupling constant.
$\pi$: Mathematical constant pi. |
61 | 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. | 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,
\begin{align*}
& \frac{\mathrm{d}}{\mathrm{~d} \log Q} f_{q}(x, Q)= \frac{3 Q_{q}^{2} \alpha}{\... | [
"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}]"
] | Expression | Others | $f_q(x, Q)$: 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 s... |
62 | 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 rea... | 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,
\begin... | [
"\\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... | Expression | Others | $\sigma$: 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... |
63 | 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 rea... | 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,
\begin... | [
"\\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}^{... | Expression | Others | $\sigma$: 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, typic... |
64 | Find the couplings of the physical charged Higgs boson $\phi^{+}$ to the quark mass eigenstates, given the Lagrangian $$\mathcal{L}_{m} = -\lambda_{d}^{ij} \bar{Q}_{L}^{i} \cdot \phi_{1} d_{R}^{j} - \lambda_{u}^{ij} \epsilon^{ab} \bar{Q}_{La}^{i} \phi_{2b}^{\dagger} u_{R}^{j} + \text{h.c.}$$ | Assuming that the Yukawa interactions between quarks and scalars take the following form:
\begin{align}
\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}
\en... | [
"\\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.}"
] | Expression | Others | $v$: 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... |
65 | 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 ... | 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.
... | [
"\\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}"
] | Expression | Others | $\Gamma$: 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 |
66 | 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 \Gamm... | 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 acc... | [
"\\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}"
] | Expression | Others | $\Gamma$: 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 electrom... |
67 | Derive the commutator $[M^{\mu \nu}, M^{\rho \sigma}]$. Hints :
- Denote $\Lambda^{\prime} \mu_{\nu} \approx \delta^{\mu}{ }_{v}+\chi^{\mu}{ }_{v}$. Check $(\Lambda^{-1} \Lambda^{\prime} \Lambda)_{\rho \sigma} \approx \delta_{\rho \sigma}+\chi_{\mu \nu} \Lambda^{\mu}{ }_{\rho} \Lambda^{v}{ }_{\sigma}$.
- Denote $\Lambd... | We can write
\begin{aligned}
(\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} \\
& =\delta_{\rho \sigma}+\Lambda_{\rho}^{\mu} \Lamb... | [
"[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})"
] | Expression | Others | $M^{\mu \nu}$: Generators of the Lorentz algebra.
$\mathfrak{i}$: Imaginary unit, as used in the final expression for the commutator.
$g^{\nu \sigma}$: Metric tensor. |
68 | What are the commutation relations for $[\mathrm{M}^{\mu \nu}, \mathrm{P}^{\rho}]$ and $[\mathrm{P}^{\mu}, \mathrm{P}^{\nu}]$ in the Poincaré algebra?
You 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] = ... | The action of $\Lambda^{-1} \mathrm{a} \Lambda$ on a point $x$ can be written explicitly as follows,
\begin{aligned}
{[(\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} \\
& =x_{\rho}+a_{\mu}... | [
"[\\mathrm{M}^{\\mu \\nu}, \\mathrm{P}^{\\rho}] = \\mathfrak{i}(g^{\\rho \\mu} \\mathrm{P}^{\\nu} - g^{\\rho \\nu} \\mathrm{P}^{\\mu})"
] | Expression | Others | $\mathrm{M}^{\mu \nu}$: 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 s... |
69 | Calculate the expression in coordinate space of the retarded propagator given in eq. \begin{align}
\tilde{G}_R^0(\kappa) = \frac{i}{(\kappa_0 + i0^+)^2 - (\kappa^2 + m^2)}.
