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501 |
Superconductivity
|
Please discuss the interaction problem of adding two electrons outside a filled Fermi sea at $T=0K$. Approximation: You can assume that the electrons inside the Fermi sea are free electrons. Please calculate the interaction between two electrons in the weak coupling case.
|
[] |
Expression
|
{"$C_{\\mathbf{k}}$": "creation operator for an electron with momentum $\\mathbf{k}$", "$C_{-\\mathbf{k}}$": "creation operator for an electron with momentum $-\\mathbf{k}$", "$|F\\rangle$": "ground state of the normal electrons with a filled Fermi sea", "$\\overline{H}$": "reduced Hamiltonian", "$\\varepsilon_{\\mathbf{k}}$": "energy of an electron with momentum $\\mathbf{k}$", "$V$": "interaction potential", "$k_F$": "Fermi wave vector", "$a(\\mathbf{k})$": "coefficient for electron pair state with momentum $\\mathbf{k}$", "$E$": "energy of the system state", "$\\lambda$": "Lagrange multiplier in the variational method", "$\\hbar\\omega_D$": "cutoff energy determined by Debye frequency $\\omega_D$", "$g(0)$": "density of states at the Fermi surface", "$A$": "sum of coefficients $a(\\mathbf{k})$ over allowed states"}
| |||
502 |
Superconductivity
|
It is known that the BCS superconducting Hamiltonian can be written as:
\begin{equation} \label{eq:6.4.23_again}
\bar{H} = E_s(0) + \sum_k \sqrt{\epsilon_k^2 + \Delta^2} (\alpha_k^\dagger \alpha_k + \alpha_{-k}^\dagger \alpha_{-k}),
\end{equation}
where $\alpha$ is the quasiparticle operator, $\Delta$ represents the superconducting energy gap, and $E_s(0)$ denotes the ground state energy, given by the expression:
\begin{equation}
E_s(0) = 2 \sum_k \epsilon_k v_k^2 - 2\Delta \sum_k u_k v_k + \frac{\Delta^2}{V},
\end{equation}
with:
\begin{equation*}
\xi_k = \sqrt{\epsilon_k^2 + \Delta^2} \quad (\epsilon_k = E_k - E_F),
\end{equation*}
Please solve the following problem based on the Hamiltonian: Calculate the superconducting gap $\Delta$ at zero temperature;
|
[] |
Expression
|
{"$\\Delta$": "superconducting gap", "$T$": "temperature", "$V$": "interaction potential", "$\\alpha_k$": "quasiparticle operator", "$\\alpha_{-k}$": "quasiparticle operator", "$u_k$": "Bogoliubov transformation coefficient", "$v_k$": "Bogoliubov transformation coefficient", "$\\epsilon_k$": "quasiparticle energy", "$\\xi_k$": "quasiparticle excitation energy", "$g(0)$": "density of states at the Fermi surface", "$\\omega_D$": "Debye frequency"}
| |||
503 |
Superconductivity
|
The known Hamiltonian of BCS superconductors can be expressed as:
\begin{equation} \label{eq:6.4.23_again}
\bar{H} = E_s(0) + \sum_k \sqrt{\epsilon_k^2 + \Delta^2} (\alpha_k^\dagger \alpha_k + \alpha_{-k}^\dagger \alpha_{-k}),
\end{equation}
Here, $\alpha$ is the quasiparticle operator, $\Delta$ represents the energy gap of the quasiparticles, and $E_s(0)$ is the ground state energy, expressed as:
\begin{equation}
E_s(0) = 2 \sum_k \epsilon_k v_k^2 - 2\Delta \sum_k u_k v_k + \frac{\Delta^2}{V},
\end{equation}
where:
\begin{equation*}
\xi_k = \sqrt{\epsilon_k^2 + \Delta^2} \quad (\epsilon_k = E_k - E_F),
\end{equation*}
Please solve the following problem starting from the Hamiltonian: Please calculate the superconducting critical temperature $T_c$ based on the gap equation.
