id
int64 1
520
| topic
stringclasses 6
values | question
stringlengths 59
4.96k
| final_answer
listlengths 0
2
| answer_type
stringclasses 5
values | answer
stringclasses 101
values | symbol
stringlengths 2
2.16k
| chapter
stringclasses 100
values | section
stringlengths 0
86
|
|---|---|---|---|---|---|---|---|---|
201
|
Semiconductors
|
It is known that the electronic band of a one-dimensional crystal can be written as
$$
E(k)=\frac{h^{2}}{m_{0} a^{2}}(\frac{7}{8}-\cos 2 \pi k a+\frac{1}{8} \cos 6 \pi k a)
$$
In this equation, $a$ is the lattice constant. Try to find the effective mass of electrons at the top of the band.
|
[] |
Expression
|
{"$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", "$h$": "Planck's constant", "$E$": "energy", "$k$": "wave vector", "$m$": "mass", "$m_{0}$": "electron rest mass"}
|
Electronic states in semiconductors
|
Electronic states in semiconductors
|
|
202
|
Semiconductors
|
Given the lattice constant of a two-dimensional square lattice is $a$, if the electron energy can be expressed as
$$
E(k)=\frac{h^{2}(k_{x}^{2}+k_{y}^{2})}{2 m_{\mathrm{n}}^{*}}
$$
Try to find the density of states.
|
[] |
Expression
|
{"$a$": "lattice constant", "$E$": "energy", "$k_{x}$": "wave vector component in x-direction", "$k_{y}$": "wave vector component in y-direction", "$m_{\\mathrm{n}}^{*}$": "effective mass of electron", "$h$": "Planck's constant", "$S$": "area of the crystal"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
203
|
Semiconductors
|
Calculate the number of quantum states per unit volume between the energy $E=E_{\mathrm{c}}$ and $E=E_{\mathrm{c}}+100(\frac{h^{2}}{8 m_{\mathrm{n}}^{*} L^{2}})$.
|
[] |
Expression
|
{"$E$": "energy", "$E_{\\mathrm{c}}$": "conduction band minimum energy", "$h$": "Planck's constant", "$m_{\\mathrm{n}}^{*}$": "effective mass of an electron", "$L$": "length or characteristic dimension", "$g_{\\mathrm{c}}$": "density of states", "$m_{\\mathrm{dn}}$": "some effective mass (context-dependent)"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
204
|
Semiconductors
|
For two pieces of n-type silicon material, at a certain temperature $T$, the ratio of the electron densities of the first piece to the second piece is $n_{1} / n_{2}=\mathrm{e}$ (e is the base of the natural logarithm). If the Fermi level of the first piece of material is $3 k_{0} T$ below the conduction band edge, find the position of the Fermi level in the second piece of material.
|
[] |
Expression
|
{"$E_{\\mathrm{F} 1}$": "Fermi level of the first piece of material", "$E_{\\mathrm{F} 2}$": "Fermi level of the second piece of material", "$k_{0}$": "Boltzmann constant or a specific constant related to temperature", "$T$": "temperature", "$E_{\\mathrm{c}}$": "conduction band edge", "$n_{1}$": "carrier concentration in the first piece of material", "$n_{2}$": "carrier concentration in the second piece of material"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
205
|
Semiconductors
|
For a p-type semiconductor, in the ionization region of impurities, the known relation is $\frac{p_{0}(p_{0}+N_{\mathrm{D}})}{N_{\mathrm{A}}-N_{\mathrm{D}}-p_{0}}=\frac{N_{\mathrm{v}}}{g} \exp (-\frac{E_{\mathrm{A}}-E_{\mathrm{v}}}{k_{0} T})$. When the condition $p_{0} \ll N_{\mathrm{D}}$ is satisfied, find the expression for hole density $p_{0}$. In the formula, $g$ is the spin degeneracy of the acceptor level.
|
[] |
Expression
|
{"$p_{0}$": "hole density", "$N_{\\mathrm{D}}$": "donor density", "$N_{\\mathrm{A}}$": "acceptor density", "$N_{\\mathrm{v}}$": "effective density of states in the valence band", "$g$": "spin degeneracy of the acceptor level", "$E_{\\mathrm{A}}$": "energy level of the acceptor", "$E_{\\mathrm{v}}$": "energy level at the valence band edge", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$E_{\\mathrm{F}}$": "Fermi energy"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
206
|
Semiconductors
|
There is an n-type semiconductor, in addition to the donor concentration $N_{\mathrm{D}}$, it also contains a small amount of acceptors, with a concentration of $N_{\mathrm{A}}$. Find the expression for the electron concentration under weak ionization conditions.
|
[] |
Expression
|
{"$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{A}}$": "acceptor concentration", "$E_{\\mathrm{A}}$": "energy level of the acceptor", "$n_{0}$": "electron concentration in the conduction band", "$n_{\\mathrm{D}}^{+}$": "ionized donor concentration", "$E_{\\mathrm{D}}$": "energy level of the donor", "$E_{\\mathrm{F}}$": "Fermi level", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$E_{\\mathrm{c}}$": "energy of the conduction band edge", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$\\Delta E_{\\mathrm{D}}$": "ionization energy of the donor"}
|
Statistical Distribution of Charge Carriers in Semiconductors
|
Statistical Distribution of Charge Carriers in Semiconductors
|
|
207
|
Semiconductors
|
Please explain why at room temperature, for a certain semiconductor, the electron concentration $n=n_{i} \sqrt{\mu_{\mathrm{p}} / \mu_{\mathrm{n}}}$ results in the minimum electrical conductivity $\sigma$. In this equation, $n_{\mathrm{i}}$ is the intrinsic carrier concentration, and $\mu_{\mathrm{p}}, ~ \mu_{\mathrm{n}}$ are the mobilities of holes and electrons respectively. Find the hole concentration under the above condition.
|
[] |
Expression
|
{"$n$": "electron concentration", "$n_{i}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{p}}$": "mobility of holes", "$\\mu_{\\mathrm{n}}$": "mobility of electrons", "$\\sigma$": "electrical conductivity", "$n_{0}$": "electron concentration when conductivity is minimized", "$p_{0}$": "hole concentration", "$q$": "elementary charge"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
208
|
Semiconductors
|
Suppose a semiconductor crystal is subjected to an electric field $\boldsymbol{E}$ and a magnetic field $\boldsymbol{B}$, with $\boldsymbol{E}$ in the $x-y$ plane and $\boldsymbol{B}$ along the $z$ direction, try to derive the distribution function of semiconductor electrons in the electromagnetic field considering the multiple interactions of the magnetic field with electrons.
|
[] |
Expression
|
{"$\\boldsymbol{E}$": "electric field vector", "$\\boldsymbol{B}$": "magnetic field vector", "$E_{x}$": "electric field component along x-axis", "$E_{y}$": "electric field component along y-axis", "$B_{z}$": "magnetic field component along z-axis", "$a_{x}$": "acceleration component along x-axis", "$a_{y}$": "acceleration component along y-axis", "$a_{z}$": "acceleration component along z-axis", "$v_{x}$": "velocity component along x-axis", "$v_{y}$": "velocity component along y-axis", "$v_{z}$": "velocity component along z-axis", "$m$": "mass", "$q$": "charge", "$\\mathscr{E}_{x}$": "electric field perturbation along x-axis", "$\\mathscr{E}_{y}$": "electric field perturbation along y-axis", "$\\varphi_{1}$": "perturbation function related to velocity component x", "$\\varphi_{2}$": "perturbation function related to velocity component y", "$f_{0}$": "equilibrium distribution function", "$\\tau$": "relaxation time", "$l$": "mean free path", "$f_{1}$": "perturbation function related to the component of distribution function along x", "$f_{2}$": "perturbation function related to the component of distribution function along y", "$k$": "dimensionless parameter involving charge, mean free path, mass, and magnetic field", "$\\boldsymbol{v}$": "velocity vector"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
209
|
Semiconductors
|
Assuming $\tau_{\mathrm{n}}=\tau_{\mathrm{p}}=\tau_{0}$ is a constant that does not change with the doping density in the sample, find the value of conductivity when the small-signal lifetime of the sample reaches its maximum.
|
[] |
Expression
|
{"$\\tau_{\\mathrm{n}}$": "electron lifetime", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$\\tau_{0}$": "constant lifetime", "$N_{\\mathrm{t}}$": "trap density", "$r_{\\mathrm{p}}$": "recombination rate for holes", "$r_{\\mathrm{n}}$": "recombination rate for electrons", "$n_{0}$": "initial electron density", "$p_{0}$": "initial hole density", "$n_{1}$": "perturbation electron density", "$p_{1}$": "perturbation hole density", "$n_{\\mathrm{i}}$": "intrinsic carrier density", "$e_{\\mathrm{ni}}$": "elementary charge (in intrinsic condition)", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\tau_{\\max}$": "maximum lifetime", "$E_{\\mathrm{t}}$": "trap energy level", "$E_{\\mathrm{i}}$": "intrinsic energy level", "$k$": "Boltzmann constant", "$T$": "temperature"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
210
|
Semiconductors
|
Assume $\tau_{\mathrm{p}}=\tau_{\mathrm{n}}=\tau_{0}$, and based on the small signal lifetime formula
\tau=\tau_{\mathrm{p}} \frac{n_{0}+n_{1}}{n_{0}+p_{0}}+\tau_{\mathrm{n}} \frac{p_{0}+p_{1}}{n_{0}+p_{0}}
discuss the relationship between the lifetime $\tau$ and the position of the recombination center level $E_{\mathrm{t}}$ in the band gap, and briefly explain its physical significance.
|
[] |
Expression
|
{"$\\tau_{\\mathrm{p}}$": "hole lifetime", "$\\tau_{\\mathrm{n}}$": "electron lifetime", "$\\tau_{0}$": "reference lifetime", "$\\tau$": "total carrier lifetime", "$n_{0}$": "equilibrium electron concentration", "$n_{1}$": "excess electron concentration", "$p_{0}$": "equilibrium hole concentration", "$p_{1}$": "excess hole concentration", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$E_{\\mathrm{t}}$": "recombination center energy level", "$E_{\\mathrm{i}}$": "intrinsic Fermi level", "$R_{0}$": "universal gas constant", "$T$": "temperature", "$k$": "Boltzmann constant", "$E_{\\mathrm{e}}$": "conduction band energy", "$E_{\\mathrm{v}}$": "valence band energy", "$E_{\\mathrm{c}}$": "energy at conduction band minimum"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
211
|
Semiconductors
|
Let $f_{\mathrm{t}}$ be the probability that the composite center energy level $E_{\mathrm{t}}$ is occupied by an electron, $N_{\mathrm{c}}$ and $N_{\mathrm{v}}$ are the effective density of states of the conduction band and the valence band, respectively. Let $f_{\mathrm{t}}$ be the probability that the composite center energy level $E_{\mathrm{t}}$ is occupied by an electron, and $N_{\mathrm{c}}$ be the effective density of states of the conduction band. Consider the rate equation for electrons:
\frac{\mathrm{d} n}{\mathrm{~d} t}=-\frac{n(1-f_{\mathrm{t}})}{\tau_{\mathrm{n}}}+\frac{N_{\mathrm{c}} f_{\mathrm{t}}}{\tau_{\mathrm{n}}^{\prime}}
where $\tau_{\mathrm{n}}$ and $\tau_{\mathrm{n}}^{\prime}$ are the characteristic time constants related to electron capture and emission, respectively. Under thermal equilibrium conditions, $\tau_{\mathrm{n}}^{\prime}$ can be expressed in terms of $\tau_{\mathrm{n}}$, $N_{\mathrm{c}}$, and parameter $n_1$ (where $n_1 = N_{\mathrm{c}} \exp (-\frac{E_{\mathrm{c}}-E_{\mathrm{t}}}{k_{0} T})$, $E_c$ is the conduction band edge energy, $k_0$ is the Boltzmann constant, and $T$ is the temperature). Find the expression for $\tau_{\mathrm{n}}^{\prime}$.
|
[] |
Expression
|
{"$f_{\\mathrm{t}}$": "probability that the composite center energy level is occupied by an electron", "$E_{\\mathrm{t}}$": "composite center energy level", "$N_{\\mathrm{c}}$": "effective density of states of the conduction band", "$n$": "electron density in the conduction band", "$\\tau_{\\mathrm{n}}$": "characteristic time constant for electron capture", "$\\tau_{\\mathrm{n}}^{\\prime}$": "characteristic time constant for electron emission", "$n_1$": "factor related to the effective density of states and energy difference", "$E_{\\mathrm{c}}$": "conduction band edge energy", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$f_{10}$": "probability that the composite center energy level is occupied by electrons in thermal equilibrium", "$E_{\\mathrm{f}}$": "Fermi level energy", "$\\tau_{\\mathrm{p}}^{\\prime}$": "characteristic time constant for hole emission", "$\\tau_{\\mathrm{p}}$": "characteristic time constant for hole capture", "$p_{1}$": "factor for hole-related processes", "$N_{\\mathrm{v}}$": "effective density of states of the valence band"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
212
|
Semiconductors
|
Let $f_{\mathrm{t}}$ be the probability that the composite center energy level $E_{\mathrm{t}}$ is occupied by electrons, and $N_{\mathrm{c}}$ and $N_{\mathrm{v}}$ be the effective density of states for the conduction band and the valence band, respectively. Derive the small-signal lifetime formula.
