id int64 | question string | answer string | final_answer list | answer_type string | topic string | symbol string |
|---|---|---|---|---|---|---|
301 | Assume there is a spherical cavity within an infinite anisotropic medium, and the uniform electric field far from the cavity inside the medium is known to be $E^{(e)}$. Try to find the $y$ component of the electric field inside the cavity $E_y^{(i)}$, expressed in terms of $E_y^{(e)}$ and the dielectric and geometric p... | [] | Expression | Magnetism | $E_y^{(i)}$: y component of the electric field inside the cavity.
$\varepsilon^{(y)}$: Dielectric constant of the anisotropic medium along the y-direction.
$n^{(y)}$: Depolarization coefficient of the transformed ellipsoid along the y-direction.
$E_y^{(e)}$: y component of the uniform electric field far from the cavity... | |
302 | Consider a spherical cavity in an infinite anisotropic medium, with a known uniform electric field $E^{(e)}$ far away from the cavity. Determine the $z$ component $E_z^{(i)}$ of the electric field inside the cavity, expressed in terms of $E_z^{(e)}$ and the medium and geometric parameters. | [] | Expression | Magnetism | $E_{z}^{(i)}$: Z-component of the electric field inside the cavity.
$\varepsilon^{(z)}$: Dielectric constant of the anisotropic medium in the z-direction.
$n^{(z)}$: Depolarization factor of the transformed ellipsoid in the z-direction.
$E_{z}^{(e)}$: Z-component of the electric field far away from the cavity. | |
303 | For antiferromagnetic ferrous carbonate (whose structure belongs to the magnetic class $\boldsymbol{D}_{3 d}$), derive the expression for the x-component of magnetization $M_x$ under applied stress based on its magnetoelastic effect. | [] | Expression | Magnetism | $M_x$: x-component of magnetization
$\lambda_{1}$: Magnetoelastic coupling constant (first coefficient)
$\sigma_{x x}$: x-x component of the stress tensor
$\sigma_{y y}$: y-y component of the stress tensor
$\lambda_{2}$: Magnetoelastic coupling constant (second coefficient)
$\sigma_{y z}$: y-z component of the stress t... | |
304 | For the antiferromagnet iron carbonate (whose structure belongs to the magnetic class $\boldsymbol{D}_{3 d}$), derive the expression for the magnetization component $M_y$ under applied stress based on its piezomagnetic effect. You should return your answer as an equation. | [] | Equation | Magnetism | $M_{y}$: Magnetization component along the $y$-axis.
$\lambda_{1}$: Piezomagnetic coefficient.
$\sigma_{x y}$: Component of the stress tensor.
$\lambda_{2}$: Piezomagnetic coefficient.
$\sigma_{x z}$: Component of the stress tensor. | |
305 | For a crystal belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, determine the magnetization $x$ component $M_x$ induced by the magnetoelastic effect. | [] | Expression | Magnetism | $M_x$: x component of the magnetization
$\lambda_{1}$: First magnetoelastic coupling constant
$\sigma_{y z}$: yz component of the stress tensor | |
306 | For crystal belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, find the $y$ component of the magnetization $M_y$ induced by the piezomagnetic effect. | [] | Expression | Magnetism | $M_y$: y component of the magnetization.
$\lambda_1$: Piezomagnetic coefficient.
$\sigma_{xz}$: Component of the stress tensor. | |
307 | For crystals belonging to the magnetic crystal class $\boldsymbol{D}_{4 h}(\boldsymbol{D}_{2 h})$, determine the $z$ component $M_z$ of the magnetization induced by magnetoelastic effects. | [] | Expression | Magnetism | $M_z$: z component of the magnetization
$\lambda_2$: Magnetoelastic coupling constant
$\sigma_{xy}$: Component of the stress tensor | |
308 | Ttry to find the magnetic susceptibility $\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic field perpendicular to its axis. | [] | Expression | Magnetism | $\alpha$: Magnetic susceptibility of the conductive cylinder.
$\pi$: The mathematical constant pi.
$a$: Radius of the conductive cylinder.
$k$: A constant related to the wave number, appearing in the Bessel function argument and the wave equation.
$\mathrm{J}_{1}$: Bessel function of the first kind, order 1.
$\mathrm{J... | |
309 | Ttry to find the magnetic susceptibility $\alpha$ of a conductive cylinder (with radius $a$) in a uniform periodic external magnetic, but the magnetic field is parallel to the cylinder axis. | [] | Expression | Magnetism | $\alpha$: Magnetic susceptibility of the conductive cylinder.
$k$: A constant related to the magnetic field's spatial variation inside the cylinder, appearing in the equation $\Delta f+k^{2} f=0$.
$a$: Radius of the conductive cylinder. | |
310 | The surface of a uniaxial metallic crystal is cut so that its normal forms an angle $\theta$ with the crystal's main axis of symmetry. Considering the thermoelectric effect, under isothermal boundary conditions ($\tau=0$) and assuming $a \ll 1$, find the $xx$ component of the surface impedance $\zeta_{x x}^{(\mathrm{is... | [] | Expression | Magnetism | $\zeta_{x x}^{(\mathrm{is})}$: xx component of the surface impedance under isothermal boundary conditions
$\zeta_{0}$: Surface impedance without considering the thermoelectric effect, $\zeta_{0}=(\omega \rho_{y y} / 8 \pi)^{1 / 2}(1-\mathrm{i})$
$a$: Dimensionless parameter related to the thermoelectric effect, $a=\fra... | |
311 | Determine the x-component of the magnetic moment $\mathscr{M}_{x}$ of a conducting sphere ($\mu=1$), rotating uniformly in a uniform constant magnetic field whose components are $(\mathfrak{H}_{x}, 0, \mathfrak{H}_{z})$. | [] | Expression | Magnetism | $\mathscr{M}_{x}$: x-component of the magnetic moment of the conducting sphere in the stationary reference frame.