\end{align}
Hint : perform the $\mathrm{k}_{0}$ integral in the complex plane with the theorem of residues. The remaining integrals are elementary... | The free retarded propagator in coordinate space is given by the following Fourier integral:
$$ G_{R}^{0}(x, y)=i \int \frac{d^{4} k}{(2 \pi)^{4}} \frac{e^{-i k \cdot(x-y)}}{(k^{0}+i 0^{+})^{2}-k^{2}}. $$
The integrand has two poles in the complex plane of the variable $k^{0}$, located at $k^{0}= \pm|\mathbf{k}|-\math... | [
"G_{R}^{0}(x, y) = -\\frac{i}{2 \\pi} \\theta(r^{0}) \\delta(r_{0}^{2}-r^{2})"
] | Expression | Others | $G_{R}^{0}$: 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 ... |
70 | 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?
Convention: Use metric $\eta_{\... | 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
$$\mathcal{G}_{\mathrm{R}}^{0}(p)=-\frac{\mathfrak{i} \mu^{2}}{((p^{0}+\mathfrak{i} 0^{+})^{2}-\mathbf{p}^{2}-m^{2})^{2}} $$
where the $\mathrm{i}^{+}$prescrip... | [
"2 \\pi \\mu^{2} \\frac{\\partial}{\\partial M^{2}} \\delta(M^{2}-m^{2})"
] | Expression | Others | $\mu$: 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. |
71 | 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_{... | 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
$$ \mathcal{G}_{\mathrm{R}}^{0}(p)=-\frac{\mathfrak{i} \mu^{2}}{((p^{0}+\mathfrak{i} 0^{+})^{2}-\mathbf{p}^{2}-m^{2})^{2}} $$
where the $\mathrm{i}^{+}$prescri... | [
"\\mathcal{G}_{R}^{0}(p)=-\\mu^{2} \\frac{\\partial}{\\partial m^{2}} G_{R}^{0}(p)"
] | Expression | Others | $\mathcal{G}_{R}^{0}(p)$: 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 |
72 | 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? | The free propagator is obtained from the inverse of the operator between the two fields.
The free retarded propagator reads
\begin{aligned}
\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{... | [
"\\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}}]"
] | Expression | Others | $\mu$: 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,... |
73 | 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?
Define 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 spectr... | 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
\mathcal{G}_{\mathrm{R}}^{0}(p)=-\frac{\mathfrak{i} \mu^{2}}{((p^{0}+\mathfrak{i} 0^{+})^{2}-\mathbf{p}^{2}-m^{2})^{2}}
where the $\mathrm{i}^{+}$prescription ... | [
"\\frac{2 \\pi \\mu^{2}}{m_{1}^{2}-m_{2}^{2}}[\\delta(M^{2}-m_{2}^{2})-\\delta(M^{2}-m_{1}^{2})]"
] | Expression | Others | $\pi$: 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. |
74 | 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
$$p=k_1+k_2,\qquad q=k_3+k_4,\qquad r=k_5+k_6,\qquad p+q+r=0.$$
Use Minkowski Feynman rules: propagator $\frac{i}{p^2-m^2+i0^+}$, vertex fa... | (1) The 1-loop triangle topology integral is given by:
\begin{equation*}
\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^{+})}
\end{equation*}
(2) The 2-loop coupled topology integral is given by:
\be... | [
"(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 int... | Expression | Others | $i$: 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
$\mathfrak{i}$: Imagina... |
75 | Calculate $\operatorname{tr}(\gamma^{\mu} \gamma^{v} \gamma^{\rho} \gamma^{\sigma})$. | With four Dirac matrices, we can write
\begin{align}
\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}) \\
= & +\operatorname{tr}(\gamma^{\mu} \gamma^{\sigma... | [
"4(g^{\\rho \\sigma} g^{\\mu \\nu}-g^{\\nu \\sigma} g^{\\mu \\rho}+g^{\\mu \\sigma} g^{\\nu \\rho})"
] | Expression | Others | $g^{\rho \sigma}$: 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... |
76 | 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(\math... | 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 o... | [
"\\exp (-\\frac{V|\\chi_{0}-\\vartheta_{0}|^{2}}{4 m})"
] | Expression | Others | $V$: 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 $\vartheta_0$. |
77 | Using Weyl's prescription for quantization, where a classical quantity $f(q, p)$ is mapped to an operator $F(Q,P)$ by
$$ F(Q, P) \equiv \int \frac{d p d q d \mu d v}{(2 \pi)^{2}} f(q, p) e^{i(\mu(Q-q)+v(P-p))}, $$
calculate the quantum operator corresponding to $f(q,p) = qp$.