|
[] |
Expression
|
{"$n$": "electron density", "$p$": "hole density", "$\\Delta p$": "change in hole density", "$\\tau$": "carrier lifetime", "$n_{\\mathrm{i}}$": "intrinsic carrier density", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$n_{1}$": "electron density under recombination center condition", "$\\tau_{\\mathrm{n}}$": "electron lifetime", "$p_{1}$": "hole density under recombination center condition", "$n_{0}$": "equilibrium electron density", "$p_{0}$": "equilibrium hole density", "$\\Delta n$": "change in electron density"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
213
|
Semiconductors
|
A square pulse with an appropriate frequency is irradiated onto an n-type semiconductor sample and is uniformly absorbed inside the sample to generate nonequilibrium carriers at a generation rate of $g_{\mathrm{p}}$. The lifetime of nonequilibrium holes is $\tau_{\mathrm{p}}$, and the pulse width is $\Delta t=3 \tau_{\mathrm{p}}$. Assume the moment the pulse light starts irradiating is $t=0$, find the expression for the concentration of nonequilibrium holes $\Delta p(\Delta t)$ at the moment the light pulse ends ($t=\Delta t$).
|
[] |
Expression
|
{"$t$": "time", "$\\Delta t$": "duration of the light pulse", "$\\Delta p$": "concentration of nonequilibrium holes", "$g_{\\mathrm{p}}$": "generation rate of holes", "$\\tau_{\\mathrm{p}}$": "lifetime of holes"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
214
|
Semiconductors
|
Using the bipolar diffusion theory, consider the case of intrinsic semiconductors (i.e., the electron concentration $n$ is equal to the hole concentration $p$). It is known that the general expression for the bipolar diffusion coefficient is $D^{*}=\frac{(n+p) D_{\mathrm{n}} D_{\mathrm{p}}}{n D_{\mathrm{n}}+p D_{\mathrm{p}}}$. Try to derive the specific expression for the bipolar diffusion coefficient $D^{*}$ in intrinsic semiconductors.
|
[] |
Expression
|
{"$n$": "electron concentration", "$p$": "hole concentration", "$D^{*}$": "bipolar diffusion coefficient", "$D_{\\mathrm{n}}$": "electron diffusion coefficient", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$\\mu^{*}$": "bipolar mobility", "$\\Delta n$": "change in electron concentration", "$\\Delta p$": "change in hole concentration"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
215
|
Semiconductors
|
For a silicon pn junction, the doping concentrations of the p and n regions are $N_{\mathrm{A}}=9 \times 10^{15} \mathrm{~cm}^{-3}$ and $N_{\mathrm{D}}=2 \times 10^{16} \mathrm{~cm}^{-3}$, respectively; the hole and electron mobilities in the p region are $350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, respectively, while in the n region, they are $300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. Assume the lifetime of the non-equilibrium carriers in both regions is $1 \mu \mathrm{~s}$, and the cross-sectional area of the pn junction is $10^{-2} \mathrm{~cm}^{2} ; \frac{q}{k_{0} T}=38.7(\frac{1}{V})$. When a forward voltage $V_{\mathrm{F}}=0.65 \mathrm{~V}$ is applied, calculate: Determine the expression for the hole diffusion current variation with $x$ in the n region.
|
[] |
Expression
|
{"$I_{\\mathrm{pD}}$": "hole diffusion current", "$x$": "position in the n region", "$A$": "cross-sectional area", "$q$": "elementary charge", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$p$": "hole concentration", "$I_{\\mathrm{nD}}$": "electron diffusion current", "$D_{\\mathrm{n}}^{\\prime}$": "modified electron diffusion coefficient", "$n$": "electron concentration", "$k_{0}$": "Boltzmann constant relation factor", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$I_{\\mathrm{nt}}$": "electron drift current", "$I$": "total current in the n region", "$E$": "electric field due to external forward voltage", "$E^{\\prime}$": "electric field due to diffusion imbalance"}
|
pn junction
|
pn junction
|
|
216
|
Semiconductors
|
There is a silicon pn junction, with doping concentrations in the p-region and n-region of $N_{\mathrm{A}}=9 \times 10^{15} \mathrm{~cm}^{-3}$ and $N_{\mathrm{D}}=2 \times 10^{16} \mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ respectively, and in the n-region, the hole and electron mobilities are $300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ respectively; assume the lifetime of non-equilibrium carriers in both regions is $1 \mu \mathrm{~s}, \mathrm{pn}$ junction cross-sectional area is $10^{-2} \mathrm{~cm}^{2} ; \frac{q}{k_{0} T}=38.7(\frac{1}{V})$ . When the applied forward voltage $V_{\mathrm{F}}=0.65 \mathrm{~V}$, try to find: Determine the expression for the electron diffusion current variation with $x$ in the n-region.
|
[] |
Expression
|
{"$x$": "position in the n-region", "$I_{\\mathrm{pD}}$": "hole diffusion current", "$I_{\\mathrm{nD}}$": "electron diffusion current", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$D_{\\mathrm{n}}^{\\prime}$": "modified electron diffusion coefficient", "$k_{0}$": "constant related to diffusion", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$I_{\\mathrm{nt}}$": "electron drift current", "$I$": "total current", "$E$": "electric field in the n-region from voltage drop", "$E^{\\prime}$": "electric field from faster electron diffusion"}
|
pn junction
|
pn junction
|
|
217
|
Semiconductors
|
There is a silicon pn junction, with doping concentrations in the p-region and n-region being $N_{\mathrm{A}}=9 \times 10^{15} \mathrm{~cm}^{-3}$ and $N_{\mathrm{D}}=2 \times 10^{16} \mathrm{~cm}^{-3}$, respectively. The hole and electron mobilities in the p-region are $350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, respectively, and in the n-region, the hole and electron mobilities are $300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, respectively. Assume the lifetime of non-equilibrium carriers in both regions is $1 \mu \mathrm{~s}$, the area of the pn junction is $10^{-2} \mathrm{~cm}^{2}$; $\frac{q}{k_{0} T}=38.7(\frac{1}{V})$. When a forward bias voltage of $V_{\mathrm{F}}=0.65 \mathrm{~V}$ is applied, determine: Determine the expression for the variation of electron drift current with $x$ in the n-region.
|
[] |
Expression
|
{"$I_{\\mathrm{pD}}$": "hole diffusion current", "$I_{\\mathrm{nD}}$": "electron diffusion current", "$I_{\\mathrm{nt}}$": "electron drift current", "$I$": "total current", "$A$": "cross-sectional area", "$q$": "elementary charge", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$D_{\\mathrm{n}}^{\\prime}$": "effective electron diffusion coefficient", "$k_{0}$": "Boltzmann constant", "$T$": "absolute temperature", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$x$": "position in the n-region", "$E$": "electric field due to voltage drop", "$E^{\\prime}$": "electric field due to faster electron diffusion"}
|
pn junction
|
pn junction
|
|
218
|
Semiconductors
|
There is a silicon pn junction, with the doping concentrations of the p-region and n-region being $N_{\mathrm{A}}=9 \times 10^{15} \mathrm{~cm}^{-3}$ and $N_{\mathrm{D}}=2 \times 10^{16} \mathrm{~cm}^{-3}$ respectively; the hole and electron mobilities in the p-region are $350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ respectively, while in the n-region the hole and electron mobilities are $300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ respectively; assuming the minority carrier lifetime in both regions is $1 \mu \mathrm{~s}, \mathrm{pn}$ junction cross-sectional area is $10^{-2} \mathrm{~cm}^{2} ; \frac{q}{k_{0} T}=38.7(\frac{1}{V})$. When a forward bias voltage $V_{\mathrm{F}}=0.65 \mathrm{~V}$ is applied, find: Determine the expression for the total electron current in the n-region as a function of $x$.
|
[] |
Expression
|
{"$x$": "position in the n-region", "$I_{\\mathrm{pD}}$": "hole diffusion current", "$I_{\\mathrm{nD}}$": "electron diffusion current", "$D_{\\mathrm{n}}^{\\prime}$": "effective electron diffusion coefficient", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$q$": "elementary charge", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$I_{\\mathrm{nt}}$": "electron drift current", "$I$": "total current in the n-region", "$E$": "electric field due to voltage drop", "$E^{\\prime}$": "electric field due to electron diffusion", "$I_{\\mathrm{n}}$": "total electron current"}
|
pn junction
|
pn junction
|
|
219
|
Semiconductors
|
A metal contacts uniformly doped $n-Si$ material to form a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\mathrm{D}}=0.6 \mathrm{eV}, N_{\mathrm{D}}=5 \times 10^{16} \mathrm{~cm}^{-3}$, find the relationship curve of $1 / C^{2}$ versus $(V_{\mathrm{D}}-V)$ under a 5V reverse bias voltage.
|
[] |
Expression
|
{"$q$": "elementary charge", "$V_{\\mathrm{D}}$": "barrier potential difference on the semiconductor side", "$N_{\\mathrm{D}}$": "semiconductor dopant concentration", "$C$": "capacitance", "$d$": "space charge region width", "$E_{\\mathrm{M}}$": "maximum electric field at the semiconductor interface", "$\\varepsilon_0$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$N$": "dopant concentration in the semiconductor", "$V$": "applied voltage"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
220
|
Semiconductors
|
A metal plate is 0.4 $\mu \mathrm{~m}$ away from n-type silicon, forming a parallel plate capacitor, with dry air in between having a relative permittivity $\varepsilon_{\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state. Find the expression for depletion layer width $X_{\mathrm{d}}$ when $V_{\mathrm{s}}=0.4 \mathrm{~V}$;
|
[] |
Expression
|
{"$X_{\\mathrm{d}}$": "depletion layer width", "$V_{\\mathrm{s}}$": "surface potential", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$\\varepsilon_{0}$": "vacuum permittivity", "$q$": "elementary charge", "$N_{\\mathrm{D}}$": "donor concentration", "$V_{\\mathrm{sm}}$": "surface potential at maximum depletion layer width", "$V_{\\mathrm{B}}$": "built-in potential", "$k_{0}$": "Boltzmann constant in appropriate units", "$T$": "temperature", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$X_{\\mathrm{dm}}$": "maximum depletion layer width"}
|
Semiconductor Surface and MIS Structure
|
Semiconductor Surface and MIS Structure
|
|
221
|
Semiconductors
|
A metal plate is separated from n-type silicon by a distance of $0.4 \mu \mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air in between is $\varepsilon_{\mathrm{ra}}=1$. When a negative voltage is applied to the metal end, the semiconductor is in a depletion state. Find the expression for the maximum depletion layer width $X_{\mathrm{dm}}$ when $V_{\mathrm{s}}=0.4 \mathrm{~V}$;
|
[] |
Expression
|
{"$X_{\\mathrm{dm}}$": "maximum depletion layer width", "$V_{\\mathrm{s}}$": "surface potential", "$X_{\\mathrm{d}}$": "depletion layer width", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$\\varepsilon_{0}$": "vacuum permittivity", "$q$": "elementary charge", "$N_{\\mathrm{D}}$": "donor concentration", "$V_{\\mathrm{sm}}$": "surface potential at maximum depletion layer width", "$V_{\\mathrm{B}}$": "built-in potential", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration"}
|
Semiconductor Surface and MIS Structure
|
Semiconductor Surface and MIS Structure
|
|
222
|
Semiconductors
|
For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is applied in the $x$ direction, and a magnetic field of 0.5 T is applied in the $z$ direction, resulting in a Hall voltage of -0.4 mV, find: Given the resistivity of the material is $1.5 \times 10^{-3} \Omega \cdot \mathrm{~cm}$, find the carrier mobility.
|
[] |
Expression
|
{"$\\mu_{\\mathrm{H}}$": "Hall mobility", "$R_{\\mathrm{H}}$": "Hall coefficient", "$\\sigma_{0}$": "conductivity", "$\\rho_{0}$": "resistivity"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
223
|
Semiconductors
|
Assume the relaxation time $\tau$ is constant, and try to calculate the Hall coefficient of an n-type semiconductor.
|
[] |
Expression
|
{"$\\tau$": "relaxation time", "$\\mathscr{E}$": "external electric field", "$\\boldsymbol{B}$": "magnetic field", "$e$": "elementary charge", "$m_{\\mathrm{n}}$": "effective mass of the electron", "$\\omega$": "cyclotron frequency", "$v_{x}$": "velocity component in x direction", "$v_{y}$": "velocity component in y direction", "$v_{x 0}$": "initial velocity component in x direction", "$v_{y 0}$": "initial velocity component in y direction", "$n$": "electron density", "$j_{x}$": "current density in x direction", "$j_{y}$": "current density in y direction", "$\\mathscr{E}_{x}$": "electric field component in x direction", "$\\mathscr{E}_{y}$": "electric field component in y direction", "$\\mathscr{C}_{y}$": "unknown field component in y direction", "$R$": "Hall coefficient"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
224
|
Semiconductors
|
In the experiment of measuring the Hall coefficient of a semiconductor, the current $J_{x}$ induced by the external electric field $\varepsilon_{x}$ is called the original current. The transverse current generated under the action of the Lorentz force and the Hall electric field $\varepsilon_{y}$ includes the electron current component $J_{\mathrm{n} y}$ and the hole current component $J_{\mathrm{p} y}$. Find the ratio $f_{\mathrm{c}}=J_{\mathrm{ny}} / J_{x}$.