|
[] |
Expression
|
{"$T_c$": "critical temperature for superconductivity", "$T$": "temperature", "$\\Delta$": "superconducting energy gap", "$V$": "interaction potential", "$k$": "wave number or momentum index", "$\\alpha_k$": "quasiparticle annihilation operator", "$\\alpha_k^\\dagger$": "quasiparticle creation operator", "$\\xi_k$": "quasiparticle excitation energy", "$\\epsilon_k$": "single-particle energy level", "$u_k$": "transformation coefficient for the quasiparticle state", "$v_k$": "transformation coefficient for the quasiparticle state", "$\\beta$": "inverse temperature factor", "$k_B$": "Boltzmann constant", "$g(0)$": "density of states at the Fermi level", "$\\omega_D$": "Debye frequency", "$M$": "ionic mass", "$e$": "base of the natural logarithm", "$\\gamma$": "Euler's constant"}
| |||
504 |
Superconductivity
|
It is known that BCS superconducting Hamiltonian can be written as: \begin{equation} \label{eq:6.4.23_context} \bar{H} = E_s(0) + \sum_k \sqrt{\epsilon_k^2 + \Delta^2} (\alpha_k^\dagger \alpha_k + \alpha_{-k}^\dagger \alpha_{-k}), \end{equation} where $\alpha$ is the quasiparticle operator, $\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ denotes the ground state energy, given by: \begin{equation} E_s(0) = 2 \sum_k \epsilon_k v_k^2 - 2\Delta \sum_k u_k v_k + \frac{\Delta^2}{V}, \end{equation} where: \begin{equation*} \xi_k = \sqrt{\epsilon_k^2 + \Delta^2} \quad (\epsilon_k = E_k - E_F), \end{equation*} Please solve the following problem starting from the Hamiltonian: At $T \ll T_c$, calculate the approximate expression for $\Delta(T)$ according to the previous results
|
[] |
Expression
|
{"$T$": "temperature", "$T_c$": "critical temperature", "$\\Delta(T)$": "energy gap at temperature T", "$\\Delta(0)$": "energy gap at absolute zero", "$\\hbar$": "reduced Planck's constant", "$\\omega_D$": "Debye frequency", "$\\beta$": "inverse of thermal energy (1/kB*T)", "$\\epsilon$": "energy variable", "$k_B$": "Boltzmann constant"}
| |||
505 |
Superconductivity
|
The Hamiltonian of known BCS superconductivity can be written as:
\begin{equation} \label{eq:6.4.23_context}
\bar{H} = E_s(0) + \sum_k \sqrt{\epsilon_k^2 + \Delta^2} (\alpha_k^\dagger \alpha_k + \alpha_{-k}^\dagger \alpha_{-k}),
\end{equation}
where $\alpha$ is the quasiparticle operator, $\Delta$ represents the energy gap of the quasiparticles, $E_s(0)$ represents the ground state energy, expressed as:
\begin{equation}
E_s(0) = 2 \sum_k \epsilon_k v_k^2 - 2\Delta \sum_k u_k v_k + \frac{\Delta^2}{V},
\end{equation}
where:
\begin{equation*}
\xi_k = \sqrt{\epsilon_k^2 + \Delta^2} \quad (\epsilon_k = E_k - E_F),
\end{equation*}
Please complete the following problem starting from the Hamiltonian: Calculate the approximate expression for $\Delta(T)$ as $T \rightarrow T_c$
|
[] |
Expression
|
{"$\\Delta(T)$": "energy gap as function of temperature T", "$T$": "temperature", "$T_c$": "critical temperature", "$g(0)$": "density of states at the Fermi surface", "$V$": "interaction potential", "$\\hbar \\omega_D$": "Debye energy", "$\\epsilon$": "energy variable for integration", "$\\beta$": "inverse temperature in units of energy", "$k_B$": "Boltzmann constant", "$\\omega_n$": "Matsubara frequency", "$\\xi$": "generic energy variable", "$\\gamma$": "Euler constant", "$\\zeta(3)$": "Riemann zeta function at 3"}
| |||
506 |
Superconductivity
|
It is known that the Hamiltonian for BCS superconductors is
\begin{equation}
\bar{H} = E_s(0) + \sum_k \sqrt{\epsilon_k^2 + \Delta^2} (\alpha_k^\dagger \alpha_k + \alpha_{-k}^\dagger \alpha_{-k}),
\end{equation}
where $\alpha$ is the quasiparticle operator, $\Delta$ indicates the quasiparticle energy gap, $E_s(0)$ represents the ground state energy, and the expression is:
\begin{equation}
E_s(0) = 2 \sum_k \epsilon_k v_k^2 - 2\Delta \sum_k u_k v_k + \frac{\Delta^2}{V},
\end{equation}
where:
\begin{equation*}
\xi_k = \sqrt{\epsilon_k^2 + \Delta^2} \quad (\epsilon_k = E_k - E_F),
\end{equation*}
Please calculate: In the case $T \ll T_c$, the low-temperature approximation of the electronic specific heat
|
[] |
Expression
|
{"$T$": "temperature", "$T_c$": "critical temperature", "$c_{es}$": "electronic specific heat capacity", "$\\Delta (0)$": "energy gap at zero temperature", "$\\Delta (T)$": "energy gap at temperature $T$", "$\\beta$": "inverse temperature factor (1/kBT)", "$f$": "distribution function", "$\\xi$": "variable related to energy", "$k_B$": "Boltzmann constant", "$g(0)$": "density of states at the Fermi level", "$\\epsilon$": "energy variable", "$\\hbar$": "reduced Planck's constant"}
| |||
507 |
Magnetism
|
Given the ferromagnetic Heisenberg model $H = -J \sum_{l,\delta}S_l \cdot S_{l+\delta}$, where $\delta$ represents the difference in position between neighboring lattice sites, discuss the case of $J>0$. Please perform the following calculations: Solve for the ground state energy of the Heisenberg model.