(You may use the results from previous p... | To obtain the Weyl mapping of $qp$, we use the polarization identity: $q p=\frac{1}{2}((q+p)^{2}-q^{2}-p^{2})$.
From the previous result $(\alpha q+\beta p)^{n} \rightarrow (\alpha Q+\beta P)^{n}$, we have the following specific mappings:
For $q^2$: set $\alpha=1, \beta=0, n=2$. So, $q^{2} \rightarrow Q^{2}$.
For $p^2... | [
"\\frac{1}{2}(QP+PQ)"
] | Expression | Others | $Q$: Quantum position operator.
$P$: Quantum momentum operator. |
78 | Consider the fermionic integral,
$$\langle\chi_{j_{1}} \cdots \chi_{j_{p}} \bar{\chi}_{i_{1}} \cdots \bar{\chi}_{i_{q}}\rangle \equiv \operatorname{det}^{-1}(\boldsymbol{M}) \int \prod_{k=1}^{n}[d \chi_{k} d \bar{\chi}_{k}] \chi_{j_{1}} \cdots \chi_{j_{p}} \bar{\chi}_{i_{1}} \cdots \bar{\chi}_{i_{q}} \exp (\bar{\chi}^... | To calculate these integrals, we introduce the generating function:
$$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... | [
"-M_{j_1 i_1}^{-1}"
] | Expression | Others | $M_{j_1 i_1}^{-1}$: 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. |
79 | Consider the fermionic integral,
$$
\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}^... | 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[\b... | [
"M_{j_1 i_2}^{-1} M_{j_2 i_1}^{-1} - M_{j_1 i_1}^{-1} M_{j_2 i_2}^{-1}"
] | Expression | Others | $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... |
80 | 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... | 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).
The integral is:
\begin{align}
\widetilde{f}(\eta) & =\int d \chi e^{\... | [
"\\eta a+b"
] | Expression | Others | $\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). |
81 | Consider two Grassmann variables $\theta_\pm$.
For 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 norma... | From the eigenvalue equation $\tau_3 f = \lambda f$, setting $\lambda=1/2$:
$0 = \frac{1}{2} a \implies a=0$
$(\frac{1}{2}-\frac{1}{2})b=0 \implies 0 \cdot b = 0$ (no constraint on $b$)
$(-\frac{1}{2}-\frac{1}{2})c=0 \implies -c=0 \implies c=0$
$0 = \frac{1}{2} d \implies d=0$
So, eigenfunctions for $\lambda=1/2$ ... | [
"$\\theta_{+}"
] | Expression | Others | $\theta_+$: One of the two Grassmann variables. |
82 | Consider two Grassmann variables $\theta_\pm$.
For 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 norma... | 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_-$. | [
"$\\theta_{+} + \\theta_{-}"
] | Expression | Others | $\theta_+$: One of the two Grassmann variables.
$\theta_-$: One of the two Grassmann variables. |
83 | Consider two Grassmann variables $\theta_\pm$.