|
[] |
Expression
|
{"$f_{\\mathrm{c}}$": "ratio of transverse electron current to the original current", "$J_{x}$": "original current density along the x axis", "$J_{\\mathrm{ny}}$": "transverse electron current density", "$J_{\\mathrm{py}}$": "transverse hole current density", "$n$": "electron concentration", "$p$": "hole concentration", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\sigma_{\\mathrm{n}}$": "electron conductivity", "$\\sigma_{\\mathrm{p}}$": "hole conductivity", "$\\varepsilon_{x}$": "electric field component along the x axis", "$\\varepsilon_{y}$": "electric field component along the y axis", "$B_{z}$": "magnetic field component along the z axis", "$J_{y}$": "total transverse current density"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
225
|
Semiconductors
|
Try to prove that in the Hall effect under the conditions of simultaneous presence of two types of charge carriers and a weak magnetic field, the Hall angle $\theta$ and the Hall coefficient $R$ can be expressed as
\begin{aligned}
& \theta=\arctan \frac{p \mu_{\mathrm{p}}^{2}-n \mu_{\mathrm{n}}^{2}}{p \mu_{\mathrm{p}}+n \mu_{\mathrm{n}}} B_{z} \\
& R=\frac{1}{q} \frac{p \mu_{\mathrm{p}}^{2}-n \mu_{\mathrm{n}}^{2}}{(p \mu_{\mathrm{p}}+n \mu_{\mathrm{n}})^{2}}
\end{aligned} If the Hall angle of a sample is measured to be $\theta=0$, find the corresponding electrical conductivity;
|
[] |
Expression
|
{"$\\theta$": "Hall angle", "$p$": "hole concentration", "$\\mu_{\\text{p}}$": "hole mobility", "$n$": "electron concentration", "$\\mu_{\\text{n}}$": "electron mobility", "$B_z$": "magnetic field component in the z-direction", "$\\sigma$": "electrical conductivity", "$q$": "elementary charge", "$b$": "mobility ratio between electrons and holes"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
226
|
Semiconductors
|
In the experiment of measuring the Hall coefficient of semiconductors, the current induced by the external electric field $\mathscr{E}_{x}$ is called the primary current. Under the action of Lorentz force and Hall electric field, a transverse electron current $J_{\mathrm{ey}}$ and a transverse hole current $J_{\mathrm{py}}$ are generated. Find the ratio of the transverse electron current to the primary current at equilibrium $f_{\mathrm{e}}=\frac{J_{\mathrm{ey}}}{J_{x}}$.
|
[] |
Expression
|
{"$f_{\\mathrm{e}}$": "ratio of transverse electron current to primary current", "$J_{\\mathrm{ey}}$": "transverse electron current", "$J_{x}$": "primary current", "$n$": "electron concentration", "$p$": "hole concentration", "$\\mu_{\\mathrm{e}}$": "electron mobility", "$\\mu_{\\mathrm{h}}$": "hole mobility", "$\\sigma_{\\mathrm{e}}$": "electron conductivity", "$\\sigma_{\\mathrm{h}}$": "hole conductivity", "$\\mathscr{E}_{x}$": "external electric field along x-axis", "$B$": "magnetic field", "$\\mathscr{E}_{y}$": "transverse Hall electric field", "$v_{e x}$": "electron drift velocity along x-axis", "$R_{\\mathrm{H}}$": "Hall coefficient", "$J_{y}$": "transverse current", "$f_{\\mathrm{h}}$": "ratio of transverse hole current to primary current", "$T$": "temperature"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
227
|
Others
|
N atoms form a two-dimensional square lattice, with each atom contributing one electron to form a two-dimensional free electron gas. The electron energy expression is
$$
E(k)=\frac{\hbar^{2} k_{x}^{2}}{2 m}+\frac{\hbar^{2} k_{y}^{2}}{2 m}
$$ Derive the formula for the density of st ates of a two-dimensional free gas.
|
[] |
Expression
|
{"$E$": "energy", "$k$": "wave vector magnitude", "$k_x$": "wave vector component in x-direction", "$k_y$": "wave vector component in y-direction", "$m$": "mass", "$\\hbar$": "reduced Planck's constant", "$g(E)$": "density of states per unit area for a two-dimensional free electron gas"}
|
Movement of electrons in a crystal in electric and magnetic fields
|
Movement of electrons in a crystal in electric and magnetic fields
|
|
228
|
Others
|
A two-dimensional square lattice composed of N atoms, each contributing 1 electron to form a two-dimensional free electron gas. The expression for the electron energy is
$$
E(k)=\frac{\hbar^{2} k_{x}^{2}}{2 m}+\frac{\hbar^{2} k_{y}^{2}}{2 m}
$$ At this time, a magnetic field B is applied perpendicular to the square lattice. The energy levels of the free electron gas will condense into Landau levels. What is the degeneracy of these levels?
|
[] |
Expression
|
{"$B$": "magnetic field", "$E$": "energy", "$k$": "wave vector", "$m$": "mass", "$\\hbar$": "reduced Planck's constant", "$\\omega_{c}$": "cyclotron frequency", "$E_{n}$": "energy of the n-th Landau level", "$n$": "quantum number", "$D$": "degeneracy of the Landau levels", "$e$": "elementary charge"}
|
Movement of electrons in a crystal in electric and magnetic fields
|
Movement of electrons in a crystal in electric and magnetic fields
|
|
229
|
Theoretical Foundations
|
A particle is incident with kinetic energy $E$, subjected to the following double $\delta$ potential barriers:
V(x)=V_{0}[\delta(x)+\delta(x-a)]
Find the expression for the conditions under which complete transmission occurs. You should return your answer as an equation.
|
[] |
Equation
|
{"$E$": "kinetic energy", "$V_0$": "potential barrier height", "$x$": "position", "$a$": "distance between delta potential barriers", "$k$": "wave number", "$m$": "mass of the particle", "$\\hbar$": "reduced Planck's constant", "$C$": "dimensionless parameter related to potential and mass", "$R$": "reflection coefficient", "$D$": "transmission coefficient", "$A$": "amplitude of wave function in region 0 < x < a", "$B$": "amplitude of wave function in region 0 < x < a", "$\\theta$": "dimensionless parameter related to wave number and potential"}
|
Schrödinger equation one-dimensional motion
| ||
230
|
Theoretical Foundations
|
For the energy eigenstate $|n\rangle$ of the harmonic oscillator, calculate the expression for the uncertainty product $\Delta x \cdot \Delta p$.
|
[] |
Expression
|
{"$|n\\rangle$": "ket vector corresponding to state n", "$\\Delta x$": "uncertainty in position", "$\\Delta p$": "uncertainty in momentum", "$\\langle n|$": "bra vector corresponding to state n", "$\\delta_{n n^{\\prime}}$": "Kronecker delta function", "$\\hbar$": "reduced Planck's constant", "$m$": "mass", "$\\omega$": "angular frequency", "$a$": "annihilation operator", "$a^{+}$": "creation operator", "$n$": "quantum number", "$\\bar{x}$": "average position", "$\\bar{p}$": "average momentum", "$\\overline{x^{2}}$": "average of the square of position", "$\\overline{p^{2}}$": "average of the square of momentum", "$\\hat{n}$": "number operator"}
|
Schrödinger equation one-dimensional motion
| ||
231
|
Theoretical Foundations
|
In the coherent state $|\alpha\rangle$, calculate the uncertainty product $\Delta x \cdot \Delta p$.
|
[] |
Expression
|
{"$|\\alpha\\rangle$": "coherent state represented by the parameter alpha", "$\\Delta x$": "uncertainty in position", "$\\Delta p$": "uncertainty in momentum", "$\\alpha$": "parameter of the coherent state", "$\\alpha^{*}$": "complex conjugate of alpha", "$\\hbar$": "reduced Planck's constant", "$m$": "mass", "$\\omega$": "angular frequency", "$\\bar{n}$": "average photon number", "$\\bar{E}$": "average energy", "$a$": "annihilation operator", "$a^{+}$": "creation operator", "$n$": "number operator"}
|
Schrödinger equation one-dimensional motion
| ||
232
|
Theoretical Foundations
|
Define the radial momentum operator
\begin{equation*}
\boldsymbol{p}_{r}=\frac{1}{2}(\frac{\boldsymbol{r}}{r} \cdot \boldsymbol{p}+\boldsymbol{p} \cdot \frac{\boldsymbol{r}}{r}) \tag{1}
\end{equation*}
Find the commutation relation $[r, p_{r}]$.
|
[] |
Expression
|
{"$r$": "radial distance", "$p_{r}$": "radial momentum operator", "$\\boldsymbol{r}$": "position vector", "$\\boldsymbol{p}$": "momentum operator", "$\\hbar$": "reduced Planck's constant"}
|
Schrödinger equation one-dimensional motion
| ||
233
|
Theoretical Foundations
|
A particle with mass $\mu$ moves in a central potential field
\begin{equation*}
V(r)=\lambda r^{\nu}, \quad-2<\nu<\infty \tag{1}
\end{equation*}
We only discuss the case where bound states can exist, i.e., when $\lambda \nu > 0$. The radial wave function $u(r)=rR(r)$ satisfies the following radial Schrödinger equation:
\begin{equation*}
\frac{\hbar^{2}}{2 \mu} \frac{\mathrm{~d}^{2} u}{\mathrm{~d} r^{2}}+[E-\lambda r^{\nu}-l(l+1) \frac{\hbar^{2}}{2 \mu r^{2}}] u=0 \tag{2}
\end{equation*}
By introducing the dimensionless radial distance $\rho$ and energy $\varepsilon$, and denoting the radial function as $w(\rho)=u(r)$, the above equation can be non-dimensionalized. Please write down the dimensionless radial equation in terms of $w(\rho)$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mu$": "mass of the particle", "$\\lambda$": "parameter of the potential field", "$\\nu$": "exponent in the potential energy", "$\\hbar$": "reduced Planck's constant", "$E$": "energy", "$l$": "angular momentum quantum number", "$r$": "radial distance", "$u$": "radial wave function", "$\\rho$": "dimensionless radial distance", "$\\varepsilon$": "dimensionless energy", "$w$": "radial function in dimensionless form", "$n$": "quantum number for energy levels", "$N$": "quantum number for energy levels in harmonic oscillator potential"}
|
Schrödinger equation one-dimensional motion 5.4 p104
| ||
234
|
Theoretical Foundations
|
For the hydrogen atom's s states $(n l m=n 00)$, calculate the expression for the uncertainty product $\Delta x \cdot \Delta p_{x}$.
|
[] |
Expression
|
{"$n$": "principal quantum number", "$l$": "azimuthal quantum number", "$m$": "magnetic quantum number", "$\\Delta x$": "position uncertainty", "$\\Delta p_{x}$": "momentum uncertainty in the x-direction", "$\\langle x\\rangle$": "average position", "$\\langle p_{x}\\rangle$": "average momentum in the x-direction", "$\\langle x^{2}\\rangle$": "average square of the position", "$\\langle p_{x}^{2}\\rangle$": "average square of the momentum in the x-direction", "$\\langle r^{2}\\rangle$": "average square of the radial distance", "$a_{0}$": "Bohr radius", "$\\mu$": "reduced mass", "$E_{n}$": "energy at quantum number n", "$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$\\boldsymbol{p}$": "momentum vector"}
|
Schrödinger equation one-dimensional motion 5.17 p121
| ||
235
|
Theoretical Foundations
|
For electrons and other spin $1 / 2$ particles, the eigenstates of $s_{z}$ are often denoted by $\alpha$ and $\beta$, where $\alpha$ is equivalent to $\chi_{\frac{1}{2}}$, and $\beta$ is equivalent to $\chi_{-\frac{1}{2}}$. Given the electron wave function
\begin{equation*}
\psi(r, \theta, \varphi, s_{z})=\alpha Y_{l 0}(\theta, \varphi) R(r)
\end{equation*}
find the only possible measurement value of the total angular momentum $j_{z}$ (taking $\hbar=1$).
|
[] |
Numeric
|
{"$s_{z}$": "z-component of spin angular momentum", "$\\alpha$": "spin-up eigenstate", "$\\beta$": "spin-down eigenstate", "$\\chi_{\\frac{1}{2}}$": "irreducible representation for spin-up state", "$\\chi_{-\\frac{1}{2}}$": "irreducible representation for spin-down state", "$r$": "radial coordinate", "$\\theta$": "polar angle coordinate", "$\\varphi$": "azimuthal angle coordinate", "$Y_{l 0}$": "spherical harmonic (azimuthal quantum number zero)", "$R(r)$": "radial wave function", "$j_{z}$": "z-component of total angular momentum", "$l$": "orbital angular momentum quantum number", "$l^{2}$": "squared orbital angular momentum", "$l_{z}$": "z-component of orbital angular momentum", "$\\sigma_{z}$": "z-component of spin", "$j^{2}$": "squared total angular momentum", "$C_{1}$": "coefficient for state with total angular momentum j=l+1/2", "$C_{2}$": "coefficient for state with total angular momentum j=l-1/2", "$\\phi_{l, l+\\frac{1}{2}, \\frac{1}{2}}$": "eigenstate with total angular momentum quantum numbers (l, l+1/2, 1/2)", "$\\phi_{l, l-\\frac{1}{2}, \\frac{1}{2}}$": "eigenstate with total angular momentum quantum numbers (l, l-1/2, 1/2)"}
|
Schrödinger equation one-dimensional motion 6.34 p180
| ||
236
|
Theoretical Foundations
|
If the operator $\hat{f}(x)$ commutes with $\hat{D}_{x}(a)$, find the general solution for $\hat{f}(x)$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\hat{f}(x)$": "a general operator function of x", "$x$": "a variable representing position or an argument", "$\\hat{D}_{x}(a)$": "a displacement operator acting on x", "$a$": "the period of the function or displacement amount"}
|
Schrödinger equation one-dimensional motion
| ||
237
|
Theoretical Foundations
|
For a spin $1 / 2$ particle, find the effect of the operator $\sigma_{r}=\boldsymbol{\sigma} \cdot \boldsymbol{r} / \boldsymbol{r}$ acting on the common eigenfunctions $\phi_{l j m_{j}}$ of $(l^{2}, j^{2}, j_{z})$ (taking $\hbar=1$). You should return your answer as an equation.