V: Volume of the conducting sphere.
$\alpha^{\prime}$: Component of the complex magnetic polarizability, determining the magnetic moment in the plane of vector $\boldsymbol{\Omega}$ and $\mathfrak{H}$.
$\ma... | |
312 | Determine the y-component $\mathscr{M}_{y}$ of the magnetic moment of a conducting sphere ($\mu=1$) uniformly rotating in a uniform constant magnetic field whose components are $(\mathfrak{H}_{x}, 0, \mathfrak{H}_{z})$. | [] | Expression | Magnetism | $\mathscr{M}_{y}$: y-component of the magnetic moment of the conducting sphere.
$V$: Volume of the conducting sphere.
$\alpha^{\prime \prime}$: Parameter determining the component of the magnetic moment perpendicular to the plane of the vector $\boldsymbol{\Omega}$ and $\mathfrak{H}$.
$\mathfrak{H}_{x}$: x-component of... | |
313 | Try to determine the magnetic moment of a non-uniformly rotating sphere (sphere radius is $a$). Assume the rotation speed is very low so that the penetration depth $\delta \gg a$. | [] | Expression | Magnetism | $\mathscr{M}$: Magnetic moment acquired by the sphere.
$m$: Mass of the sphere.
$a$: Radius of the sphere.
$\sigma$: Electrical conductivity of the sphere.
$c$: Speed of light.
$e$: Elementary charge.
$\Omega$: Angular velocity of rotation of the sphere.
$t$: Time.
$\mathcal M$: Magnetic moment acquired by the sphere. | |
314 | The tangential magnetic field before the shock wave $\boldsymbol{H}_{t 1}=0$, while it is $\boldsymbol{H}_{t 2} \neq 0$ after the shock wave (this kind of shock wave is called a switch-on shock wave) . Determine the range of values $v_{n 1}$ for such a shock wave in a ideal gas with thermodynamic properties $(c_p/c_v =... | [] | Interval | Magnetism | $u_{\mathrm{A} 1}$: Alfvén velocity in front of the shock wave.
$u_{01}$: A characteristic velocity related to the gas in front of the shock wave, appearing in the boundary conditions for a switch-on shock wave. | |
315 | Determine the direction of the extraordinary ray when light is refracted from vacuum into the surface of a uniaxial crystal, assuming the crystal surface is perpendicular to its optical axis. Hint: calculate $\tan(\vartheta^{\prime})$ where $\vartheta^{\prime}$ is the refraction angle. | [] | Expression | Magnetism | $\varepsilon_{\perp}$: Perpendicular component of the permittivity (or dielectric constant) of the uniaxial crystal
$\vartheta$: Incidence angle
$\varepsilon_{\|}$: Parallel component of the permittivity (or dielectric constant) of the uniaxial crystal | |
316 | Determine the direction of the extraordinary ray when vertically incident on the surface of a uniaxial crystal, assuming the optical axis of the uniaxial crystal is in an arbitrary direction. Hint: calculate $\tan(\vartheta^{\prime})$ where $\vartheta^{\prime}$ is the refraction angle. | [] | Expression | Magnetism | $\vartheta^{\prime}$: Refraction angle.
$\varepsilon_{\|}$: Permittivity parallel to the optical axis.
$\varepsilon_{\perp}$: Permittivity perpendicular to the optical axis.
$\alpha$: Angle between the optical axis and the surface normal. | |
317 | Determine the asymptotic form of the gyration vector's frequency dependence in the high-frequency regime. | [] | Expression | Magnetism | $f$: Function describing the gyration vector's frequency dependence, $f(\omega)$.
$\omega$: Angular frequency.
$\pi$: Mathematical constant pi.
$N$: Number density of electrons.
$e$: Charge of an electron, $e = -|e|$.
$c$: Speed of light.
$m$: Mass of the electron.
$|e|$: Magnitude of the electron charge.
$\varepsilon$... | |
318 | Determine the intensity distribution within the diffraction spot around the main maximum when diffraction occurs on a spherical crystal with radius $a$. | [] | Expression | Magnetism | {\mathrm{d}\sigma}: Differential scattering cross-section.
$\pi$: Mathematical constant pi.
$e$: Elementary charge.
$m$: Mass of the electron.
$c$: Speed of light in vacuum.
$\vartheta$: Scattering angle.
$n_b$: Number density of scattering centers in the crystal.
$\varkappa$: Magnitude of the wave vector transfer $\bo... | |
319 | Assume energy levels depend only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, where $n, ~ n^{\prime}, ~ l, ~ m$ are given. Find the branching ratios for transitions to $l^{\prime}=l+1$, $m^{\prime}=m+1, m, m-1$. You should return yo... | [] | Tuple | Theoretical Foundations | $l$: Azimuthal or orbital angular momentum quantum number of the initial state
$m$: Magnetic quantum number of the initial state | |
320 | Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, with $n, ~ n^{\prime}, ~ l, ~ m$ all given. Find the branching ratios for transitions to $l^{\prime}=l-1$, $m^{\prime}=m+1, m, m-1$. You should return... | [] | Tuple | Theoretical Foundations | $l$: Azimuthal quantum number of the initial state.