For 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 norm... | 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_-$. | [
"$\\theta_{+} - \\theta_{-}"
] | Expression | Others | $\theta_+$: Grassmann variable
$\theta_-$: Grassmann variable |
84 | 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. | 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_-$. | [
"$\\theta_{+} - i\\theta_{-}"
] | Expression | Others | $\theta_+$: Grassmann variable
$i$: Imaginary unit
$\theta_-$: Grassmann variable |
85 | 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 $(\ta... | $\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_-$.} | [
"\\theta_{-}"
] | Expression | Others | $\theta_-$: Grassmann variable |
86 | 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})$. | In this case, we have
$$
\operatorname{ad}_{x}(Y)=-i[X, Y]=c Y, \quad e^{\operatorname{ad}_{X}} Y=e^{c} Y
$$
Since $Y$ commutes with itself, this implies that
$$
e^{\operatorname{tad}_{Y}} e^{\mathrm{ad}_{x}} \mathrm{Y}=e^{\mathrm{c}} \mathrm{Y}
$$
Using this in the general Baker-Campbell-Hausdorff formula gives
$$\... | [
"i X+\\frac{i c}{e^{c}-1} Y"
] | Expression | Others | $i$: The imaginary unit.
$X$: An operator.
$c$: A numerical constant.
$e$: Euler's number, the base of the natural logarithm.
$Y$: An operator. |
87 | Calculate the one-loop $\beta$-function of a scalar field theory with cubic interactions in six spacetime dimensions. | 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 n... | [
"\\beta=-\\frac{3 \\lambda^{3}}{4(4 \\pi)^{3}}"
] | Expression | Others | $\beta$: The one-loop beta-function of the scalar field theory.
$\lambda$: The coupling constant of the cubic interaction. |
88 | 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? | 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}^{... | [
"\\beta(e)=\\frac{e^{3}}{12 \\pi^{2}}"
] | Expression | Others | $\beta(e)$: Beta function for the electromagnetic coupling constant $e$, describing its running with the renormalization scale $M$.
$e$: Electromagnetic coupling constant.
$\pi$: Pi, mathematical constant. |
89 | 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}... | Given the infinitesimal transformation for $y^{\mu}$:
$$y^{\mu}=\frac{x^{\mu}+b^{\mu} x^{2}}{1+2 b \cdot x+b^{2} x^{2}}$$
For infinitesimal $b^{\mu}$, we expand $(1+2 b \cdot x+b^{2} x^{2})^{-1} \approx 1 - (2 b \cdot x+b^{2} x^{2}) + \mathcal{O}(b^2) \approx 1 - 2 b \cdot x + \mathcal{O}(b^2)$.
So, $y^{\mu} \approx (x... | [
"T^{\\mu}=-\\mathfrak{i}(x^{2} g^{\\mu \\rho}-2 x^{\\rho} x^{\\mu}) \\partial_{\\rho}"
] | Expression | Others | $T^{\mu}$: 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 ... |
90 | 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 $A_a^i$? | 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
$$\Pi_{a}^{i} \equiv \frac{\par... | [
"\\Pi_{a}^{i} = \\partial_{0} A_{\\mathrm{a}}^{i}"
] | Expression | Others | $\Pi_{a}^{i}$: Conjugate momentum of the gauge field component $A_a^i$, defined as $\Pi_{a}^{i} \equiv \frac{\partial \mathcal{L}_{\mathrm{YM}}}{\partial \partial_{0} \mathcal{A}_{\mathrm{a}}^{i}}$.
$\partial_{0}$: Partial derivative with respect to the time coordinate.
$A_{\mathrm{a}}^{i}$: Gauge field component. |
91 | For Yang-Mills theory in the temporal gauge $A^{0}=0$ (with $g=1$), derive the Hamiltonian density $\mathcal{H}$. Express it first in terms of the chromo-electric fields $E_a^i = \Pi_a^i$ and chromo-magnetic fields $B_a^i = \frac{1}{2} \epsilon_{ijk} F_a^{jk}$, and then in terms of $\Pi_a^i$ and $F_a^{ij}$. | It is convenient to introduce the chromo-electric and chromo-magnetic fields by
$$
\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}
$$
in terms of which the Lagrangian density can be... | [
"\\mathcal{H}=\\frac{1}{2}(E_{a}^{i} E_{a}^{i}+B_{a}^{i} B_{a}^{i}) = \\frac{1}{2} \\Pi_{a}^{i} \\Pi_{a}^{i}+\\frac{1}{4} F_{a}^{i j} F_{a}^{i j}"
] | Expression | Others | $\mathcal{H}$: Hamiltonian density of the Yang-Mills theory.