|
[] |
Equation
|
{"$\\sigma_{r}$": "radial component of the Pauli spin operator", "$\\boldsymbol{\\sigma}$": "Pauli spin operator", "$\\boldsymbol{r}$": "position vector", "$\\phi_{l j m_{j}}$": "common eigenfunction of angular momentum operators", "$j$": "total angular momentum quantum number", "$m_{j}$": "magnetic quantum number associated with total angular momentum", "$l$": "orbital angular momentum quantum number", "$j_{z}$": "z-component of total angular momentum", "$\\hbar$": "reduced Planck's constant", "$l_{z}$": "z-component of orbital angular momentum", "$x$": "x-coordinate of the position vector", "$y$": "y-coordinate of the position vector", "$z$": "z-coordinate of the position vector", "$C$": "transition coefficient between eigenstates for $\\sigma_{r}$ operator", "$C^{\\prime}$": "inverse of the transition coefficient $C$", "$\\theta$": "polar angle in spherical coordinates", "$\\varphi$": "azimuthal angle in spherical coordinates"}
|
Schrödinger equation one-dimensional motion
| ||
238
|
Theoretical Foundations
|
For a system composed of two spin $1/2$ particles, where $\boldsymbol{s}_{1}, ~ \boldsymbol{\sigma}_{1}$ and $\boldsymbol{s}_{2}, ~ \boldsymbol{\sigma}_{2}$ represent the spin angular momentum and Pauli operators for particles 1 and 2, respectively, $\boldsymbol{s}_{1}=\frac{1}{2} \boldsymbol{\sigma}_{1}, \boldsymbol{s}_{2}=\frac{1}{2} \boldsymbol{\sigma}_{2}$ (taking $\hbar=1$). Find the simplest algebraic equation satisfied by $\boldsymbol{\sigma}_{1} \cdot \boldsymbol{\sigma}_{2}$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\boldsymbol{s}_{1}$": "spin angular momentum for particle 1", "$\\boldsymbol{\\sigma}_{1}$": "Pauli operators for particle 1", "$\\boldsymbol{s}_{2}$": "spin angular momentum for particle 2", "$\\boldsymbol{\\sigma}_{2}$": "Pauli operators for particle 2", "$\\hbar$": "reduced Planck's constant", "$\\boldsymbol{S}$": "total spin angular momentum", "$\\boldsymbol{S}^{2}$": "square of the total spin angular momentum", "$S$": "spin quantum number"}
|
Schrödinger equation one-dimensional motion
| ||
239
|
Others
|
Majorana fermions. One can write a relativistic equation for a massless 2 -component fermion field that transforms as the upper two components of a Dirac spinor $(\psi_{L})$. Call such a 2-component field $\chi_{a}(x), a=1,2.$ Let us write a 4 -component Dirac field as
\psi(x)=\binom{\psi_{L}}{\psi_{R}}
and recall that the lower components of $\psi$ transform in a way equivalent by a unitary transformation to the complex conjugate of the representation $\psi_{L}$. In this way, we can rewrite the 4 -component Dirac field in terms of two 2 -component spinors:
\psi_{L}(x)=\chi_{1}(x), \quad \psi_{R}(x)=i \sigma^{2} \chi_{2}^{*}(x)
From the Dirac Lagrangian $\mathcal{L} = \bar{\psi}(\mathrm{i} \not \partial-m) \psi$ rewritten in terms of $\chi_{1}$ and $\chi_{2}$, identify and state the form of the mass term.
|
[] |
Expression
|
{"$\\psi$": "4-component Dirac field", "$\\psi_L$": "2-component left-handed spinor", "$\\psi_R$": "2-component right-handed spinor", "$\\chi_1$": "2-component spinor associated with left-handed projection", "$\\chi_2$": "2-component spinor associated with right-handed projection", "$\\mathcal{L}$": "Lagrangian", "$m$": "mass parameter", "$\\sigma^2$": "second Pauli matrix"}
|
The Dirac Field
|
Majorana fermions
|
|
240
|
Others
|
Fierz transformations. Let $u_{i}, i=1, \ldots, 4$, be four 4 -component Dirac spinors. In the text, we proved the Fierz rearrangement formulaes. The first of these formulae can be written in 4 -component notation as
$$
\bar{u}_{1} \gamma^{\mu}(\frac{1+\gamma^{5}}{2}) u_{2} \bar{u}_{3} \gamma_{\mu}(\frac{1+\gamma^{5}}{2}) u_{4}=-\bar{u}_{1} \gamma^{\mu}(\frac{1+\gamma^{5}}{2}) u_{4} \bar{u}_{3} \gamma_{\mu}(\frac{1+\gamma^{5}}{2}) u_{2} .
$$
In fact, there are similar rearrangement formulae for any product
$$
(\bar{u}_{1} \Gamma^{A} u_{2})(\bar{u}_{3} \Gamma^{B} u_{4}),
$$
where $\Gamma^{A}, \Gamma^{B}$ are any of the 16 combinations of Dirac matrices. To begin, normalize the 16 matrices $\Gamma^{A}$ to the convention
$$
\operatorname{tr}[\Gamma^{A} \Gamma^{B}]=4 \delta^{A B} .
$$
This gives $\Gamma^{A}={1, \gamma^{0}, i \gamma^{j}, \ldots}$; write all 16 elements of this set. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\Gamma^{A}$": "general Dirac matrix", "$\\Gamma^{B}$": "general Dirac matrix", "$\\Gamma^{C}$": "general Dirac matrix", "$\\Gamma^{D}$": "general Dirac matrix", "$\\gamma^{0}$": "gamma matrix for time component (0)", "$\\gamma^{\\mu}$": "gamma matrix for spacetime component (μ)", "$\\gamma^{5}$": "gamma matrix (fifth gamma matrix in 5D)", "$\\sigma^{\\mu \\nu}$": "antisymmetric spin tensor", "$\\sigma^{0 i}$": "spin tensor for time and spatial component", "$\\sigma^{i j}$": "spin tensor for spatial components", "$\\delta^{A B}$": "Kronecker delta", "$\\bar{u}_{1}$": "Dirac spinor (1)", "$\\bar{u}_{3}$": "Dirac spinor (3)", "$u_{2}$": "Dirac spinor (2)", "$u_{4}$": "Dirac spinor (4)"}
|
The Dirac Field
|
Fierz transformations
|
|
241
|
Others
|
This problem concerns the discrete symmetries $P, C$, and $T$. Compute the transformation property under $C$ of the antisymmetric tensor fermion bilinear $\bar{\psi} \sigma^{\mu \nu} \psi$, with $\sigma^{\mu \nu}=\frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}]$. This completes the table of the transformation properties of bilinears at the end of the chapter. You should return your answer as an equation.
|
[] |
Equation
|
{"$C$": "charge conjugation transformation", "$\\bar{\\psi}$": "Dirac adjoint of psi", "$\\psi$": "Dirac spinor field", "$\\sigma^{\\mu \\nu}$": "antisymmetric tensor", "$\\gamma^{\\mu}$": "gamma matrices component (mu)", "$\\gamma^{\\nu}$": "gamma matrices component (nu)", "$\\gamma^{0}$": "gamma matrices (time component)", "$\\gamma^{2}$": "gamma matrices (spatial component)", "$\\gamma^{1}$": "gamma matrices (spatial component)", "$\\gamma^{3}$": "gamma matrices (spatial component)"}
|
The Dirac Field
|
The discrete symmetries $P, C$ and $T$
|
|
242
|
Others
|
Exotic contributions to $\boldsymbol{g} \mathbf{- 2}$. Any particle that couples to the electron can produce a correction to the electron-photon form factors and, in particular, a correction to $g-2$. Because the electron $g-2$ agrees with QED to high accuracy, these corrections allow us to constrain the properties of hypothetical new particles.
The unified theory of weak and electromagnetic interactions contains a scalar particle $h$ called the Higgs boson, which couples to the electron according to
\begin{align*}
H_{\text{int}} = \int d^3 x \frac{\lambda}{\sqrt{2}} h \bar{\psi} \psi.
\end{align*}
One can study the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\lambda$ and the mass $m_h$ of the Higgs boson. QED accounts extremely well for the electron's anomalous magnetic moment. If $a=(g-2) / 2$,
$$
|a_{\text {expt. }}-a_{\mathrm{QED}}|<1 \times 10^{-10}
$$
What limits does this place on $\lambda$ and $m_{h}$ ? In the simplest version of the electroweak theory, $\lambda=3 \times 10^{-6}$ and $m_{h}>60 \mathrm{GeV}$. Show that these values are not excluded.
Hint: You can find the contribution of a virtual Higgs boson to the electron $(g - 2)$, in terms of $\lambda$ and the mass $m_h$ of the Higgs boson and check its value with $1 \times 10^{-10}$. You should return your answer as an equation.
|
[] |
Equation
|
{"$a$": "anomalous part of the magnetic moment", "$g$": "gyromagnetic ratio", "$\\lambda$": "coupling constant in electroweak theory", "$m_{h}$": "mass of the Higgs boson", "$m$": "mass of the electron or muon in context", "$\\delta F_{2}$": "form factor contribution", "$q$": "momentum transfer parameter"}
|
Radiative Corrections: Introduction
|
Exotic contributions to $g-2$
|
|
243
|
Others
|
Although we have discussed QED radiative corrections at length in the last two chapters, so far we have made no attempt to compute a full radiatively corrected cross section. The reason is of course that such calculations are quite lengthy. Nevertheless it would be dishonest to pretend that one understands radiative corrections after computing only isolated effects as we have done. This "final project" is an attempt to remedy this situation. The project is the computation of one of the simplest, but most important, radiatively corrected cross sections.
Strongly interacting particles-pions, kaons, and protons-are produced in $e^{+} e^{-}$annihilation when the virtual photon creates a pair of quarks. If one ignores the effects of the strong interactions, it is easy to calculate the total cross section for quark pair production. In this final project, we will analyze the first corrections to this formula due to the strong interactions.
Let us represent the strong interactions by the following simple model: Introduce a new massless vector particle, the gluon, which couples universally to quarks:
\Delta H=\int d^{3} x g \bar{\psi}_{f i} \gamma^{\mu} \psi_{f i} B_{\mu}
Here $f$ labels the type ("flavor") of the quark ( $u, d, s, c$, etc.) and $i=1,2,3$ labels the color. The strong coupling constant $g$ is independent of flavor and color. The electromagnetic coupling of quarks depends on the flavor, since the $u$ and $c$ quarks have charge $Q_{f}=+2 / 3$ while the $d$ and $s$ quarks have charge $Q_{f}=-1 / 3$. By analogy to $\alpha$, let us define
\alpha_{g}=\frac{g^{2}}{4 \pi}
In this exercise, we will compute the radiative corrections to quark pair production proportional to $\alpha_{g}$.
This model of the strong interactions of quarks does not quite agree with the currently accepted theory of the strong interactions, quantum chromodynamics (QCD). However, all of the results that we will derive here are also
correct in QCD with the replacement
$$
\alpha_{g} \rightarrow \frac{4}{3} \alpha_{s} .
$$
Throughout this exercise, you may ignore the masses of quarks. You may also ignore the mass of the electron, and average over electron and positron polarizations. To control infrared divergences, it will be necessary to assume that the gluons have a small nonzero mass $\mu$, which can be taken to zero only at the end of the calculation. However, it is consistent to sum over polarization states of this massive boson by the replacement:
$$
\sum \epsilon^{\mu} \epsilon^{\nu *} \rightarrow-g^{\mu \nu}
$$
this also implies that we may use the propagator
$$
\widehat{B^{\mu} B^{\nu}}=\frac{-i g^{\mu \nu}}{k^{2}-\mu^{2}+i \epsilon}
$$ In the analysis of the 3-body final state process $e^{+} e^{-} \rightarrow \bar{q} q g$, the total 4-momentum is $q$. The final quark (4-momentum $k_1$) and antiquark (4-momentum $k_2$) are massless, while the gluon (4-momentum $k_3$) has mass $\mu$. Dimensionless energy fractions are defined as $x_i = \frac{2 k_i \cdot q}{q^{2}}$. The physical integration region for $x_1$ and $x_2$ is bounded. One of these boundaries corresponds to the kinematic configuration where the 3-momenta of the quark ($\mathbf{k}_1$) and the antiquark ($\mathbf{k}_2$) are parallel. Determine the equation for this specific boundary in terms of $x_1$, $x_2$, $\mu^2$, and $q^2$ (the square of the total 4-momentum). You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$e^{+}$": "positron", "$e^{-}$": "electron", "$\\bar{q}$": "antiquark", "$q$": "quark or total 4-momentum", "$g$": "gluon", "$k_1$": "4-momentum of the final quark", "$k_2$": "4-momentum of the final antiquark", "$k_3$": "4-momentum of the gluon", "$\\mu$": "mass of the gluon", "$x_1$": "dimensionless energy fraction for the quark", "$x_2$": "dimensionless energy fraction for the antiquark", "$x_3$": "dimensionless energy fraction for the gluon", "$\\mathbf{k}_1$": "3-momentum of the final quark", "$\\mathbf{k}_2$": "3-momentum of the final antiquark", "$q^2$": "square of the total 4-momentum", "$E_1$": "energy of the final quark", "$E_2$": "energy of the final antiquark", "$E_3$": "energy of the gluon"}
|
Final Project I
|
Radiation of Gluon Jets
|
|
244
|
Others
|
Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,
$$
\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^{2}-\frac{\lambda}{4!} \phi^{4}+\bar{\psi}(i \not \partial) \psi-i g \bar{\psi} \gamma^{5} \psi \phi,
$$
compute the Callan-Symanzik $\beta$ function for $g$:
$$
\beta_{g}(\lambda, g),
$$
to leading order in coupling constants, assuming that $\lambda$ and $g^{2}$ are of the same order.