|
[] |
Expression
|
{"$\\hat{\\mathbf{S}}_l$": "vector spin operator at lattice site", "$\\hat{S}_l^x$": "x-component of spin operator at lattice site", "$\\hat{S}_l^y$": "y-component of spin operator at lattice site", "$\\hat{S}_l^z$": "z-component of spin operator at lattice site", "$H$": "Hamiltonian", "$\\hbar$": "reduced Planck's constant", "$S$": "ion spin quantum number", "$m$": "eigenvalue of the z-component of spin", "$\\hat{S}_l^+$": "spin raising operator at lattice site", "$\\hat{S}_l^-$": "spin lowering operator at lattice site", "$l$": "lattice site index", "$\\hat{S}_{l+\\delta}$": "spin operator at neighboring lattice site", "$J$": "exchange interaction constant", "$\\delta$": "neighbor index", "$Z$": "coordination number of the lattice", "$E_0$": "ground state energy"}
|
Electromagnetic properties
| ||
508 |
Strongly Correlated Systems
|
Consider the action of the Hubbard model: \begin{align}
S_{\text{loc}}[\Phi^\dagger, \Phi] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger(\partial_\tau - \mu - i\Delta_c - \Delta\sigma^z)\Phi_{\mathbf{r}}, \nonumber \\
S_1[\Phi^\dagger, \Phi, \Omega] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger \dot{R}_{\mathbf{r}} \Phi_{\mathbf{r}}, \nonumber \\
S_2[\Phi^\dagger, \Phi, \Omega] &= - \int_0^\beta d\tau \sum_{\mathbf{r},\mathbf{r}'} t_{\mathbf{r}\mathbf{r}'} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger R_{\mathbf{r}'} \Phi_{\mathbf{r}'}.
\label{eq:6.141}
\end{align}
Where: \begin{equation} R_{\mathbf{r}} = e^{-\frac{i}{2}\varphi_{\mathbf{r}}\sigma^z} e^{-\frac{i}{2}\theta_{\mathbf{r}}\sigma^y} e^{-\frac{i}{2}\psi_{\mathbf{r}}\sigma^z} = \begin{pmatrix} \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} & -\sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} \\ \sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} & \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} \end{pmatrix}, \end{equation}
We consider the fields $\delta_c$ and $\delta$ at the saddle-point approximation: $\delta_c = i(U/2)\langle\phi_{\mathbf{r}}^\dagger\phi_{\mathbf{r}}\rangle$ and $\delta = (U/2)\langle\phi_{\mathbf{r}}^\dagger\sigma^z\phi_{\mathbf{r}}\rangle$, for the half-filled case, $\mu + i\delta_c = 0$. Please calculate First-order moment $\langle S_2 \rangle$
|
[] |
Numeric
|
{"$\\langle S_2 \\rangle$": "first-order moment", "$\\langle \\Phi^*_{\\mathbf{r\\sigma}}\\Phi_{\\mathbf{r\\sigma}}\\rangle$": "average value of the product of field functions at location and spin", "$\\mathbf{r}$": "position vector", "$\\mathbf{r'}$": "another position vector", "$\\sigma$": "spin index"}
|
Strong correlation basis
| ||
509 |
Others
|
The first-order perturbation calculation for the Hamiltonian of interacting electron systems \begin{equation}H = H_0 + H' = \sum_{\mathbf{k},\sigma} \frac{\hbar^2 k^2}{2m} C_{\mathbf{k}\sigma}^{\dagger} C_{\mathbf{k}\sigma} + \frac{1}{2V} \sum_{\mathbf{q}} v(q) (\rho_{\mathbf{q}}^{\dagger} \rho_{\mathbf{q}} - N) \label{eq:4.9.1}\end{equation} is called the Hartree-Fock approximation. Therefore, within the Hartree-Fock approximation, one only needs to take the diagonal average of $H$ with the ground state of non-interacting electrons (Fermi surface state) \begin{equation}|0\rangle_0 = \prod_{k \le k_F, \sigma} C_{\mathbf{k}\sigma}^{\dagger} |Vac\rangle \label{eq:4.9.2}\end{equation}. Here, $|Vac\rangle$ represents the state where all $\mathbf{k}$ spaces are unoccupied, i.e., the true vacuum state. The average ground state energy per electron under the Hartree-Fock approximation (expressed in terms of Fermi wave vector);
|
[] |
Expression
|