$m$: Magnetic quantum number of the initial state. | |
321 | Assume the energy level depends only on the principal quantum number $n$. Let the initial state be $(n l m)$ and the final state be $(n^{\prime} l^{\prime} m^{\prime})$, with $n, ~ n^{\prime}, ~ l, ~ m$ all given. Calculate the branching ratio for transitions to $l^{\prime}=l+1$ and $l^{\prime}=l-1$. You should return ... | [] | Tuple | Theoretical Foundations | $l$: Azimuthal quantum number of the initial state | |
322 | Irradiate atoms with right circularly polarized light propagating along the positive $z$ direction, causing stimulated transitions of electrons in the atom ($E_n < E_{n'}$). Find the selection rules. You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | $\Delta l$: Change in the orbital angular momentum quantum number during transition.
$\Delta m$: Change in the magnetic quantum number during transition. | |
323 | According to experimental measurements, the energy level of the hydrogen atom's $2\mathrm{s}_{1 / 2}$ is higher than the $2 \mathrm{p}_{1 / 2}$ level by 1058 MHz (Lamb shift). Find the average lifetime of the electron's spontaneous transition from the $2 \mathrm{s}_{1 / 2}$ level to the $2 \mathrm{p}_{1 / 2}$ level, in... | [] | Numeric | Theoretical Foundations | ||
324 | For two electrons at the $n \mathrm{p}$ energy level $(l=1)$ of an atom, attempt to determine the number of all possible total angular momentum eigenstates using both $L-S$ coupling and $j-j$ coupling. | [] | Numeric | Theoretical Foundations | ||
325 | Estimate the mass of a Uranium nucleus in micrograms, knowing that it contains 92 protons and 143 neutrons.
Hint: $m_{p} c^{2} \simeq m_{n} c^{2} \simeq 939 \mathrm{MeV}$. | [] | Numeric | Theoretical Foundations | ||
326 | Calculate the electric field in volt per meter that a muon feels in the $1 s$-state of muonic lead.
Hints Bohr radius $a_{B}=\hbar c /(Z \alpha m_{\mu} c^{2}), m_{\mu} c^{2}=105.6 \mathrm{MeV}$. | [] | Numeric | Theoretical Foundations | ||
327 | For the case of decay into two identical particles, when $V<v_{0}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion. | [] | Interval | Theoretical Foundations | $v_0$: Speed of the two identical decay particles in the C frame (Center of mass frame)
$V$: Velocity of the L frame (Laboratory frame) relative to the C frame (Center of mass frame)
$\pi$: Mathematical constant pi, approximately 3.14159 | |
328 | For the case of decay into two identical particles, when $v_{0}<V<\frac{v_{0}}{\sqrt{1-v_{0}^{2}}}$, determine the range of the angle (the opening angle) between the two decay particles in the $L$ system. You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint incl... | [] | Interval | Theoretical Foundations | $V$: The speed of the center-of-mass (C) system relative to the laboratory (L) system.
$v_0$: The speed of each decay particle in the center-of-mass (C) system. | |
329 | For the case of decay into two identical particles, when $V>\frac{v_{0}}{\sqrt{1-v_{0}^{2}}}$, find the range of the angle between the two decay particles in the $L$ frame (their separation angle). You should return your answer as an interval like [a, b], (a, b), [a, b), or (a, b], depending on the endpoint inclusion. | [] | Interval | Theoretical Foundations | $v_0$: Speed of each decay particle in the $C$ frame
$V$: Velocity of the $C$ frame relative to the $L$ frame | |
330 | Identify appropriately normalized coefficients in the expansion of the fields in terms of plane wave solutions with annihilation and/or creation operators. You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | $f(\mathbf{p})$: The coefficient function for annihilation operators, which is determined to be $\frac{1}{(2 \pi)^{3 / 2}}$.
$g(\mathbf{p})$: The coefficient function for creation operators, which is determined to be $0$.
$\pi$: Mathematical constant pi. | |
331 | The set of all coherent states for all values of $z$ is obviously complete. Indeed, it is overcomplete: The energy eigenstates can all be constructed by taking successive derivatives at $z=0$, so the coherent states
\footnotetext{
${ }^{1}$ [Eds.] Roy J. Glauber, "Photon correlations", Phys. Rev. Lett. 10 (1963) 83-86... | [] | Tuple | Theoretical Foundations | $\alpha$: Real constant in the completeness relation $1=\alpha \int d(\operatorname{Re} z) d(\operatorname{Im} z) e^{-\beta z^{*} z}|z\rangle\langle z|$
$\beta$: Real constant in the completeness relation $1=\alpha \int d(\operatorname{Re} z) d(\operatorname{Im} z) e^{-\beta z^{*} z}|z\rangle\langle z|$
$\pi$: Pi, the ... | |
332 | In Model 3, compute, to lowest non-vanishing order in $g$, the center-of-momentum differential cross-section and the total cross section for "nucleon"-"antinucleon" elastic scattering. | [] | Expression | Theoretical Foundations | $\sigma$: Total cross-section.
$g$: Coupling constant for the interaction between $\phi$ and $\psi$.
$\pi$: Mathematical constant pi.
$E_T$: Total energy in the center-of-momentum frame.
$\mu$: Mass parameter of the scalar field $\phi$.
$|\mathbf{p}_i|$: Magnitude of the initial 3-momentum of a particle. | |
333 | The two-particle density of states factor, $D$, in the center-of-momentum frame, $\mathbf{P}_{T}=\mathbf{0}$:
\begin{equation*}
D=\frac{1}{16 \pi^{2}} \frac{|\mathbf{p}_{f}| d \Omega_{f}}{E_{T}}.