|
[] |
Expression
|
{"$\\mathcal{L}$": "Lagrangian", "$\\phi$": "scalar field", "$\\lambda$": "quartic coupling constant", "$\\bar{\\psi}$": "Dirac adjoint spinor field", "$\\psi$": "spinor field", "$g$": "Yukawa coupling constant", "$\\gamma^{5}$": "gamma matrix", "$\\beta$": "beta function", "$\\beta_{g}$": "beta function for coupling constant g", "$\\delta_{\\psi}$": "wave function renormalization constant for the spinor field", "$\\delta_{\\phi}$": "wave function renormalization constant for the scalar field", "$\\delta_{g}$": "coupling constant renormalization", "$\\Lambda$": "UV cutoff", "$M$": "renormalization scale"}
|
The Renormalization Group
|
Beta Function in Yukawa Theory
|
|
245
|
Others
|
Beta functions in Yukawa theory. In the pseudoscalar Yukawa theory with masses set to zero,
$$
\mathcal{L}=\frac{1}{2}(\partial_{\mu} \phi)^{2}-\frac{\lambda}{4!} \phi^{4}+\bar{\psi}(i \not \partial) \psi-i g \bar{\psi} \gamma^{5} \psi \phi,
$$
compute the Callan-Symanzik $\beta$ function for $\lambda$:
\beta_{\lambda}(\lambda, g),
to leading order in coupling constants, assuming that $\lambda$ and $g^{2}$ are of the same order.
|
[] |
Expression
|
{"$\\beta$": "beta function", "$\\lambda$": "coupling constant in the theory", "$g$": "coupling constant in the theory", "$\\phi$": "scalar field", "$\\psi$": "fermionic field", "$\\gamma^{5}$": "gamma matrix with pseudoscalar property", "$\\delta_{\\phi}$": "field renormalization counterterm for the scalar field", "$\\delta_{\\lambda}$": "renormalization counterterm for the coupling constant", "$M$": "renormalization scale", "$\\Lambda$": "ultraviolet cutoff"}
|
The Renormalization Group
|
Beta Function in Yukawa Theory
|
|
246
|
Others
|
Asymptotic symmetry. Consider the following Lagrangian, with two scalar fields $\phi_{1}$ and $\phi_{2}$ :
\mathcal{L}=\frac{1}{2}((\partial_{\mu} \phi_{1})^{2}+(\partial_{\mu} \phi_{2})^{2})-\frac{\lambda}{4!}(\phi_{1}^{4}+\phi_{2}^{4})-\frac{2 \rho}{4!}(\phi_{1}^{2} \phi_{2}^{2}) .
Notice that, for the special value $\rho=\lambda$, this Lagrangian has an $O(2)$ invariance rotating the two fields into one another. For a theory with two scalar fields $\phi_1$ and $\phi_2$ described by the Lagrangian:
$$
\mathcal{L}=\frac{1}{2}((\partial_{\mu} \phi_{1})^{2}+(\partial_{\mu} \phi_{2})^{2})-\frac{\lambda}{4!}(\phi_{1}^{4}+\phi_{2}^{4})-\frac{2\rho}{4!} \phi_{1}^{2} \phi_{2}^{2}
$$
Working in four dimensions, what is the $\beta$ function for the coupling constant $\lambda$, denoted as $\beta_{\lambda}$, to leading order in the coupling constants?
|
[] |
Expression
|
{"$\\phi_1$": "first scalar field", "$\\phi_2$": "second scalar field", "$\\lambda$": "coupling constant for self-interactions", "$\\rho$": "coupling constant for interaction between fields", "$\\beta$": "beta function symbol", "$\\beta_{\\lambda}$": "beta function for the coupling constant \\lambda", "$\\beta_{\\rho}$": "beta function for the coupling constant \\rho", "$\\mu$": "renormalization scale", "$\\delta_{\\lambda}$": "variation in coupling constant \\lambda", "$\\delta_{\\rho}$": "variation in coupling constant \\rho"}
|
The Renormalization Group
|
Asymptotic Symmetry
|
|
247
|
Others
|
State the leading term in $\gamma(\lambda)$ for $\phi^{4}$ theory.
|
[] |
Expression
|
{"$\\gamma$": "anomalous dimension", "$\\lambda$": "coupling constant", "$\\phi$": "field"}
|
Critical Exponents and Scalar Field Theory
|
The exponent $\eta$
|
|
248
|
Others
|
Compute the anomalous dimension \(\gamma\) in an \(O(N)\)-symmetric \(\phi^{4}\) theory for \(N = 3\) and coupling constant \(\lambda = 0.5\), using the formula
\[\gamma = (N+2)\,\frac{\lambda^{2}}{(4\pi)^{4}}.\]
|
[] |
Numeric
|
{"$\\gamma$": "anomalous dimension", "$N$": "symmetry index of O(N) theory", "$\\phi$": "field in \\(\\phi^{4}\\) theory", "$\\lambda$": "coupling constant", "$\\pi$": "mathematical constant pi"}
|
Critical Exponents and Scalar Field Theory
|
The exponent $\eta$
|
|
249
|
Others
|
Brute-force computations in $\boldsymbol{S U ( 3 )}$. The standard basis for the fundamental representation of $S U(3)$ is
\begin{array}{rlrl}
t^{1} & =\frac{1}{2}(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}), \quad t^{2}=\frac{1}{2}(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0
\end{array}), \quad t^{3}=\frac{1}{2}(\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{array}), \\
t^{4} & =\frac{1}{2}(\begin{array}{lll}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}), & t^{5}=\frac{1}{2}(\begin{array}{ccc}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0
\end{array}), \\
t^{6} & =\frac{1}{2}(\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}), \quad t^{7}=\frac{1}{2}(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0
\end{array}), \quad t^{8}=\frac{1}{2 \sqrt{3}}(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2
\end{array}) .
\end{array} Write down the dimension $d$ of $S U(N)$ group. You should return your answer as an equation.
|
[] |
Equation
|
{"$d$": "dimension of the $SU(N)$ group", "$N$": "parameter of the $SU(N)$ group"}
|
Non-Abelian Gauge Invariance
|
Brute-force computations in $S U(3)$
|
|
250
|
Others
|
Matrix element for proton decay. Some advanced theories of particle interactions include heavy particles $X$ whose couplings violate the conservation of baryon number. Integrating out these particles produces an effective interaction that allows the proton to decay to a positron and a photon or a pion. This effective interaction is most easily written using the definite-helicity components of the quark and electron fields: If $u_{L}, d_{L}, u_{R}, e_{R}$ are two-component spinors, then this effective interaction is
\Delta \mathcal{L}=\frac{2}{m_{X}^{2}} \epsilon_{a b c} \epsilon^{\alpha \beta} \epsilon^{\gamma \delta} e_{R \alpha} u_{R a \beta} u_{L b \gamma} d_{L c \delta} .
A typical value for the mass of the $X$ boson is $m_{X}=10^{16} \mathrm{GeV}$. Estimate, in order of magnitude, the value of the proton lifetime if the proton is allowed to decay through this interaction.
|
[] |
Numeric
|
{"$p$": "proton", "$e^{+}$": "positron", "$\\pi^{0}$": "neutral pion", "$\\mathcal{O}_{X}$": "operator for the decay process", "$m_{X}$": "scale of the higher dimensional operator", "$i$": "color index for quark", "$j$": "color index for quark", "$k$": "color index for quark", "$\\alpha$": "spinor index", "$\\beta$": "spinor index", "$\\gamma$": "spinor index", "$\\delta$": "spinor index", "$e_{R \\alpha}$": "right-handed electron field with spinor index", "$u_{R i \\beta}$": "right-handed up-quark field with color and spinor indices", "$u_{L j \\gamma}$": "left-handed up-quark field with color and spinor indices", "$d_{L k \\delta}$": "left-handed down-quark field with color and spinor indices", "$\\mathcal{M}$": "amplitude of the decay process", "$m_{p}$": "proton mass", "$\\Gamma$": "decay width"}
|
Operator Products and Effective Vertices
|
Matrix element for proton decay
|
|
251
|
Others
|
A model with two Higgs fields. Assume that the two Higgs fields couple to quarks by the set of fundamental couplings
\mathcal{L}_{m}=-\lambda_{d}^{i j} \bar{Q}_{L}^{i} \cdot \phi_{1} d_{R}^{j}-\lambda_{u}^{i j} \epsilon^{a b} \bar{Q}_{L a}^{i} \phi_{2 b}^{\dagger} u_{R}^{j}+\text { h.c. }
Find the couplings of the physical charged Higgs boson of part (c) to the mass eigenstates of quarks. These couplings depend only on the values of the quark masses and $\tan \beta$ and on the elements of the CKM matrix.
|
[] |
Expression
|
{"$\\lambda_{d}^{ij}$": "Yukawa coupling matrix for down-type quarks", "$\\lambda_{u}^{ij}$": "Yukawa coupling matrix for up-type quarks", "$\\bar{Q}_{L}^{i}$": "left-handed quark doublet", "$\\phi_{1}$": "Higgs field component coupling to down-type quarks", "$\\phi_{2}$": "Higgs field component coupling to up-type quarks", "$d_{R}^{j}$": "right-handed down-type quark", "$u_{R}^{j}$": "right-handed up-type quark", "$\\tan \\beta$": "ratio of the two Higgs doublet vacuum expectation values", "$\\phi^{+}$": "physical charged Higgs boson", "$v_{1}$": "vacuum expectation value associated with $\\phi_{1}$", "$v_{2}$": "vacuum expectation value associated with $\\phi_{2}$", "$v$": "total vacuum expectation value", "$U_{u}$": "unitary matrix for left-handed up-type quark fields", "$U_{d}$": "unitary matrix for left-handed down-type quark fields", "$W_{u}$": "unitary matrix for right-handed up-type quark fields", "$W_{d}$": "unitary matrix for right-handed down-type quark fields", "$D_{d}$": "diagonal mass matrix for down-type quarks", "$D_{u}$": "diagonal mass matrix for up-type quarks", "$V_{\\mathrm{CKM}}$": "Cabibbo-Kobayashi-Maskawa (CKM) matrix", "$m_{u}$": "mass matrix for up-type quarks", "$m_{d}$": "mass matrix for down-type quarks", "$\\pi_{1}^{+}$": "charged Higgs boson component", "$\\pi_{2}^{+}$": "charged Higgs boson component"}
|
Gauge Theories with Spontaneous Symmetry Breaking
|
A model with two Higgs fields
|
|
252
|
Others
|
Dependence of radiative corrections on the Higgs boson mass. In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\Pi_{WW}(q^2)$ (specifically the part proportional to $g^{\mu\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_W^2$. Provide the derivation steps and the final expression for $\Pi_{WW}(q^2)$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\Pi_{WW}(q^2)$": "vacuum polarization amplitude for W bosons", "$\\mu$": "index representing a spacetime dimension", "$\\nu$": "index representing a spacetime dimension", "$m_h$": "Higgs boson mass", "$M$": "subtraction scale", "$m_W$": "W boson mass", "$g$": "coupling constant", "$q^2$": "momentum transfer squared", "$E$": "measure of divergence in dimensional regularization"}
|
Quantization of Spontaneously Broken Gauge Theories
|
Dependence of radiative corrections on the Higgs boson mass
|
|
253
|
Others
|
Dependence of radiative corrections on the Higgs boson mass. In Feynman-'t Hooft gauge, compute the dependence of the vacuum polarization amplitude $\Pi_{ZZ}(q^2)$ (specifically the part proportional to $g^{\mu\nu}$) on the Higgs boson mass $m_h$. Consider the diagrams involving the Higgs boson, work in the large $m_h$ limit, use dimensional regularization with $M$ as the subtraction scale, and fix the subtraction point at $M^2=m_Z^2$. Provide the final expression for $\Pi_{ZZ}(q^2)$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\Pi_{ZZ}(q^2)$": "vacuum polarization amplitude for Z bosons", "$g$": "coupling constant", "$\\theta_w$": "Weinberg angle", "$m_h$": "Higgs boson mass", "$m_Z$": "Z boson mass", "$q^2$": "momentum transfer squared", "$M$": "subtraction scale"}
|
Quantization of Spontaneously Broken Gauge Theories
|
Dependence of radiative corrections on the Higgs boson mass
|
|
254
|
Theoretical Foundations
|
Consider the wave equation $\nabla^{2} u+k^{2} n(r)^{2} u=0$ with slowly varying $n(r)$. If we introduce the eikonal function $S(r)$ by substituting $u = \mathrm{e}^{\frac{2 \pi i}{\lambda} S(r)}$ (where $\lambda=2 \pi / k$) into the wave equation, what is the resulting differential equation for $S(r)$ before any series expansion of $S(r)$ is performed (this is known as the Riccati equation in this context)? You should return your answer as an equation.
|
[] |
Equation
|
{"$u$": "wave function", "$k$": "wave number", "$n(r)$": "refractive index as a function of position", "$S(r)$": "eikonal function as a function of position", "$\\lambda$": "wavelength", "$\\hbar$": "reduced Planck's constant", "$p$": "momentum", "$m$": "mass", "$E$": "energy", "$V(r)$": "potential energy as a function of position"}
|
One-Body Problems without Spin
|
OneDimensional Problems
|
|
255
|
Theoretical Foundations
|
The tensor force between two particles 1 and 2 of spin $1/2$ is associated with the operator $T_{12}$ given by:
T_{12}=\frac{(\boldsymbol{\sigma}_{1} \cdot \boldsymbol{r})(\boldsymbol{\sigma}_{2} \cdot \boldsymbol{r})}{r^{2}}-\frac{1}{3}(\boldsymbol{\sigma}_{1} \cdot \boldsymbol{\sigma}_{2})
where $\boldsymbol{\sigma}_{1}$ and $\boldsymbol{\sigma}_{2}$ are the Pauli spin matrices for particle 1 and 2 respectively, and $\boldsymbol{r}$ is the relative position vector between the particles.