$E_a^i$: Chromo-electric field, defined as $E_a^i \equiv F_a^{0i} = \Pi_a^i$ in the $A^0=0$ gauge.
$B_a^i$: Chromo-magnetic field, defined as $B_a^i \equiv \frac{1}{2} \epsilon_{ijk} F_a^{jk}$.
$\Pi_a^i$: Canonical momentum conjugate to the spatial component ... |
92 | A point charge $e$ is located at point $O$ near a system of grounded conductors, inducing charges $e_{a}$ on these conductors. If the charge $e$ is absent and one of the conductors (the $a$-th) has a potential $\varphi_{a}^{\prime}$ (with the remaining conductors still grounded), then the potential at point $O$ is $\va... | If the charge $e_{a}$ on the conductor gives the conductor a potential $\varphi_{a}$, while the charge $e_{a}^{\prime}$ gives the conductor a potential $\varphi_{a}^{\prime}$, then we have:
$$\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}$$
... | [
"e_a = -\\frac{e \\varphi_0'}{\\varphi_a'}"
] | Expression | Magnetism | $e_a$: Induced charge on the $a$-th conductor.
$e$: A point charge.
$\varphi_0'$: Potential at point $O$ in a specific state: the point charge $e$ is absent, the $a$-th conductor has potential $\varphi_a'$, and remaining conductors are grounded.
$\varphi_a'$: Potential of the $a$-th conductor in a specific state: the p... |
93 | Determine the capacitance $C$ of a circular ring made of a thin wire with a circular cross-section (ring radius is $b$, and the radius of the wire cross-section is $a$, where $b \gg a)$. | Since the ring is very thin, the electric field near its surface is equal to that produced by the same charge distribution along the axis of the ring (this is exact for a straight cylinder). Therefore, the potential of the ring is
$$
\varphi_{a}=\frac{e}{2 \pi b} \oint \frac{\mathrm{~d} l}{r},
$$
where $r$ is the dist... | [
"C=\\frac{\\pi b}{\\ln (8 b / a)}"
] | Expression | Magnetism | $C$: Capacitance of the circular ring
$b$: Radius of the circular ring
$a$: Radius of the wire cross-section |
94 | An infinitely long cylindrical conductor with radius $R$ is immersed in a uniform transverse electric field with strength $\mathfrak{C}$. Find the potential distribution $\varphi(r, \theta)$ outside the cylinder. | Using polar coordinates in the plane perpendicular to the axis of the cylinder. Thus, the solution to the two-dimensional Laplace equation involving a constant vector is
$$
\varphi_{1}=\text { const } \cdot \mathfrak{C} \cdot \nabla \ln r=\text { const } \cdot \frac{\mathfrak{C} \cdot \boldsymbol{r}}{r^{2}} .
$$
Addin... | [
"\\varphi=-\\mathfrak{C} r \\cos \\theta(1-\\frac{R^{2}}{r^{2}})"
] | Expression | Magnetism | $\varphi$: Total potential distribution outside the cylinder.
$\mathfrak{C}$: Uniform transverse electric field vector.
$r$: Radial coordinate in polar coordinates.
$\theta$: Angular coordinate in polar coordinates.
$R$: Radius of the infinitely long cylindrical conductor. |
95 | An infinitely long conducting cylinder with a radius $R$ is immersed in a uniform transverse electric field with a strength of $\mathfrak{C}$. Find the induced surface charge density $\sigma(\theta)$ on the surface of the cylinder. | The potential is
\begin{equation}
\varphi=-\mathfrak{C} r \cos \theta(1-\frac{R^{2}}{r^{2}}).