\end{equation*}
where we have used the notation: the final particles' momenta $|\mathbf{p_f|$ in the center-of-momentum fra... | [] | Expression | Theoretical Foundations | $E_{k}$: Energy of the particle with momentum $\mathbf{k}$
$\phi$: Azimuthal angle of $\mathbf{k}$ about the $\mathbf{P}_{T}$ axis
$|\mathbf{P}_{T}|$: Magnitude of the total momentum $\mathbf{P}_{T}$ | |
334 | When we attempted to quantize the free Dirac theory
\begin{equation*}
\mathscr{L}= \pm \bar{\psi}(i \partial \!\!\!/-m) \psi \tag{P12.1}
\end{equation*}
with canonical commutation relations, we encountered a disastrous contradiction with the positivity of energy. We succeeded when we used canonical anticommutators (i... | [] | Tuple | Theoretical Foundations | $b$: Imaginary part of $\lambda$, $b = \mathrm{Im}\lambda$
$\delta^{(3)}$: Three-dimensional Dirac delta function
$\mathbf{p}$: Three-momentum vector
$\mathbf{p}^{\prime}$: Another three-momentum vector
$b_{\mathbf{p}}$: Annihilation operator for a particle with momentum $\mathbf{p}$
$b_{\mathbf{q}}^{\dagger}$: Creatio... | |
335 | Under what conditions on the $g^{\prime}$ s is $\mathscr{H}^{\prime}(0)$ invariant under $P$. Hint: $\Omega_{P T}$ is anti-unitary. You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | $g_2$: Coupling constant for the second interaction term in $\mathscr{H}'$
$g_3$: Coupling constant for the third interaction term in $\mathscr{H}'$ | |
336 | Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:
\mathscr{L}=\mathscr{L}_{\mathrm{em}}-\frac{1}{2} \lambda(\partial_{\mu} A^{\mu}+\sigma A_{\mu} A^{\mu})^{2}+\mathscr{L}_{\mathrm{ghos... | [] | Expression | Theoretical Foundations | $\sigma$: Arbitrary real number, a parameter in the gauge-fixing term.
$k^{\mu}$: Contravariant four-momentum. | |
337 | Even in quantum electrodynamics, it is possible (though not usual) to work in a gauge where ghost fields are needed. For example, this is a valid form of the electrodynamic Lagrangian:
$$
\mathscr{L}=\mathscr{L}_{\mathrm{em}}-\frac{1}{2} \lambda(\partial_{\mu} A^{\mu}+\sigma A_{\mu} A^{\mu})^{2}+\mathscr{L}_{\mathrm{gh... | [] | Expression | Theoretical Foundations | $\mathscr{L}_{\mathrm{ghost}}$: Ghost Lagrangian.
$\bar{\eta}$: Complex anti-ghost field.
$\square^{2}$: D'Alembert operator, defined as $\square^{2} = \partial_{\mu}\partial^{\mu}$.
$\sigma$: Arbitrary real number, a gauge-fixing parameter.
$A_{\mu}$: Covariant four-vector potential.
$\partial^{\mu}$: Contravariant fo... | |
338 | $\mathrm{SU}(3)$ allows only one possible coupling of the electromagnetic current to a quark and an antiquark. Thus (by the same reasoning used for the decuplet in the previous problem), in the limit of perfect $\mathrm{SU}(3)$ symmetry, if quarks are observable, their magnetic moments would be proportional to their ch... | [] | Numeric | Theoretical Foundations | ||
339 | The model
\begin{equation*}
\mathscr{L}=\frac{1}{2}(\partial_{\mu} \boldsymbol{\Phi}) \cdot(\partial^{\mu} \boldsymbol{\Phi})-U(\boldsymbol{\Phi})
\end{equation*}
was a theory with spontaneous breakdown of $\mathrm{U}(1)$ internal symmetry. The particle spectrum of the theory consisted of a massless Goldstone boson an... | [] | Expression | Theoretical Foundations | $\mu_{0}$: Mass of the photon before symmetry breaking.
$a$: Vacuum expectation value (VEV) of the scalar field, or a constant related to symmetry breaking.
$e$: Electric charge or gauge coupling constant. | |
340 | Attempt to derive the elementary excitation spectrum in a nearly ideal Bose gas, where the elementary excitation spectrum is considered as the dispersion relation of the collective wave function fluctuations. You should return your answer as an equation. | [] | Equation | Superconductivity | $\hbar$: Reduced Planck's constant.
$\omega$: Angular frequency of the collective wave.
$p$: Momentum, defined as $p=\boldsymbol{\hbar} \boldsymbol{k}$. It is treated as a scalar magnitude in the equations.
$m$: Mass of the particle.
$n$: Constant mean value related to the Bose gas density.
$U_0$: Interaction strength ... | |
341 | Calculate the probability of a quasiparticle with momentum $p$ (close to the threshold $p_{\mathrm{c}}$) emitting a phonon, when the quasiparticle speed reaches the speed of sound. | [] | Expression | Superconductivity | $w$: Total probability of phonon emission
$A$: Constant related to the interaction strength, defined as $A=p_{\mathrm{c}}+\frac{\rho}{u} \frac{\partial \varepsilon}{\partial \rho}|_{p=p_{c}}$
$p$: Momentum of a quasiparticle
$p_{\mathrm{o}}$: Threshold momentum, appears as a typo for $p_{\mathrm{c}}$ in the final expre... | |
342 | There is a planar film with thickness $d \ll \xi, \delta$. Find the critical value of the magnetic field parallel to the planar film, which can destroy superconductivity. | [] | Expression | Superconductivity | $H_{\mathrm{c}}^{f}$: Critical magnetic field of the film.