{"$E^{\\text{HF}}_0$": "ground state energy in the Hartree-Fock approximation", "$E_k$": "total kinetic energy of the electron system", "$E_{\\text{ex}}$": "exchange energy", "$k$": "wave vector", "$\\sigma$": "spin index", "$\\hbar$": "reduced Planck's constant", "$m$": "electron mass", "$k_F$": "Fermi wave vector", "$E_F$": "Fermi energy", "$q$": "momentum transfer", "$v(q)$": "interaction potential", "$N$": "number of electrons", "$e$": "elementary charge", "$\\mathscr{E}_{\\text{ex}}$": "self-energy correction of a single electron", "$x$": "ratio of wave vector $k_1$ to Fermi wave vector $k_F$", "$S(x)$": "exchange integral function", "$V$": "volume"}
|
Basics of Condensed Matter Theory
| ||
510 |
Others
|
In ionic crystals, long-wavelength optical modes represent the opposite motion of positive and negative ions within a unit cell, accompanied by polarization and interacting strongly with electromagnetic waves, thus having an important impact on the electrical and optical properties of ionic crystals. For simplicity, assume each unit cell contains only two ions with equal charge magnitude but opposite sign, still confined to an isotropic continuum model. Since, in the long-wavelength limit, the relative displacement $(u_+ - u_-)$ of the positive and negative ions within each unit cell is almost the same, a vector $\mathbf{W}$ can be used to describe the optical branch vibration \begin{equation} \mathbf{W} \equiv \rho^{1/2} (u_+ - u_-) \label{eq:2.6.1} \end{equation} here $\rho$ represents the reduced mass density \begin{equation} \rho = \frac{M}{\omega}, \quad M = \frac{M_+ M_-}{M_+ + M_-} \label{eq:2.6.2} \end{equation} $M_\pm$ are the masses of positive and negative ions, $M$ is the reduced mass. The vector $\mathbf{W}$ can be called the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, and satisfy $P = \gamma_{12} W + \gamma_{22}E$, with elastic energy as $\frac{1}{2}\gamma_{11}\mathbf{W}\cdot \mathbf{W}$ Please complete the following question: Calculate the Hamiltonian density
|
[] |
Expression
|
{"$\\mathcal{T}$": "kinetic energy density", "$\\phi$": "total potential energy density", "$\\phi_{\\text{elastic}}$": "elastic potential energy density", "$\\phi_{\\text{polarization}}$": "polarization potential energy density", "$\\gamma_{11}$": "elastic coefficient", "$\\gamma_{12}$": "coupling coefficient between ionic displacement and electric field", "$\\gamma_{22}$": "electric field coefficient", "$\\mathbf{W}$": "displacement field", "$\\mathbf{E}$": "macroscopic electric field", "$\\mathbf{P}$": "polarization intensity", "$\\mathcal{L}$": "Lagrangian density", "$\\mathcal{H}$": "Hamiltonian density"}
|
Basics of Condensed Matter Theory
| ||
511 |
Others
|
In an ionic crystal, long-wavelength optical modes represent the opposite movement of positive and negative ions within the unit cell, accompanied by polarization and strong interaction with electromagnetic waves, thus having an important impact on the electrical and optical properties of the ionic crystal. For simplicity, assume that each unit cell contains only two ions with equal and opposite charges, still restricted to the isotropic continuous model. Because the relative displacement of positive and negative ions $(u_+ - u_-)$ in each unit cell at the long-wavelength limit is nearly the same, a vector $\mathbf{W}$ can describe the optical branch vibration \begin{equation} \mathbf{W} \equiv \rho^{1/2} (u_+ - u_-) \label{eq:2.6.1} \end{equation} where $\rho$ represents the density of the reduced mass \begin{equation} \rho = \frac{M}{\Omega}, \quad M = \frac{M_+ M_-}{M_+ + M_-} \label{eq:2.6.2} \end{equation} $M_\pm$ are the masses of the positive and negative ions, and $M$ is the reduced mass. The vector $\mathbf{W}$ can be termed as the reduced displacement. Assume the polarization intensity of the crystal is $P$, the macroscopic field is $E$, satisfying $P = \gamma_{12} W + \gamma_{22}E$, and the elastic energy is $\frac{1}{2}\gamma_{11}\mathbf{W}\cdot \mathbf{W}$. Please complete the following question: Calculate the squared transverse vibration frequency $\omega^2_L$