\end{equation}
Thus, the surface charge density is
$$\sigma= -\frac{1}{4\pi} \frac{\partial \varphi}{\partial r} = \frac{\mathfrak{C}}{2 \pi} \cos \theta. $$ | [
"\\sigma=\\frac{\\mathfrak{C}}{2 \\pi} \\cos \\theta"
] | Expression | Magnetism | $\sigma$: Induced surface charge density on the surface of the cylinder
$\mathfrak{C}$: Strength of the uniform transverse electric field
$\pi$: Mathematical constant pi
$\theta$: Angular position (polar angle) in cylindrical coordinates |
96 | An infinitely long conducting cylinder with a radius of $R$ is immersed in a uniform transverse electric field with a strength of $\mathfrak{C}$. Find the induced dipole moment $\mathscr{P}$ per unit length of the cylinder. | The potential is
\begin{equation*}
\varphi=-\mathfrak{C} r \cos \theta(1-\frac{R^{2}}{r^{2}}).
\end{equation*}
The dipole moment $\mathscr{P}$ per unit length of the cylinder can be determined by comparing $\varphi$ with the potential of a two-dimensional dipole field, which is given by
$$2 \mathscr{P} \cdot \na... | [
"\\mathscr{P}=\\mathfrak{C} R^{2} / 2"
] | Expression | Magnetism | $\mathscr{P}$: 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. |
97 | Determine the attraction energy between an electric dipole and a planar conductor surface. | 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{U}=-\\frac{1}{16 x^{3}}(2 \\mathscr{P}_{x}^{2}+\\mathscr{P}_{y}^{2})"
] | Expression | Magnetism | $\mathscr{U}$: 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 $\mathscr{P}$. |
98 | 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.
You should work in Gaussi... | 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
$$-\mathfrak{C} z+\frac{1}{2 \pi} \int_{0}^{2 \pi} \int_{-l}... | [
"\\mathscr{P} = \\mathfrak{C} \\frac{l^{3}}{3 L}[1+\\frac{1}{L}(\\frac{4}{3}-\\ln 2)]"
] | Expression | Magnetism | $\mathscr{P}$: Electric dipole moment of the rod.
$\mathfrak{C}$: External electric field, parallel to the axis of the rod.
$l$: Half-length of the cylindrical rod.
$L$: Dimensionless parameter related to the rod's geometry, defined as $L=\ln (2 l / a)-1$. |
99 | Attempt to find another approximate expression for the electric dipole moment of a conductive thin cylindrical rod (length $2l$, radius $a$, and $a \ll l$) in an electric field $\mathfrak{C}$, expressed directly in the logarithmic form of $l$ and $a$ $(\ln (4l/a))$. The field is parallel to the rod's axis.
You should... | Let $\tau(z)$ be the induced charge per unit length on the surface of the rod; $z$ is the coordinate along the axis of the cylinder, with the origin chosen at the midpoint of the rod. The condition of constant potential on the conductor's surface is
$$-\mathfrak{C} z+\frac{1}{2 \pi} \int_{0}^{2 \pi} \int_{-l}^{l} \fra... | [
"\\mathscr{P}=\\frac{\\mathfrak{C} l^{3}}{3 \\ln (4 l / a)} (1 + \\frac{7}{3 \\ln (4 l / a)})"
] | Expression | Magnetism | $\mathscr{P}$: 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. |
100 | 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 $\sigma$. | We first have
$$
\sigma=-\frac{1}{4 \pi} \frac{\partial \varphi}{\partial n}|_{\xi=0}=-(\frac{1}{4 \pi h_{1}}-\frac{\partial \varphi}{\partial \xi})_{\xi=0}
$$
(According to equation
\begin{align}
d l^2 &= h_1^2 d \xi^2 + h_2^2 d \eta^2 + h_3^2 d \zeta^2, \\
h_1 &= \frac{\sqrt{(\xi - \eta)(\xi - \zeta)}}{2 R_\xi}, \qu... | [
"\\sigma=\\mathfrak{C} \\frac{\\nu_{x}}{4 \\pi n^{(x)}}"
] | Expression | Magnetism | $\sigma$: 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.
... |
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