$H_{\mathrm{c}}$: Critical magnetic field of large-scale superconductors.
$\delta$: Penetration depth.
$d$: Thickness of the planar film. | |
343 | If the average magnetic induction intensity of the cylindrical sample cross-section is $\bar{B}$, and in the mixed state with an external magnetic field $\mathfrak{S}$, all vortices are distributed at distances $d \gg \delta$ from each other, forming an equilateral triangular lattice in the sample cross-section. Try to... | [] | Expression | Superconductivity | $\mathfrak{S}$: External magnetic field
$H_{\mathrm{c} 1}$: Lower critical field
$\phi_{0}$: Magnetic flux quantum
$\delta$: Penetration depth
$a$: Dimensionless vortex spacing, $a = d/\delta$ | |
344 | If the average magnetic induction intensity of the cross-section of a cylindrical sample is $\bar{B}$, and in the mixed state with an applied magnetic field of $\mathfrak{S}$, each vortex line is distributed at a distance $d \gg \delta$ from each other and forms an equilateral triangular lattice within the sample cross... | [] | Expression | Superconductivity | $a$: Dimensionless vortex spacing, $a = d/\delta$
$\phi_0$: Magnetic flux quantum
$\delta$: London penetration depth
$\bar{B}$: Average magnetic induction intensity of the cross-section of a cylindrical sample | |
345 | Ignoring interactions between spins, calculate the magnetization of a paramagnet when the ratio of $\beta \mathscr{G}$ to $T$ is arbitrary. | [] | Expression | Magnetism | M: Magnetization of the paramagnet.
$\beta$: A constant that, when multiplied by the magnetic field $\mathscr{G}$, gives an energy scale.
N: Total number of spins.
V: Volume.
S: Total spin quantum number.
$\mathscr{G}$: Magnetic field strength.
T: Absolute temperature. | |
346 | Taking $m=2, n=10, r_{0}=0.3 \mathrm{~nm}, W=4 \mathrm{eV}$, calculate the value of $\beta$, unit $\mathrm{eV} \cdot \mathrm{m}^{10}$ | [] | Numeric | Theoretical Foundations | $\beta$: Constant coefficient for the repulsive term in the interaction energy | |
347 | For $\mathrm{H}_{2}$, the Lennard-Jones potential parameters obtained from gas measurements are $\varepsilon=50 \times 10^{-6} J, \sigma=2.96 \stackrel{\circ}{\mathrm{~A}}$. Calculate the binding energy of $\mathrm{H}_{2}$ when it forms a face-centered cubic solid molecular hydrogen (in units of $\mathbf{K J} / \mathrm... | [] | Numeric | Theoretical Foundations | ||
348 | Consider the lattice vibrations of a diatomic chain where the force constants between nearest neighbor atoms alternate as $c$ and $10c$. The two types of atoms have the same mass, and the nearest neighbor distance is $\frac{a}{2}$. Find the vibrational frequency $\omega(k)$ at $k=0$. You should return your answer as a ... | [] | Tuple | Theoretical Foundations | $c$: Force constant between nearest neighbor atoms.
$M$: Mass of the atoms in the diatomic chain. | |
349 | Consider the lattice vibrations of a diatomic chain where the force constants alternate as $c$ and $10c$ between nearest neighbor atoms on the chain. The two kinds of atoms have the same mass and the nearest neighbor distance is $\frac{a}{2}$. Find the vibration frequency $\omega(k)$ at $k=\frac{\pi}{a}$. You should re... | [] | Tuple | Theoretical Foundations | $c$: Force constant between nearest neighbor atoms.
$M$: Mass of the atoms in the diatomic chain. | |
350 | Find the optical frequencies $\omega_{\max }^{0}, \omega_{\min }^{0}$, and present the answer in tuple form. You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | $\omega_{\max}^0$: Maximum optical frequency
$\omega_{\min}^0$: Minimum optical frequency
$\beta$: Spring constant, $\beta=1.5 \times 10^{1} \mathrm{~N} / \mathrm{m}$
$m$: Mass of the lighter atom, $m=5 \times 1.67 \times 10^{-24} \mathrm{~g}$
$M$: Mass of the heavier atom, such that $\frac{M}{m}=4$ | |
351 | What is the corresponding phonon energy in electron volts for the optical wave $\omega_{\min }^{o}$? | [] | Numeric | Theoretical Foundations | ||
352 | Determine the wavelength band of the electromagnetic wave corresponding to the optical wave $\omega_{\max }^{0}$. | [] | Numeric | Theoretical Foundations | ||
353 | For a one-dimensional lattice with a lattice constant of $2.5 A$, estimate the time required for an electron to move from the bottom of the energy band to the top under an external electric field of $10^{7} \mathrm{~V} / \mathrm{m}$ | [] | Numeric | Theoretical Foundations | ||
354 | The spin of $\mathrm{He}^{3}$ is $1 / 2$, making it a fermion. The density of liquid $\mathrm{He}^{3}$ near absolute zero is $0.081 \mathrm{gcm}^{-3}$. Calculate the Fermi temperature $\mathbf{T}^{\mathbf{F}}$ | [] | Numeric | Theoretical Foundations | ||
355 | Find the Fermi energy and Fermi temperature, and provide the answer as a tuple You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | ||
356 | Find the average free path of electrons at room temperature and low temperature, represented as a tuple You should return your answer as a tuple format. | [] | Tuple | Theoretical Foundations | ||
357 | Find the ground state orbital radius; | [] | Numeric | Theoretical Foundations | ||
358 | Given the expression for the free energy of an object, how can the average kinetic energy of the object's particles be calculated? | [] | Expression | Others | $m$: Mass of the object's identical particles.