|
[] |
Expression
|
{"$\\omega_L^2$": "squared transverse vibration frequency", "$\\mathbf{E}_L$": "longitudinal electric field", "$\\mathbf{W}_L$": "longitudinal wave amplitude", "$\\mathbf{D}$": "electric displacement field", "$\\mathbf{P}_L$": "longitudinal polarization", "$\\mathbf{P}$": "polarization", "$\\gamma_{12}$": "coupling coefficient (12)", "$\\gamma_{22}$": "coupling coefficient (22)", "$\\gamma_{11}$": "coupling coefficient (11)"}
|
Basics of Condensed Matter Theory
| ||
512 |
Theoretical Foundations
|
The problem of motion in the Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\xi, \eta, \varphi$ is defined by: \begin{align} x &= \sqrt{\xi \eta} \cos \varphi, \quad y = \sqrt{\xi \eta} \sin \varphi, \quad z = \frac{1}{2}(\xi - \eta), r &= \sqrt{x^2 + y^2 + z^2} = \frac{1}{2}(\xi + \eta). \label{eq:37.1} \end{align} Or conversely: \begin{equation} \xi = r + z, \quad \eta = r - z, \quad \varphi = \arctan\left(\frac{y}{x}\right); \label{eq:37.2} \end{equation} The values of $\xi$ and $\eta$ can range from 0 to $\infty$, and $\varphi$ ranges from 0 to $2\pi$. The surfaces $\xi = $ constant and $\eta = $ constant are rotational paraboloids around the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following question: Write the line element in parabolic coordinates
|
[] |
Expression
|
{"$\\xi$": "parabolic coordinate", "$\\eta$": "parabolic coordinate", "$\\varphi$": "azimuthal angle in parabolic coordinates"}
|
Other physical foundations
| ||
513 |
Theoretical Foundations
|
The problem of motion in a Coulomb field can be well handled in parabolic coordinates. The parabolic coordinate system $\xi, \eta, \varphi$ is defined by the following equations: \begin{align} x &= \sqrt{\xi \eta} \cos \varphi, \quad y = \sqrt{\xi \eta} \sin \varphi, \quad z = \frac{1}{2}(\xi - \eta), \ r &= \sqrt{x^2 + y^2 + z^2} = \frac{1}{2}(\xi + \eta). \label{eq:37.1} \end{align} Or conversely: \begin{equation} \xi = r + z, \quad \eta = r - z, \quad \varphi = \arctan\left(\frac{y}{x}\right); \label{eq:37.2} \end{equation} The values of $\xi$ and $\eta$ range from 0 to $\infty$, and $\varphi$ ranges from 0 to $2\pi$. The surfaces $\xi =$ constant and $\eta =$ constant are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem: Write out the Laplacian in parabolic coordinates
|
[] |
Expression
|
{"$\\xi$": "parabolic coordinate (xi)", "$\\eta$": "parabolic coordinate (eta)", "$\\varphi$": "angular coordinate"}
|
Other physical foundations
| ||
514 |
Theoretical Foundations
|
The problem of motion in a Coulomb field can be well handled in parabolic coordinates, where the parabolic coordinate system $\xi, \eta, \varphi$ is defined by the following: \begin{align} x &= \sqrt{\xi \eta} \cos \varphi, \quad y = \sqrt{\xi \eta} \sin \varphi, \quad z = \frac{1}{2}(\xi - \eta), \ r &= \sqrt{x^2 + y^2 + z^2} = \frac{1}{2}(\xi + \eta). \label{eq:37.1} \end{align} or conversely: \begin{equation} \xi = r + z, \quad \eta = r - z, \quad \varphi = \arctan\left(\frac{y}{x}\right); \label{eq:37.2} \end{equation} The values of $\xi$ and $\eta$ can range from 0 to $\infty$, and $\varphi$ ranges from 0 to $2\pi$. Surfaces of constant $\xi$ and constant $\eta$ are rotational paraboloids about the $z$-axis with the origin as the focus. This is an orthogonal coordinate system. Please complete the following problem: Write down the single-particle Schrödinger equation You should return your answer as an equation.