If $\chi_{0,0}$ represents the spin singlet state (an eigenfunction of total spin $S=0$, defined as $\chi_{0,0}=\frac{1}{\sqrt{2}}(\alpha_{1} \beta_{2}-\beta_{1} \alpha_{2})$) for the two-particle system, calculate the result of applying the operator $T_{12}$ to $\chi_{0,0}$. In other words, find $T_{12} \chi_{0,0}$.
|
[] |
Numeric
|
{"$T_{12}$": "tensor force operator between two particles", "$\\boldsymbol{\\sigma}_{1}$": "Pauli spin matrix for particle 1", "$\\boldsymbol{\\sigma}_{2}$": "Pauli spin matrix for particle 2", "$\\boldsymbol{r}$": "relative position vector between the particles", "$r$": "magnitude of the relative position vector", "$\\chi_{0,0}$": "spin singlet state of total spin S=0", "$\\alpha_{1}$": "spin-up state of particle 1", "$\\beta_{1}$": "spin-down state of particle 1", "$\\alpha_{2}$": "spin-up state of particle 2", "$\\beta_{2}$": "spin-down state of particle 2", "$\\chi_{1,1}$": "triplet state with total spin projection m_s=1", "$\\chi_{1,0}$": "triplet state with total spin projection m_s=0", "$\\chi_{1,-1}$": "triplet state with total spin projection m_s=-1", "$\\vartheta$": "polar angle in spherical coordinates", "$\\varphi$": "azimuthal angle in spherical coordinates", "$Y_{2,0}$": "spherical harmonic function for l=2, m=0", "$Y_{2,1}$": "spherical harmonic function for l=2, m=1", "$Y_{2,-1}$": "spherical harmonic function for l=2, m=-1", "$Y_{2,2}$": "spherical harmonic function for l=2, m=2", "$Y_{2,-2}$": "spherical harmonic function for l=2, m=-2"}
|
Particles with Spin
|
OneDimensional Problems
|
|
256
|
Theoretical Foundations
|
In a neutral helium atom, one electron is in the $1s$ ground state and the other is in the $2p$ excited state ($n=2, l=1$). Using a theoretical model based on hydrogen-like wave functions with screening of one nuclear charge by the $1s$ electron, calculate the ionization energy (in eV) for the $2p$ electron if the atom is in the parahelium state (give a number and keep three decimal places).
|
[] |
Numeric
|
{"$1s$": "ground state orbital for helium electron", "$2p$": "excited state orbital for helium electron", "$n$": "principal quantum number", "$l$": "azimuthal quantum number", "$E_{1}$": "energy of the electron in the ground state 1s", "$E_{n}$": "energy of the electron in the excited state np", "$R_{n l}$": "normalized radial wave function for quantum numbers n and l", "$r_{1}$": "distance of first electron from nucleus", "$r_{2}$": "distance of second electron from nucleus", "$r_{12}$": "distance between two electrons", "$\\varepsilon$": "parameter for parahelium or orthohelium state", "$\\psi$": "two-electron wave function", "$\\mathscr{C}$": "classical integral for electron-electron interaction", "$\\mathscr{E}$": "exchange integral for electron-electron interaction", "$\\Omega_{1}$": "solid angle for first electron", "$\\Omega_{2}$": "solid angle for second electron", "$P_{\\lambda}$": "Legendre polynomial of degree lambda"}
|
Many-Body Problems
|
OneDimensional Problems
|
|
257
|
Theoretical Foundations
|
The function
\begin{equation*}
\tilde{\varphi}(x)=\frac{1}{(1+\alpha x)^{2}} \tag{176.1}
\end{equation*}
with a suitable value of $\alpha$, independent of $Z$, may be used as a fair approximation to the Thomas-Fermi function $\varphi_{0}(x)$ for a neutral atom. The constant $\alpha$ shall be determined in such a way as to permit exact normalization of $\tilde{\varphi}$. Hint: the answer is a number.
|
[] |
Numeric
|
{"$\\tilde{\\varphi}$": "approximated function to the Thomas-Fermi function", "$x$": "variable in the function", "$\\alpha$": "constant independent of $Z$ for approximation", "$Z$": "atomic number", "$\\varphi_{0}$": "Thomas-Fermi function for a neutral atom", "$n(r)$": "electron density", "$r$": "radial distance in atomic units", "$V(r)$": "atomic potential", "$a$": "constant related to the atomic number, defined as $a=0.88534 Z^{-\\frac{1}{3}}$", "$y$": "integration variable, $y=\\alpha x$"}
|
Many-Body Problems
|
OneDimensional Problems
|
|
258
|
Theoretical Foundations
|
Calculate the numerical value for the mean lifetime (in seconds) of the $2P$ state in a hydrogen atom, which decays to the $1S$ state by emission of a photon. This requires first determining the total transition probability $P$ for an electron from a higher $P$ state to a lower $S$ state (summed over all photon directions and polarizations), and then specializing this for the hydrogen $2P \rightarrow 1S$ transition.
|
[] |
Numeric
|
{"$2P$": "2P state in a hydrogen atom", "$1S$": "1S state in a hydrogen atom", "$P$": "total transition probability", "$m$": "magnetic quantum number", "$e$": "elementary charge", "$\\omega$": "angular frequency of light", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light in vacuum", "$R$": "radial transition matrix element", "$\\Theta$": "polar angle", "$\\Phi$": "azimuthal angle", "$\\tau$": "mean lifetime of the 2P state"}
|
Radiation Theory
|
OneDimensional Problems
|
|
259
|
Theoretical Foundations
|
To compare the intensities of emission of the two first Lyman lines of atomic hydrogen, Ly $\alpha$ and Ly $\beta$. Numerically calculate $I_\alpha/I_\beta$ You should return your answer as an equation.
|
[] |
Equation
|
{"$I_{\\alpha}$": "intensity of Lyman alpha line", "$I_{\\beta}$": "intensity of Lyman beta line", "$E_{\\alpha}$": "energy difference for Lyman alpha transition", "$E_{\\beta}$": "energy difference for Lyman beta transition"}
|
Radiation Theory
|
OneDimensional Problems
|
|
260
|
Others
|
Consider a scalar field theory with interaction Lagrangian $\mathcal{L}_{\mathrm{I}}=-\frac{g}{3!} \phi^{3}-\frac{\lambda}{4!} \phi^{4}$. Derive the identity that relates the number of loops ($n_L$), internal lines ($n_I$), trivalent vertices ($n_3$), and tetravalent vertices ($n_4$). You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mathcal{L}_{\\mathrm{I}}$": "interaction Lagrangian", "$g$": "coupling constant for the $\\phi^3$ interaction", "$\\lambda$": "coupling constant for the $\\phi^4$ interaction", "$\\phi$": "scalar field", "$n_{L}$": "number of loops", "$n_{I}$": "number of internal lines", "$n_{3}$": "number of trivalent vertices", "$n_{4}$": "number of tetravalent vertices"}
|
Perturbation Theory
|
Perturbation Theory
|
|
261
|
Others
|
For a massless spin-one particle with four-momentum $p$, its physical helicity polarization vectors are denoted by $\epsilon_{\pm}^{\mu}(\mathbf{p})$. Under a Lorentz transformation $\Lambda$ (which transforms the momentum $p$ to $\Lambda p$), these polarization vectors transform according to the following relation derived using little group properties:
$$[\Lambda^{-1}]_{\nu}^{\mu} \epsilon_{ \pm}^{v}(\Lambda \mathbf{p}) = X$$
What is the expression for $X$ in terms of the original polarization vector $\epsilon_{ \pm}^{\mu}(\mathbf{p})$, the original four-momentum $p^{\mu}$, a phase factor $e^{\mp i \theta}$ (where $\theta$ is the Wigner rotation angle), and coefficients $\beta_{ \pm}$ which depend on the parameters of the Lorentz transformation and the reference momentum?
|
[] |
Expression
|
{"$p$": "four-momentum", "$\\epsilon_{\\pm}^{\\mu}(\\mathbf{p})$": "physical helicity polarization vector for momentum $p$", "$\\theta$": "Wigner rotation angle", "$\\beta_{\\pm}$": "coefficients depending on Lorentz transformation parameters", "$\\epsilon_{\\pm}^{\\mu}(\\mathbf{q})$": "helicity polarization vector for reference momentum $q$", "$\\omega$": "reference energy component", "$q^{\\mu}$": "reference four-momentum", "$\\mathcal{M}_{\\mu}$": "matrix element component", "$\\Lambda$": "Lorentz transformation", "$R$": "element of the little group", "$\\mathbf{p}$": "momentum vector", "$\\mathcal{R}(\\widehat{\\mathbf{p}})$": "spatial rotation aligning axis 3 with vector $\\widehat{\\mathbf{p}}$"}
|
Quantum Electrodynamics
|
Quantum Electrodynamics
|
|
262
|
Others
|
Give the expression of the one-loop self-energy in a $\phi^{4}$ theory in the Matsubara formalism. Calculate it in the limit $\beta \mathrm{m} \ll 1$.
|
[] |
Expression
|
{"$\\Sigma$": "self-energy", "$\\lambda$": "coupling constant", "$T$": "temperature", "$n$": "index for summation", "$p$": "momentum", "$m$": "mass", "$E_p$": "energy associated with momentum $p$", "$n_B$": "Bose-Einstein distribution", "$\\Lambda$": "momentum cutoff"}
|
Functional Quantization
|
Functional Quantization
|
|
263
|
Others
|
Consider two Grassmann variables $\theta_\pm$. For the operator $\tau_3 \equiv \frac{1}{2}(\theta_{+} \frac{\partial}{\partial \theta_{+}}-\theta_{-} \frac{\partial}{\partial \theta_{-}})$, find two linearly independent eigenfunctions corresponding to the eigenvalue $0$. Express them using $1, \theta_+, \theta_-, \theta_+\theta_-$ and normalize constant terms or leading $\theta$ terms to 1. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\tau_3$": "operator", "$\\theta_{+}$": "theta plus", "$\\theta_{-}$": "theta minus", "$\\lambda$": "eigenvalue", "$a$": "coefficient for constant term", "$b$": "coefficient for theta plus term", "$c$": "coefficient for theta minus term", "$d$": "coefficient for theta plus theta minus term"}
|
Path Integrals for Fermions and Photons
|
Path Integrals for Fermions and Photons
|
|
264
|
Others
|
Consider two Grassmann variables $\theta_\pm$. Consider the operator $\tau_1 \equiv \frac{1}{2}(\theta_{+} \frac{\partial}{\partial \theta_{-}}+\theta_{-} \frac{\partial}{\partial \theta_{+}})$ acting on functions of two Grassmann variables $\theta_{\pm}$. Find all distinct eigenvalues of $\tau_1$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\tau_1$": "operator", "$\\theta_{+}$": "Grassmann variable (positive)", "$\\theta_{-}$": "Grassmann variable (negative)", "$\\theta_{\\pm}$": "Grassmann variables (positive/negative)", "$a$": "constant term in function of Grassmann variables", "$b$": "coefficient of $\\theta_{+}$ in function", "$c$": "coefficient of $\\theta_{-}$ in function", "$d$": "coefficient of $\\theta_{+}\\theta_{-}$ in function", "$\\lambda$": "eigenvalue"}
|
Path Integrals for Fermions and Photons
|
Path Integrals for Fermions and Photons
|
|
265
|
Others
|
Consider two Grassmann variables $\theta_\pm$. Consider the operator $\tau_2 \equiv \frac{i}{2}(\theta_{-} \frac{\partial}{\partial \theta_{+}}-\theta_{+} \frac{\partial}{\partial \theta_{-}})$ acting on functions of two Grassmann variables $\theta_{\pm}$. Find all distinct eigenvalues of $\tau_2$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\tau_2$": "operator", "$\\theta_{-}$": "Grassmann variable (negative)", "$\\theta_{+}$": "Grassmann variable (positive)", "$\\theta_{\\pm}$": "Grassmann variables (positive and negative)"}
|
Path Integrals for Fermions and Photons
|
Path Integrals for Fermions and Photons
|
|
266
|
Others
|
Given the operators $\tau_{1} \equiv \frac{1}{2}(\theta_{+} \frac{\partial}{\partial \theta_{-}}+\theta_{-} \frac{\partial}{\partial \theta_{+}})$ and $\tau_{2} \equiv \frac{i}{2}(\theta_{-} \frac{\partial}{\partial \theta_{+}}-\theta_{+} \frac{\partial}{\partial \theta_{-}})$, calculate the action of the operator $(\tau_1 - i\tau_2)$ on the Grassmann variable $\theta_-$.