$F$: Free energy of the object.
$r$: A subscript indicating that a set of parameters, possibly related to position or other extensive variables, are held constant during the partial differentiation.
$v$: A subscript indicating that a set of parameters, possibly related to ... | |
359 | Find the expression for heat capacity $C_{\nu}$ when variables are $T, \mu, V$. | [] | Expression | Others | $C_v$: Heat capacity at constant volume
$T$: Absolute temperature
$S$: Entropy
$N$: Number of particles
$\mu$: Chemical potential | |
360 | Find the probability distribution of atomic kinetic energy. | [] | Expression | Others | $\mathrm{d} w_{\varepsilon}$: Infinitesimal probability of an atom having kinetic energy in the range $\varepsilon$ to $\varepsilon + \mathrm{d}\varepsilon$
$\varepsilon$: Atomic kinetic energy
$T$: Temperature
$\pi$: Mathematical constant pi
$\mathrm{e}$: Euler's number, the base of the natural logarithm | |
361 | Find the probability distribution of molecular rotational angular velocity. | [] | Expression | Others | $\mathrm{d} w_{\Omega}$: Differential probability distribution of angular velocity.
$\pi$: Mathematical constant pi.
$T$: Temperature (or $k_B T$ where $k_B$ is Boltzmann constant).
$I_1$: Principal moment of inertia along the first principal axis.
$I_2$: Principal moment of inertia along the second principal axis.
$I_... | |
362 | Determine the coordinate density matrix of the harmonic oscillator. | [] | Expression | Others | $\rho$: Coordinate density matrix of the harmonic oscillator.
$q$: Oscillator coordinate.
$q^{\prime}$: A different oscillator coordinate, used in the density matrix $\rho(q, q^{\prime})$.
$\omega$: Angular frequency of the harmonic oscillator.
$\pi$: Mathematical constant pi.
$\hbar$: Reduced Planck's constant.
$T$: T... | |
363 | Find the distribution of particles by momentum for a relativistic ideal gas. | [] | Expression | Others | $\mathrm{d} N_{p}$: Differential number of particles with momentum in a specific range
$N$: Total number of particles
$V$: Volume
$c$: Speed of light
$m$: Mass of the particle
$p$: Magnitude of the momentum of the particle
$T$: Temperature
$K_{1}$: Macdonald function (Hankel function of imaginary argument) of order 1
$... | |
364 | Find the number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\theta$ and $\theta+\mathrm{d} \theta$. | [] | Expression | Others | $\mathrm{d} \nu_{\theta}$: Number of gas molecules that collide with the unit area of the vessel wall per unit time, with the angle between the velocity direction and the surface normal of the vessel wall located between $\theta$ and $\theta+\mathrm{d} \theta$.
$N$: Total number of gas molecules.
$V$: Volume of the ves... | |
365 | Find the number of gas molecules with speeds between $v$ and $v+\mathrm{d} v$ that collide with a unit area of the wall per unit time. | [] | Expression | Others | $\mathrm{d} \nu_{v}$: Number of gas molecules with speeds between $v$ and $v+\mathrm{d} v$ that collide with a unit area of the wall per unit time.
$N$: Total number of gas molecules.
$V$: Volume.
$\pi$: Mathematical constant pi, approximately 3.14159.
$m$: Mass of a single gas molecule.
$T$: Temperature of the gas. In... | |
366 | Find the work and heat obtained in the process of gas under constant pressure (isobaric process), expressed as a tuple. You should return your answer as a tuple format. | [] | Tuple | Others | $R$: Work obtained in the process.
$Q$: Heat obtained in the process.
$N$: Number of moles or particles of the gas.
$T_1$: Initial temperature of the gas.
$T_2$: Final temperature of the gas.
$c_p$: Molar specific heat at constant pressure. | |
367 | If a gas is compressed from volume $V_{1}$ to volume $V_{2}$ following the law $P V^{n}=a$ (polytropic process), calculate the work done on it and the heat received by it, expressed as a tuple. You should return your answer as a tuple format. | [] | Tuple | Others | $R$: Work done on the gas, $R = -\int_{V_{1}}^{V_{2}} P \mathrm{~d} V$.
$Q$: Heat received by the gas.
$a$: Polytropic constant, defined by the process law $P V^{n}=a$.
$n$: Polytropic index.
$V_2$: Final volume of the gas.
$V_1$: Initial volume of the gas.