|
[] |
Equation
|
{"$\\xi$": "coordinate variable xi", "$\\eta$": "coordinate variable eta", "$\\psi$": "wave function", "$\\varphi$": "angle variable phi", "$E$": "energy"}
|
Other physical foundations
| ||
515 |
Theoretical Foundations
|
The problem of motion in a Coulomb field can be well-handled in parabolic coordinates. The parabolic coordinate system $\xi, \eta, \varphi$ is defined as follows: \begin{align} x &= \sqrt{\xi \eta} \cos \varphi, \quad y = \sqrt{\xi \eta} \sin \varphi, \quad z = \frac{1}{2}(\xi - \eta), \ r &= \sqrt{x^2 + y^2 + z^2} = \frac{1}{2}(\xi + \eta). \label{eq:37.1} \end{align} or conversely: \begin{equation} \xi = r + z, \quad \eta = r - z, \quad \varphi = \arctan\left(\frac{y}{x}\right); \label{eq:37.2} \end{equation} The values of $\xi$ and $\eta$ range from 0 to $\infty$, and $\varphi$ ranges from 0 to $2\pi$. The surfaces $\xi =$ constant and $\eta =$ constant are rotational paraboloids around the $z$-axis, with the origin as the focus. This is an orthogonal coordinate system. Please complete the following task: Use the method of separation of variables to solve the Schrödinger equation corresponding to the discrete spectrum;
|
[] |
Expression
|
{"$\\psi$": "eigenfunction", "$m$": "magnetic quantum number", "$E$": "energy", "$\\xi$": "parabolic coordinate (xi)", "$\\eta$": "parabolic coordinate (eta)", "$\\beta_1$": "separation parameter 1", "$\\beta_2$": "separation parameter 2", "$n$": "principal quantum number", "$\\rho_1$": "scaled parabolic coordinate 1", "$\\rho_2$": "scaled parabolic coordinate 2", "$n_1$": "parabolic quantum number 1", "$n_2$": "parabolic quantum number 2", "$\\varphi$": "azimuthal angle", "$\\psi_{n_1 n_2 m}$": "wave function for discrete spectrum", "$L_{n_1}^{|m|}(\\xi)$": "associated Laguerre polynomial for n1", "$L_{n_2}^{|m|}(\\eta)$": "associated Laguerre polynomial for n2"}
|
Other physical foundations
| ||
516 |
Theoretical Foundations
|
The problem of motion in a Coulomb field can be well treated in parabolic coordinates. The parabolic coordinate system $\xi, \eta, \varphi$ is defined by the following: \begin{align} x &= \sqrt{\xi \eta} \cos \varphi, \quad y = \sqrt{\xi \eta} \sin \varphi, \quad z = \frac{1}{2}(\xi - \eta), \ r &= \sqrt{x^2 + y^2 + z^2} = \frac{1}{2}(\xi + \eta). \label{eq:37.1} \end{align} Conversely: \begin{equation} \xi = r + z, \quad \eta = r - z, \quad \varphi = \arctan\left(\frac{y}{x}\right); \label{eq:37.2} \end{equation} The values of $\xi$ and $\eta$ can range from 0 to $\infty$, and $\varphi$ ranges from 0 to $2\pi$. The surfaces $\xi = $ constant and $\eta = $ constant are rotational paraboloids about the $z$-axis centered at the origin. This forms an orthogonal coordinate system. Please complete the following problem: Assuming a hydrogen atom in a uniform electric field, solve for the energy level correction to second order approximation (Stark effect).
