CMPhysBench: A Benchmark for Evaluating Large Language Models in Condensed Matter Physics
Paper • 2508.18124 • Published • 49
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 |
End of preview. Expand in Data Studio
CMPhysBench: A Benchmark for Evaluating Large Language Models in Condensed Matter Physics
🎉🎉🎉 This paper is accpeted by ICLR 2026.
We introduce CMPhysBench, designed to assess the proficiency of Large Language Models (LLMs) in Condensed Matter Physics, as a novel Benchmark. CMPhysBench is composed of more than 520 graduate-level meticulously curated questions covering both representative subfields and foundational theoretical frameworks of condensed matter physics, such as magnetism, superconductivity, strongly correlated systems, etc. To ensure a deep understanding of the problem-solving process,we focus exclusively on calculation problems, requiring LLMs to independently generate comprehensive solutions. Meanwhile, leveraging tree-based representations of expressions, we introduce the Scalable Expression Edit Distance (SEED) score, which provides fine-grained (non-binary) partial credit and yields a more accurate assessment of similarity between prediction and ground-truth. Our results show that even the best models, Grok-4, reach only 36 average SEED score and 28% accuracy on CMPhysBench, underscoring a significant capability gap, especially for this practical and frontier domain relative to traditional physics.
Citations
@article{wang2025cmphysbench,
title={CMPhysBench: A Benchmark for Evaluating Large Language Models in Condensed Matter Physics},
author={Wang, Weida and Huang, Dongchen and Li, Jiatong and Yang, Tengchao and Zheng, Ziyang and Zhang, Di and Han, Dong and Chen, Benteng and Luo, Binzhao and Liu, Zhiyu and others},
journal={arXiv preprint arXiv:2508.18124},
year={2025}
}
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