$c_v$: Molar specific heat at constant volume. | |
368 | Find the average value $\langle\exp (\alpha_{i} x_{i})\rangle$, where $\alpha_{i}$ is a constant and $x_{i}$ is a fluctuation following a Gaussian distribution. | [] | Expression | Others | $\langle\exp (\alpha_{i} x_{i})\rangle$: Average value of the exponential of $\alpha_{i} x_{i}$
$\alpha_{i}$: Constant coefficient
$\alpha_{k}$: Constant coefficient
$\langle x_{i} x_{k}\rangle$: Average value of the product of fluctuations $x_i$ and $x_k$
$\beta_{ik}^{-1}$: Inverse of coefficient matrix element $\beta... | |
369 | Try to find $\langle(\Delta W)^{2}\rangle$ (with $P,V,C_p,T$ and $S$ as variables). | [] | Expression | Others | $\langle(\Delta W)^{2}\rangle$: Mean square fluctuation of work, defined by $-T V^{2}(\frac{\partial P}{\partial V})+T^{2} C_{p}$
$T$: Temperature
$V$: Volume
$P$: Pressure
$C_p$: Heat capacity at constant pressure | |
370 | Try to find $\langle\Delta T \Delta P\rangle$ (where $V,C_v,P$ and $T$ are variables). | [] | Expression | Others | $\Delta T$: Fluctuation in temperature
$\Delta P$: Fluctuation in pressure
$T$: Temperature
$C_v$: Heat capacity at constant volume
$P$: Pressure
$V$: Volume | |
371 | Try to find $\langle\Delta S \Delta V\rangle$ ( $V$ and $T$ are variables). | [] | Expression | Others | $V$: Volume, specified as a variable
$T$: Temperature, specified as a variable | |
372 | Find the asymptotic behavior of $\alpha(\omega)$ as $\omega \rightarrow \infty$ (assuming $\alpha(\infty)=0$). | [] | Expression | Others | $\mathrm{i}$: Imaginary unit, $\sqrt{-1}$.
$\hbar$: Reduced Planck's constant.
$\omega$: Angular frequency.
$\dot{\hat{x}}$: Time derivative of the position operator, representing the velocity operator.
$\hat{x}$: Position operator. | |
373 | Consider a system composed of two independent oscillators (i.e., two types of phonons), with $n_{1}, ~ n_{2}$ representing their quantum numbers (phonon numbers), and $a_{1}^{+}, ~ a_{1}, ~ a_{2}^{+}, ~ a_{2}$ representing the quantum number raising and lowering operators (i.e., the creation and annihilation operators ... | [] | Tuple | Others | ||
374 | Denote the creation and annihilation operators of a single-particle state in a fermion system as $a^{+}$ and $a$, respectively, satisfying the fundamental anti-commutation relations
\begin{align*}
& {[a, a^{+}]_{+} \equiv a a^{+}+a^{+} a=1} \tag{1}\\
& a^{2}=0, \quad(a^{+})^{2}=0 \tag{2}
\end{align*}
Let $\hat{n}=a... | [] | Tuple | Others | $a^{+}$: Creation operator of a single-particle state in a fermion system
$a$: Annihilation operator of a single-particle state in a fermion system | |
375 | Introduce the total particle number operator on the energy level $\varepsilon_{j}$
\begin{align*}
\hat{N}_{j} & =\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j-m}^{+} a_{j-m}) \\
& =\sum_{m>0}(a_{j m}^{+} a_{j m}+a_{j \bar{m}}^{+} a_{j \bar{m}}) \tag{1}
\end{align*}
Calculate $[\hat{N}_{j}, C_{j}^{+}], ~[C_{j}, C_{j}^{+}], ~[C_... | [] | Tuple | Others | $C_j^{+}$: Creation operator for a pair of fermions with angular momentum coupled to 0 on level $\varepsilon_{j}$, defined as $C_{j}^{+}=\frac{1}{\sqrt{\Omega_{j}}} \sum_{m>0}(-1)^{j-m} a_{j m}^{+} a_{j-m}^{+}$.
$\Omega_j$: Normalization constant for the pair creation/annihilation operators, related to the degeneracy o... | |
376 | Starting from the Dirac equation describing the motion of electrons in an electromagnetic field, derive the electronic current flow density. | [] | Expression | Others | $\boldsymbol{j}_{e}$: Electronic current flow density.
$e$: Elementary charge (electron charge).
$m$: Electron mass.
$\varphi^{+}$: Hermitian conjugate of the large component of the Dirac spinor wavefunction.
$\boldsymbol{p}$: Momentum operator.
$c$: Speed of light.
$\boldsymbol{A}$: Magnetic vector potential.
$\varphi... | |
377 | Consider the second-order approximation of the stationary Dirac equation under the non-relativistic limit, determining the specific form of the Hamiltonian operator when an electron is moving in a central force field. | [] | Expression | Others | $H$: Hamiltonian operator.
$\boldsymbol{p}^{2}$: Square of the magnitude of the momentum operator, $(\boldsymbol{\sigma} \cdot \boldsymbol{p})^{2}$.
$m$: Mass of the electron.
$V$: Central force potential energy, defined as $V(r) = -e\phi(r)$.
$\boldsymbol{p}^{4}$: Fourth power of the magnitude of the momentum operator... | |
378 | The maximum energy value is at the corners of the first Brillouin zone.
Try to find the number of states $N(E) \mathrm{d} E$ in the unit area crystal within the energy range $E \sim(E+\mathrm{d} E)$. | [] | Expression | Semiconductors | $N(E)$: Number of states per unit energy interval in the unit area crystal.
$\mathrm{d} E$: Infinitesimal energy range.
$m_{\mathrm{p}}$: Effective mass parameter in the energy dispersion relation near the extremum.
$\hbar$: Reduced Planck's constant. | |
379 | The energy $E$ near the valence band top of a certain semiconductor crystal can be expressed as: $E(k)=E_{\text {max }}-10^{26} k^{2}(\mathrm{erg})$. Now, removing an electron with wave vector $k=10^{7} \mathrm{i} / \mathrm{cm}$, calculate the effective mass of the hole left by this electron. | [] | Numeric | Semiconductors | ||
380 | In an anisotropic crystal, its energy $E$ can be expressed in terms of the components of wave vector $\boldsymbol{k}$ as $E(k)=A k_{x}^{2}+B k_{y}^{2}+C k_{z}^{2}$. Try to derive the equation of motion of electrons where the left-hand-side is $\frac{dv{dt}$.} You should return your answer as an equation. | [] | Equation | Semiconductors | $v$: Velocity of the electron.