|
[] |
Expression
|
{"$E$": "energy", "$\\mathscr{E}$": "electric field strength", "$\\xi$": "parabolic coordinate (xi)", "$\\eta$": "parabolic coordinate (eta)", "$m$": "magnetic quantum number", "$n$": "principal quantum number", "$n_1$": "quantum number in parabolic coordinates (1)", "$n_2$": "quantum number in parabolic coordinates (2)", "$\\beta_1$": "eigenvalue of the operator for xi coordinate", "$\\beta_2$": "eigenvalue of the operator for eta coordinate", "$\\epsilon$": "parameter related to energy"}
|
Other physical foundations
| ||
517 |
Theoretical Foundations
|
We discuss the propagation of light in conductive media, considering a uniform isotropic medium whose dielectric constant is $\varepsilon$, magnetic permeability is $\mu$, and conductivity is $\sigma_0$. Utilizing the material equations $\mathbf{j} = \sigma \mathbf{E}$, $\mathbf{D} = \varepsilon \mathbf{E}$, $\mathbf{B} = \mu \mathbf{H}$, the Maxwell equations take the following form: \begin{align}
\text{curl } \mathbf{H} - \frac{\varepsilon}{c} \dot{\mathbf{E}} &= \frac{4\pi}{c} \sigma \mathbf{E}, \label{eq:1} \\
\text{curl } \mathbf{E} + \frac{\mu}{c} \dot{\mathbf{H}} &= 0, \label{eq:2} \\
\text{div } \mathbf{E} &= \frac{4\pi}{\varepsilon} \rho, \label{eq:3} \\
\text{div } \mathbf{H} &= 0. \label{eq:4}
\end{align} Please calculate: Charge density within a metal
|
[] |
Expression
|
{"$\\varepsilon$": "permittivity", "$c$": "speed of light", "$\\rho$": "charge density", "$\\dot{\\mathbf{E}}$": "time derivative of electric field", "$\\sigma$": "conductivity", "$\\rho_0$": "initial charge density", "$t$": "time", "$\\tau$": "time constant"}
|
Other physical foundations
| ||
518 |
Theoretical Foundations
|
We discuss the propagation of light in conductive media, considering a homogeneous isotropic medium with a dielectric constant $\varepsilon$, magnetic permeability $\mu$, and conductivity $\sigma_0$. Using the material equations $\mathbf{j} = \sigma \mathbf{E}$, $\mathbf{D} = \varepsilon \mathbf{E}$, $\mathbf{B} = \mu \mathbf{H}$, Maxwell's equations take the following form:
\begin{align}
\text{curl } \mathbf{H} - \frac{\varepsilon}{c} \dot{\mathbf{E}} &= \frac{4\pi}{c} \sigma \mathbf{E}, \label{eq:1} \\
\text{curl } \mathbf{E} + \frac{\mu}{c} \dot{\mathbf{H}} &= 0, \label{eq:2} \\
\text{div } \mathbf{E} &= \frac{4\pi}{\varepsilon} \rho, \label{eq:3} \\
\text{div } \mathbf{H} &= 0. \label{eq:4}
\end{align} Introduce the complex dielectric constant $\hat{\epsilon} = \epsilon + i \frac{4\pi \sigma}{\omega}$, the complex phase velocity $\hat{v} = \frac{c}{\sqrt{\mu\hat{\epsilon}}}$, and the complex refractive index $\hat{n} = \frac{c}{\hat{v}} = \frac{c}{\omega}k$, let $\hat{n}=n(1+i\kappa)$, where $\kappa$ is the attenuation coefficient. Please express the refractive index $n$ using the material constants $\epsilon,\mu,\sigma$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\hat{\\epsilon}$": "complex dielectric constant", "$\\epsilon$": "real part of the dielectric constant", "$\\sigma$": "electrical conductivity", "$\\omega$": "angular frequency", "$\\hat{v}$": "complex phase velocity", "$c$": "speed of light in vacuum", "$\\mu$": "magnetic permeability", "$\\hat{n}$": "complex refractive index", "$n$": "real refractive index", "$\\kappa$": "attenuation coefficient", "$v$": "velocity"}
|
Other physical foundations
| ||
519 |
Theoretical Foundations
|
We discuss the propagation of light in conducting media, considering a homogeneous isotropic medium with dielectric constant $\varepsilon$, magnetic permeability $\mu$, and conductivity $\sigma_0$. Using the material equations $\mathbf{j} = \sigma \mathbf{E}$, $\mathbf{D} = \varepsilon \mathbf{E}$, $\mathbf{B} = \mu \mathbf{H}$, the Maxwell equations take the following form:
\begin{align} \text{curl } \mathbf{H} - \frac{\varepsilon}{c} \dot{\mathbf{E}} &= \frac{4\pi}{c} \sigma \mathbf{E}, \\ \text{curl } \mathbf{E} + \frac{\mu}{c} \dot{\mathbf{H}} &= 0, \\ \text{div } \mathbf{E} &= \frac{4\pi}{\varepsilon} \rho, \\ \text{div } \mathbf{H} &= 0. \end{align}
Please answer the following question. If the electric field is a plane wave, calculate the energy density of the wave.