|
[] |
Numeric
|
{"$\\tau_{1}$": "operator 1", "$\\tau_{2}$": "operator 2", "$\\theta_{+}$": "Grassmann variable (positive)", "$\\theta_{-}$": "Grassmann variable (negative)"}
|
Path Integrals for Fermions and Photons
|
Path Integrals for Fermions and Photons
|
|
267
|
Others
|
Consider a non-Abelian gauge theory with the usual $\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields in the adjoint representation and $n_{f}$ Dirac fermions in the adjoint representation. Calculate the expression for $\frac{1}{g_{\mathrm{r}}^{2}(\mu)}$ for this theory, analogous to the standard one-loop running coupling constant equation. The constants for fields in the adjoint representation are given as $\mathrm{c}_{\mathrm{adj}, 0}=\frac{\mathrm{N}}{3(4 \pi)^{2}}$ for gauge fields/ghosts, $\mathrm{c}_{\mathrm{adj}, 1 / 2}=-\frac{8 \mathrm{~N}}{3(4 \pi)^{2}}$ for Dirac fermions, and we can infer the scalar contribution from the context (scalars have bosonic statistics and contribute with an opposite sign to ghosts or fermions regarding their statistical nature's impact on the beta function coefficient).
|
[] |
Expression
|
{"$\\mathfrak{s u}(N)$": "special unitary Lie algebra of degree N", "$n_{s}$": "number of complex scalar fields", "$n_{f}$": "number of Dirac fermions", "$g_{\\mathrm{r}}$": "renormalized coupling constant", "$\\mu$": "energy scale", "$\\mathrm{c}_{\\mathrm{adj}, 0}$": "coefficient for gauge fields and ghosts in adjoint representation", "$N$": "dimension of the gauge group in adjoint representation", "$\\mathrm{c}_{\\mathrm{adj}, 1 / 2}$": "coefficient for Dirac fermions in adjoint representation", "$g_{\\mathrm{b}}$": "bare coupling constant", "$\\kappa$": "reference scale", "$n_{\\mathrm{s}}$": "number of scalars in calculation", "$\\mathrm{n}_{s}$": "number of scalars", "$n_{\\mathrm{f}}$": "number of Dirac fermions in calculation"}
|
Renormalization of Gauge Theories
|
Renormalization of Gauge Theories
|
|
268
|
Others
|
For a non-Abelian gauge theory with $\mathfrak{s u}(N)$ gauge fields, $n_{s}$ complex scalar fields and $n_{f}$ Dirac fermions all in the adjoint representation, the one-loop running of the inverse squared coupling is given by $\frac{1}{g_{\mathrm{r}}^{2}(\mu)}=\frac{1}{\mathrm{~g}_{\mathrm{b}}^{2}}+\frac{\mathrm{N}}{3(4 \pi)^{2}}(11-4 n_{\mathrm{f}}-\mathrm{n}_{\mathrm{s}}) \ln \frac{\mu^{2}}{\kappa^{2}}$. Determine the condition on $n_s$ and $n_f$ for the gauge coupling to not be running at one loop. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mathfrak{s u}(N)$": "special unitary group in N dimensions", "$n_{s}$": "number of complex scalar fields", "$n_{f}$": "number of Dirac fermions", "$\\mu$": "renormalization scale", "$\\kappa$": "arbitrary scale in field theory", "$g_{\\mathrm{r}}$": "renormalized gauge coupling", "$g_{\\mathrm{b}}$": "bare gauge coupling", "$N$": "dimension of gauge group SU(N)"}
|
Renormalization of Gauge Theories
|
Renormalization of Gauge Theories
|
|
269
|
Others
|
Carry out explicitly the calculation of the functions $A$ and $B$ in
$$B(q^{2})=-i\mathrm{D}e\int\frac{d^{\mathrm{D}}l}{(2\pi)^{\mathrm{D}}}\int_{0}^{1}dx\frac{2\Delta(x)}{(l^{2}+\Delta(x))^{2}}.$$
|
[] |
Expression
|
{"$A$": "function A related to the calculation", "$B$": "function B related to the calculation", "$q$": "momentum variable", "$\\mathrm{D}$": "dimension of spacetime", "$\\Delta(x)$": "function dependent on variable x", "$e$": "mathematical constant in the expressions", "$\\ell$": "integration variable"}
|
Quantum Anomalies
|
Quantum Anomalies
|
|
270
|
Others
|
Carry out explicitly the calculation of the functions $A$ and $B$ in
$$A(q^{2})=-i\mathrm{D}e\int\frac{d^{\mathrm{D}}l}{(2\pi)^{D}}\int_{0}^{1}dx\frac{\Delta(x)+(\frac{2}{\mathrm{D}}-1)l^{2}}{(l^{2}+\Delta(x))^{2}}, $$
|
[] |
Expression
|
{"$A$": "function A", "$B$": "function B", "$q$": "momentum", "$e$": "mathematical constant e", "$\\mathrm{D}$": "dimension in dimensional regularization", "$x$": "integration variable"}
|
Quantum Anomalies
|
Quantum Anomalies
|
|
271
|
Others
|
For the time independent, $z^{3}$-dependent electrical field $E_{3}(z^{3}) \equiv \frac{E}{\cosh ^{2}(k z^{3})}$ with gauge potential $A^{4}=-i \frac{E}{k} \tanh (k z^{3})$ (other $A^i=0$), the equations of motion for the stationary solutions $z^3(u)$ and $z^4(u)$, (assuming $z^1, z^2$ are constant) are given in first-order form. What are these equations, expressed in terms of $v = \sqrt{(\dot{z}^3)^2+(\dot{z}^4)^2}$ and $\gamma \equiv mk/(eE)$? You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$z^3$": "coordinate in the third spatial dimension", "$z^4$": "coordinate in the fourth spatial dimension", "$E_3$": "electrical field component in the third dimension", "$E$": "electric field magnitude", "$k$": "wave number", "$A^4$": "gauge potential in the fourth dimension", "$A^i$": "gauge potential component", "$v$": "velocity magnitude", "$\\gamma$": "dimensionless parameter related to mass, wave number, charge, and electric field", "$m$": "mass", "$e$": "electric charge"}
|
Worldline Formalism
|
Worldline Formalism
|
|
272
|
Others
|
The Polyakov loop is defined as $\mathrm{L}(x) \equiv \mathrm{N}^{-1} \operatorname{tr} ( P \exp \int_{0}^{\beta} d \tau A^{0}(\tau, x) )$. Under a center transformation, where the gauge transformation $\Omega(\tau,x)$ obeys $\Omega(\beta, x)=\xi \Omega(0, x)$ with $\xi \in \mathbb{Z}_{N}$, how does $\mathrm{L}(x)$ transform? Express the transformed Polyakov loop $\mathrm{L}'(x)$ in terms of $\mathrm{L}(x)$ and $\xi$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mathrm{L}(x)$": "Polyakov loop at position x", "$\\mathrm{L}'(x)$": "transformed Polyakov loop at position x", "$\\xi$": "N-th root of unity from the group \\mathbb{Z}_{N}", "$N$": "number of colors or group rank in the context of SU(N)", "$A^{0}(\\tau, x)$": "temporal component of the gauge field at position x and time τ"}
|
Quantum Field Theory at Finite Temperature
|
Quantum Field Theory at Finite Temperature
|
|
273
|
Others
|
A particle with mass $m$, constrained to move freely on a ring with radius $R$, with an added perturbation
\begin{equation*}
H^{\prime}=V(\varphi)=\left\{\begin{array}{ll}
V_{1}, & -\alpha<\varphi<0 \\
V_{2}, & 0<\varphi<\alpha \\
0, & \text { other angles }
\end{array} \quad(\alpha<\pi)\right.
\end{equation*}
Find the first-order perturbation corrections for the three lowest energy levels. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$m$": "mass of the particle", "$R$": "radius of the ring", "$\\varphi$": "angle around the ring", "$V_1$": "potential value for the angle range $-\\alpha < \\varphi < 0$", "$V_2$": "potential value for the angle range $0 < \\varphi < \\alpha$", "$\\alpha$": "angle constraint less than $\\pi$", "$H_{nn}^{\\prime}$": "average value of perturbation for state $n$", "$a$": "average potential value factor, defined as $\\frac{\\alpha}{2 \\pi}(V_{1}+V_{2})$", "$E_0^{(0)}$": "ground state energy without perturbation", "$E_0^{(1)}$": "first-order perturbation correction to the ground state energy", "$H_{-1,1}^{\\prime}$": "perturbation matrix element for the first excited state", "$b_1$": "perturbation factor for the first excited state", "$E_1^{(1)}$": "first-order perturbation correction to the energy of the first excited state", "$H_{-2.2}^{\\prime}$": "perturbation matrix element for the second excited state", "$b_2$": "perturbation factor for the second excited state", "$E_2^{(1)}$": "first-order perturbation correction to the energy of the second excited state"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
274
|
Others
|
A small uniformly charged sphere acquires potential energy in an external electrostatic field
$$
\begin{equation*}
U(\boldsymbol{r})=V(\boldsymbol{r})+\frac{1}{6} r_{0}^{2} \nabla^{2} V(\boldsymbol{r})+\cdots
\end{equation*}
$$
where $r_{0}$ is the radius of the sphere, $\boldsymbol{r}$ is the position of the sphere's center, and $V(\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is
$$
\begin{equation*}
V(\boldsymbol{r})=-\frac{e^{2}}{r}
\end{equation*}
$$
If the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 1s energy levels [equivalent to the Lamb shift].
|
[] |
Numeric
|
{"$U(\\boldsymbol{r})$": "potential energy of the sphere at position $\\boldsymbol{r}$", "$V(\\boldsymbol{r})$": "electrostatic potential energy at position $\\boldsymbol{r}$", "$r_0$": "radius of the sphere", "$\\boldsymbol{r}$": "position of the sphere's center", "$e$": "elementary charge", "$m_e$": "electron mass", "$c$": "speed of light in vacuum", "$H^{\\prime}$": "perturbative correction term to the potential energy", "$E^{(1)}$": "first-order perturbative correction to the energy level", "$\\psi$": "wave function", "$\\psi_{2 p}$": "wave function for the 2p state", "$\\psi_{1 \\mathrm{~s}}$": "wave function for the 1s state", "$a_0$": "Bohr radius", "$\\hbar$": "reduced Planck's constant", "$E_{1_{s}}^{(1)}$": "first-order energy correction for the 1s state", "$\\alpha$": "fine structure constant", "$m_{\\mathrm{e}}$": "electron rest mass"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
275
|
Others
|
A small uniformly charged sphere acquires potential energy in an external electrostatic field
$$
\begin{equation*}
U(\boldsymbol{r})=V(\boldsymbol{r})+\frac{1}{6} r_{0}^{2} \nabla^{2} V(\boldsymbol{r})+\cdots
\end{equation*}
$$
where $r_{0}$ is the radius of the sphere, $\boldsymbol{r}$ is the position of the sphere's center, and $V(\boldsymbol{r})$ is the electrostatic potential energy acquired by the small charged sphere when approximated as a point charge. In a hydrogen atom, when the electron is treated as a point charge, the Coulomb potential energy between the electron and the nucleus is
$$
\begin{equation*}
V(\boldsymbol{r})=-\frac{e^{2}}{r}
\end{equation*}
$$
If the electron is treated as a charged ($-e$) sphere, and $r_{0}=e^{2} / m_{e} c^{2}$ (classical electron radius) is used, the potential energy is modified by equation (1), treating the $r_{0}^{2}$ term as a perturbation. Find the perturbative correction for the 2p energy levels [equivalent to the Lamb shift].
|
[] |
Numeric
|
{"$r_{0}$": "radius of the sphere (classical electron radius)", "$\\boldsymbol{r}$": "position of the sphere's center", "$V(\\boldsymbol{r})$": "electrostatic potential energy", "$e$": "elementary charge", "$m_{e}$": "electron mass", "$c$": "speed of light", "$H^{\\prime}$": "perturbation term in potential energy", "$E^{(1)}$": "first-order perturbative correction to the energy level", "$\\psi$": "wave function", "$\\psi_{2 p}(0)$": "wave function of the 2p state at the origin", "$\\psi_{1 \\mathrm{~s}}(0)$": "wave function of the 1s state at the origin", "$a_{0}$": "Bohr radius", "$\\alpha$": "fine structure constant", "$m_{\\mathrm{e}}$": "mass of the electron"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
276
|
Others
|
Take the ground state wave function as
$$
\psi_{0}(\boldsymbol{r}_{1}, \boldsymbol{r}_{2})=\psi_{0}(r_{1}) \psi_{0}(r_{2})
$$
where
$$
\begin{equation*}
\psi_{0}(r)=(\frac{\lambda^{3}}{\pi a_{0}^{3}})^{\frac{1}{2}} \mathrm{e}^{-x^{\prime} / a_{0}}, \quad a_{0}=\frac{\hbar^{2}}{m_{\mathrm{e}} e^{2}}.