$t$: Time.
$F$: Force acting on the electron.
$h$: Planck's constant.
$E$: Energy of the electron in the anisotropic crystal.
$k$: Wave vector (used as a general variable for differentiation in the context of $E(k)$). | |
381 | For a one-dimensional lattice with a lattice constant of $2.5 \AA$, when an external electric field of $10^{2} \mathrm{~V} / \mathrm{m}$ is applied, calculate the time required for an electron to move from the bottom of the energy band to the top. $(1 \AA=10 \mathrm{~nm}=10^{-10} \mathrm{~m})$ | [] | Numeric | Semiconductors | ||
382 | A one-dimensional lattice with a lattice constant of $2.5 \AA$, calculate the time required for an electron to move from the bottom to the top of the energy band when an external electric field of $10^{7} \mathrm{~V} / \mathrm{m}$ is applied. $(1 \AA=10 \mathrm{~nm}=10^{-10} \mathrm{~m})$ | [] | Numeric | Semiconductors | ||
383 | Donor ionization energy; | [] | Numeric | Semiconductors | ||
384 | Acceptor ionization energy; | [] | Numeric | Semiconductors | ||
385 | Ground state electron orbital radius $r_{1}$; You should return your answer as a tuple format. | [] | Tuple | Semiconductors | ||
386 | What is the acceptor concentration when the electron orbitals of adjacent impurity atoms overlap significantly? | [] | Numeric | Semiconductors | ||
387 | For silicon material doped with n-type impurity phosphorus, try to calculate the concentration of phosphorus when weak degeneracy occurs at room temperature. | [] | Numeric | Semiconductors | ||
388 | For the germanium material doped with n-type impurity phosphorus, try to calculate the numerical value of its phosphorus doping concentration at room temperature when weak degeneracy occurs. | [] | Numeric | Semiconductors | ||
389 | For silicon materials doped with n-type impurity phosphorus, try to calculate the dopant concentration when weak degeneracy occurs at room temperature. | [] | Numeric | Semiconductors | ||
390 | For a semiconductor silicon sample with a donor impurity concentration of $10^{12} \mathrm{~cm}^{-3}$, calculate the equation that the temperature value (in K) satisfies when its intrinsic carrier concentration $n_i$ equals the donor impurity concentration $N_d$. Assume $E_{\mathrm{g}}=1 \mathrm{eV}, m_{\mathrm{n}}^{*}... | [] | Equation | Semiconductors | $T$: Temperature. | |
391 | Given a silicon sample with a donor concentration $N_{\mathrm{D}}=2 \times 10^{14} \mathrm{~cm}^{-3}$ and an acceptor concentration $N_{\mathrm{A}}=10^{14} \mathrm{~cm}^{-3}$, where the donor ionization energy $\Delta E_{\mathrm{D}}=E_{\mathrm{c}}-E_{\mathrm{D}}=0.05 \mathrm{eV}$, find the equation that the temperature... | [] | Equation | Semiconductors | $T$: Temperature in Kelvin. | |
392 | In a boron-doped non-degenerate p-type silicon, containing a certain concentration of indium, the hole concentration at room temperature is measured as $p_{0}=1.1 \times 10^{16} \mathrm{~cm}^{-3}$. Given that the boron doping concentration $N_{\mathrm{A} 1}=10^{16} \mathrm{~cm}^{-3}$ and its ionization energy $\Delta E... | [] | Numeric | Semiconductors | ||
393 | At room temperature, the resistivity of intrinsic germanium is $47 \Omega \cdot \mathrm{~cm}$. If antimony impurities are added so that there is one impurity atom per $10^{6}$ germanium atoms. Assume all impurities are ionized. The concentration of germanium atoms is $4.4 \times 10^{22} / \mathrm{cm}^{3}$. Calculate th... | [] | Numeric | Semiconductors | ||
394 | When $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}, \mu_{\mathrm{p}}=1900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{n}}=3800 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, try to find the intrinsic conductivity of germanium. | [] | Numeric | Semiconductors | ||
395 | When $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}, \mu_{\mathrm{p}}=1900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{n}}=3800 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, calculate the minimum conductivity of germanium. | [] | Numeric | Semiconductors | ||
396 | Mean free time $\tau$; | [] | Numeric | Semiconductors | ||
397 | Average free path $l$; | [] | Numeric | Semiconductors | ||
398 | What should be the density of impurities to achieve a resistivity of $0.2 \Omega \cdot \mathrm{~cm}$ for n-type silicon? | [] | Numeric | Semiconductors | ||
399 | A boron-doped non-degenerate p-type silicon sample contains a certain concentration of indium, with a measured resistivity at room temperature (300K) of $\rho=2.84 \Omega \cdot \mathrm{~cm}^{2}$. Given that the doped boron concentration is $N_{\mathrm{a} 1}=10^{16} / \mathrm{cm}^{3}$, with boron ionization energy $E_{\... | [] | Numeric | Semiconductors | ||
400 | Given that the conductivity of intrinsic germanium is $3.56 \times 10^{-2} \mathrm{~S} / \mathrm{cm}$ at 310 K and $0.42 \times$ $10^{-2} \mathrm{~S} / \mathrm{cm}_{\circ}$ at 273 K, an n-type germanium sample has a donor impurity concentration of $N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$ at these two temperatures. Ca... | [] | Numeric | Semiconductors |
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