|
[] |
Expression
|
{"$\\mathbf{E}$": "electric field vector", "$\\mathbf{E}_0$": "initial electric field vector amplitude", "$\\mathbf{k}$": "wave vector", "$\\mathbf{r}$": "position vector", "$\\omega$": "angular frequency", "$t$": "time", "$\\hat{k}$": "unit wavevector", "$\\hat{n}$": "unit vector in direction of wave propagation", "$c$": "speed of light in vacuum", "$n$": "refractive index", "$\\kappa$": "extinction coefficient or absorption index", "$\\lambda$": "wavelength in the medium", "$\\mathbf{s}$": "unit vector in position direction", "$w$": "energy density of the wave", "$w_0$": "initial energy density of the wave", "$\\chi$": "absorption coefficient", "$\\lambda_0$": "wavelength in vacuum", "$v$": "frequency"}
|
Other physical foundations
| ||
520 |
Strongly Correlated Systems
|
Consider the action of the Hubbard model:
\begin{align}
S_{\text{loc}}[\Phi^\dagger, \Phi] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger(\partial_\tau - \mu - i\Delta_c - \Delta\sigma^z)\Phi_{\mathbf{r}}, \nonumber \\
S_1[\Phi^\dagger, \Phi, \Omega] &= \int_0^\beta d\tau \sum_{\mathbf{r}} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger \dot{R}_{\mathbf{r}} \Phi_{\mathbf{r}}, \nonumber \\
S_2[\Phi^\dagger, \Phi, \Omega] &= - \int_0^\beta d\tau \sum_{\mathbf{r},\mathbf{r}'} t_{\mathbf{r}\mathbf{r}'} \Phi_{\mathbf{r}}^\dagger R_{\mathbf{r}}^\dagger R_{\mathbf{r}'} \Phi_{\mathbf{r}'}.
\label{eq:6.141}
\end{align}
where:
\begin{equation}
R_{\mathbf{r}} = e^{-\frac{i}{2}\varphi_{\mathbf{r}}\sigma^z} e^{-\frac{i}{2}\theta_{\mathbf{r}}\sigma^y} e^{-\frac{i}{2}\psi_{\mathbf{r}}\sigma^z} = \begin{pmatrix} \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} & -\sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{-\frac{i}{2}\varphi_{\mathbf{r}}} \\ \sin\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} & \cos\left(\frac{\theta_{\mathbf{r}}}{2}\right)e^{\frac{i}{2}\varphi_{\mathbf{r}}} \end{pmatrix},
\end{equation}
We consider the fields $\Delta_c$ and $\Delta$ at the saddle-point approximation level: $\Delta_c = i(U/2)\langle\Phi_{\mathbf{r}}^\dagger\Phi_{\mathbf{r}}\rangle$ and $\Delta = (U/2)\langle\Phi_{\mathbf{r}}^\dagger\sigma^z\Phi_{\mathbf{r}}\rangle$. At half-filling, we have $\mu + i\Delta_c = 0$.
Please answer the problem
\begin{enumerate}
\item the effective action $S[\Omega]$ accurate to the first order in $\partial_\tau$ and the second order in $t$:
\begin{equation}
S[\Omega] = \langle S_1 + S_2 \rangle - \frac{1}{2}\langle S_2^2 \rangle_c,
\label{eq:6.142}
\end{equation}
where the expectation value $\langle \cdots \rangle$ is taken with respect to the local action $S_{\text{loc}}$.
This coincides with the action of the spin- $1/2$ Heisenberg model, what is its exchange coupling $J$?
\end{enumerate}
|
[] |
Expression
|
{"$S_{\\text{loc}}$": "local action", "$\\Phi^\\dagger$": "creation field", "$\\Phi$": "annihilation field", "$\\beta$": "inverse temperature", "$\\mathbf{r}$": "lattice site position", "$\\partial_\\tau$": "imaginary time derivative", "$\\mu$": "chemical potential", "$i$": "imaginary unit", "$\\Delta_c$": "complex field at saddle-point", "$\\Delta$": "real field at saddle-point", "$\\sigma^z$": "Pauli matrix in z-direction", "$R_{\\mathbf{r}}$": "rotation matrix at site", "$t_{\\mathbf{r}\\mathbf{r}'}$": "hopping amplitude between sites", "$\\varphi_{\\mathbf{r}}$": "azimuthal angle at site", "$\\theta_{\\mathbf{r}}$": "polar angle at site", "$\\psi_{\\mathbf{r}}$": "phase angle at site", "$U$": "interaction strength", "$t$": "hopping parameter", "$J$": "exchange coupling", "$G_\\sigma$": "single-particle propagator"}
|
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