\end{equation*}
$$
Calculate the ground state magnetic susceptibility of a helium atom, in the unit of $\mathrm{eV} /(\mathrm{Gs})^{2}$
|
[] |
Numeric
|
{"$\\psi_{0}$": "ground state wave function", "$\\boldsymbol{r}_{1}$": "position vector of electron 1", "$\\boldsymbol{r}_{2}$": "position vector of electron 2", "$r_{1}$": "radial coordinate of electron 1", "$r_{2}$": "radial coordinate of electron 2", "$\\lambda$": "variational parameter", "$a_{0}$": "Bohr radius", "$\\hbar$": "reduced Planck's constant", "$m_{\\mathrm{e}}$": "electron mass", "$e$": "elementary charge", "$x^{\\prime}$": "scaled position variable", "$c$": "speed of light", "$x_{1}$": "x-coordinate of electron 1", "$y_{1}$": "y-coordinate of electron 1", "$x_{2}$": "x-coordinate of electron 2", "$y_{2}$": "y-coordinate of electron 2", "$\\boldsymbol{B}$": "magnetic field vector", "$E^{(1)}$": "first-order energy correction", "$\\alpha_{\\beta}$": "magnetic susceptibility", "$\\mu_{\\mathrm{B}}$": "Bohr magneton", "$Z$": "atomic number for helium"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
277
|
Others
|
A hydrogen atom is situated within a certain ionic lattice, where the potential exerted by the surrounding ions on the electron in the hydrogen atom can be approximately represented as
\begin{equation*}
H^{\prime}=V_{0}(x^{4}+y^{4}+z^{4}-\frac{3}{5} r^{4})
\end{equation*}
$H^{\prime}$ can be considered a perturbation. If the 3d state wave functions of the hydrogen atom (orthonormalized) are taken as
\begin{align*}
& \psi_{1}=\frac{1}{2}(y^{2}-z^{2}) f(r) \\
& \psi_{2}=\frac{1}{2 \sqrt{3}}(2 x^{2}-y^{2}-z^{2}) f(r) \\
& \psi_{3}=y z f(r) \\
& \psi_{4}=z x f(r) \\
& \psi_{5}=x y f(r)
\end{align*}
Under the influence of $H^{\prime}$, what is the degeneracy of the first energy level after the 3d energy level splits (corresponding to $\psi_1$ and $\psi_2$)?
|
[] |
Numeric
|
{"$H^{\\prime}$": "perturbation potential", "$V_{0}$": "constant potential coefficient", "$x$": "Cartesian coordinate x", "$y$": "Cartesian coordinate y", "$z$": "Cartesian coordinate z", "$r$": "radial distance from the origin", "$\\psi_{1}$": "first wave function component", "$\\psi_{2}$": "second wave function component", "$\\psi_{3}$": "third wave function component", "$\\psi_{4}$": "fourth wave function component", "$\\psi_{5}$": "fifth wave function component", "$E_{3 d}^{(0)}$": "unperturbed 3d energy level", "$H_{11}^{\\prime}$": "perturbation matrix element (1,1)", "$H_{33}^{\\prime}$": "perturbation matrix element (3,3)"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
278
|
Others
|
A certain molecule is composed of three identical atoms ($\alpha, \beta, \gamma$), with the three atoms located at the vertices of an equilateral triangle. There is one valence electron that can move among the three atoms. Denote the unperturbed Hamiltonian of this valence electron as $H_0$, and the atomic orbitals of the electron as $|\alpha\rangle, |\beta\rangle, |\gamma\rangle$ (which are mutually orthogonal and normalized). Assume the electron's atomic energy level is $\langle k|H_0|k\rangle = E_0$ (for $k=\alpha,\beta,\gamma$). Between any two different atoms, the matrix element of $H_0$ is $\langle j|H_0|k\rangle = -a$ (where $j \neq k, a>0$). Request to solve the molecular energy levels of the unperturbed Hamiltonian $H_0$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\alpha$": "atom alpha", "$\\beta$": "atom beta", "$\\gamma$": "atom gamma", "$H_0$": "unperturbed Hamiltonian of the valence electron", "$|\\alpha\\rangle$": "atomic orbital associated with atom alpha", "$|\\beta\\rangle$": "atomic orbital associated with atom beta", "$|\\gamma\\rangle$": "atomic orbital associated with atom gamma", "$E_0$": "atomic energy level of the electron", "$a$": "matrix element of the unperturbed Hamiltonian between different atoms"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
279
|
Others
|
For the unperturbed system mentioned above (i.e., without an electric field), solve for the normalized ground state wave function $|\psi_{GS,unperturbed}\rangle$ corresponding to the lowest energy $E_0 - 2a$, and express it as a linear combination of $|\alpha\rangle, |\beta\rangle, |\gamma\rangle$.
|
[] |
Expression
|
{"$|\\psi_{GS,unperturbed}\\rangle$": "normalized ground state wave function without an electric field", "$E_0$": "initial lowest energy", "$a$": "energy parameter", "$|\\alpha\\rangle$": "basis state alpha", "$|\\beta\\rangle$": "basis state beta", "$|\\gamma\\rangle$": "basis state gamma", "$C_{\\alpha}$": "coefficient for basis state alpha in linear combination", "$C_{\\beta}$": "coefficient for basis state beta in linear combination", "$C_{\\gamma}$": "coefficient for basis state gamma in linear combination"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
280
|
Others
|
Apply a uniform weak electric field as a perturbation. Due to this electric field, the on-site energy level at atom $\alpha$ decreases by $b$, becoming $E_0-b$, while the energy levels at atoms $\beta$ and $\gamma$ remain $E_0$. Assume $b \ll a$. The hopping integral between atoms ($-a$) is not affected by the electric field. The perturbation matrix elements between different atomic orbitals are zero (i.e., $\langle j|H'|k\rangle = 0$ when $j \neq k$, and $\langle\beta|H'|\beta\rangle = \langle\gamma|H'|\gamma\rangle = 0$, $\langle\alpha|H'|\alpha\rangle = -b$). Solve for the new molecular energy levels, with results approximated to first order in $b/a$. You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$E_0$": "initial energy level", "$b$": "decrease in energy due to the electric field at atom $\\alpha$", "$a$": "hopping integral between atoms", "$\\alpha$": "first atom in the system", "$\\beta$": "second atom in the system", "$\\gamma$": "third atom in the system"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
281
|
Others
|
Initially, the electrons are in the ground state $|\psi_{GS,perturbed}\rangle$ (corresponding to the situation where an electric field is perturbing atom $\alpha$). If the field suddenly rotates so that the perturbation now acts on atom $\beta$ (with the system's new ground state being $|\psi'_{GS,perturbed}\rangle$), what is the probability that the electrons are found in this new ground state $|\psi'_{GS,perturbed}\rangle$? Approximate the result to zero order of $(b/a)$, meaning terms containing $b$ should disregard terms of order $(b/a)$ or higher.
|
[] |
Numeric
|
{"$|\\psi_{GS,perturbed}\\rangle$": "initial ground state of electrons under perturbation", "$|\\psi'_{GS,perturbed}\\rangle$": "new ground state of electrons after perturbation rotation", "$\\alpha$": "atom originally affected by the electric field", "$\\beta$": "atom affected by the electric field after rotation", "$C_{\\alpha}$": "coefficient for atom $\\alpha$ in the initial state", "$C_{\\beta}$": "coefficient for atom $\\beta$ in the initial state", "$C_{\\gamma}$": "coefficient for atom $\\gamma$ in the initial state", "$C'_{\\alpha}$": "new coefficient for atom $\\alpha$ after rotation", "$C'_{\\beta}$": "new coefficient for atom $\\beta$ after rotation", "$C'_{\\gamma}$": "new coefficient for atom $\\gamma$ after rotation", "$b$": "perturbation parameter to consider in approximation", "$a$": "characteristic parameter of the system"}
|
Steady-state perturbation theory
|
Steady-state perturbation theory
|
|
282
|
Theoretical Foundations
|
Which powers among the operators $\hat{s}$ (with any spin value $s$) are linearly independent? You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$\\hat{s}$": "spin operator", "$s$": "spin value", "$\\hat{s}_{z}$": "z-component of the spin operator"}
|
Spin
|
Spin Operator
|
|
283
|
Theoretical Foundations
|
The equation is
\begin{align}\label{eq:57.4}
(\hat s_x \psi)^1 &= \tfrac12\,\psi^2,
&(\hat s_y \psi)^1 &= -\tfrac12\,i\,\psi^2,
&(\hat s_z \psi)^1 &= \tfrac12\,\psi^1, \\
(\hat s_x \psi)^2 &= \tfrac12\,\psi^1,
&(\hat s_y \psi)^2 &= \tfrac12\,i\,\psi^1,
&(\hat s_z \psi)^2 &= -\tfrac12\,\psi^2.
\end{align} Rewrite equation in the context, expressing the operators of spin $1 / 2$ in terms of the spinor components of the vector $\hat{\boldsymbol{S}}$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\hat{\\boldsymbol{S}}$": "vector of spin", "$\\hat{s}^{\\lambda \\mu}$": "spinor component of the vector", "$\\psi^{\\nu}$": "spinor component (nu)", "$\\psi^{\\lambda}$": "spinor component (lambda)", "$g^{\\mu \\nu}$": "metric tensor component (mu nu)", "$g^{\\lambda \\nu}$": "metric tensor component (lambda nu)"}
|
Spin
|
Wave functions for particles with arbitrary spin
|
|
284
|
Theoretical Foundations
|
Determine all possible states of a three-nucleon system, where each nucleon has an angular momentum $j=3 / 2$ (with the same principal quantum number). You should return your answer as a tuple format.
|
[] |
Tuple
|
{"$j$": "angular momentum of a nucleon", "$m_{j}$": "magnetic quantum number of the nucleon's angular momentum", "$\\tau_{\\zeta}$": "isospin projection quantum number of the nucleon", "$M_{J}$": "total magnetic quantum number of the three-nucleon system", "$T_{\\zeta}$": "total isospin projection quantum number of the three-nucleon system", "$J$": "total angular momentum of the three-nucleon system", "$T$": "total isospin quantum number of the three-nucleon system"}
|
Nuclear structure
|
Shell Model
|
|
285
|
Others
|
Obtain the density matrix $\rho(q, q^{\prime})=\langle q| e^{-\beta \hat{H}}|q^{\prime}\rangle$ for the harmonic oscillator at finite temperature, $\beta=1 / T(k_{\mathrm{B}}=1)$.
|
[] |
Expression
|
{"$\\rho$": "density matrix", "$q$": "position coordinate", "$q^{\\prime}$": "position coordinate prime", "$\\beta$": "inverse temperature", "$\\hat{H}$": "Hamiltonian operator", "$m$": "mass", "$\\omega$": "angular frequency", "$\\hbar$": "reduced Planck's constant", "$T$": "temperature", "$E_{0}$": "ground state energy"}
|
Path Integral
|
Density matrix
|
|
286
|
Theoretical Foundations
|
A particle is moving in an infinite potential well $(-a<x<a)$. As an approximation for the ground state wave function, a trial wave function with three terms is used: $\psi(\lambda, x)=N[1+\lambda(\frac{x}{a})^{2}-(1+\lambda)(\frac{x}{a})^{4}] (|x|<a)$, where $N$ is the normalization constant, and $\lambda$ is the variational parameter. Use the variational method to find an approximate value for the ground state energy.
|
[] |
Expression
|
{"$a$": "half-width of the potential well", "$x$": "position", "$N$": "normalization constant", "$\\lambda$": "variational parameter", "$E(\\lambda)$": "average value of the Hamiltonian", "$\\hbar$": "reduced Planck's constant", "$m$": "mass of the particle", "$E_1$": "exact ground state energy", "$\\lambda_{1}$": "variational parameter corresponding to the ground state"}
|
Variational method
| ||
287
|
Theoretical Foundations
|
A particle of mass $m$ moves in a linear central force field
\begin{equation*}
V(r)=k r, \quad k>0 \tag{1}
\end{equation*} For the particle in the potential $V(r)=kr$ (s-wave), its exact ground state energy is $E_0 = C (\frac{\hbar^{2} k^{2}}{m})^{1 / 3}$. What is the value of the constant $C$ (accurate to four decimal places)?
The trial wave function is:
\begin{enumerate}
\item $\psi \sim e^{-\lambda r}$.
\end{enumerate}
|
[] |
Numeric
|
{"$V(r)$": "potential energy as a function of distance $r$", "$k$": "proportional constant in potential energy", "$E_0$": "exact ground state energy", "$C$": "constant to determine", "$\\hbar$": "reduced Planck's constant", "$m$": "particle mass", "$\\lambda$": "variational parameter", "$\\psi$": "trial wave function", "$T$": "kinetic energy", "$V$": "potential energy", "$r$": "distance from the origin", "$E(\\lambda)$": "energy as a function of $\\lambda$"}
|
Variational method
| ||
288
|
Theoretical Foundations
|
Assume in the deuteron, the potential between the proton and neutron is expressed as
\begin{equation*}
V(r)=-V_{0} \mathrm{e}^{-r / a} \tag{1}
\end{equation*}
Take $V_{0}=32.7 \mathrm{MeV}, a=2.16 \mathrm{fm}$ (range of force). Use the variational method to find the ground state energy level of the deuteron.
The trial function is chosen as
\begin{equation*}
\psi(\lambda, r)=N \mathrm{e}^{-\lambda r / 2 a}
\end{equation*}
|
[] |
Numeric
|
{"$V_{0}$": "potential energy constant", "$a$": "range of force", "$\\lambda$": "variational parameter", "$N$": "normalization constant", "$\\hbar$": "reduced Planck's constant", "$\\mu$": "reduced mass of the proton-neutron system", "$m_{\\mathrm{p}}$": "mass of the proton", "$m_{\\mathrm{n}}$": "mass of the neutron"}
|
Variational method
| ||
289
|
Magnetism
|
Two conductors with capacitances $C_{1}$ and $C_{2}$ respectively are separated by a distance $r$, where $r$ is greater than the dimensions of the conductors themselves. Determine the coefficient $C_{12}$.
|
[] |
Expression
|
{"$C_{1}$": "capacitance of conductor 1", "$C_{2}$": "capacitance of conductor 2", "$r$": "distance between the conductors", "$C_{12}$": "coupling coefficient between the two conductors", "$e_{1}$": "charge on conductor 1", "$\\varphi_{1}$": "potential of conductor 1", "$\\varphi_{2}$": "potential of conductor 2", "$C_{11}$": "capacitance coefficient for conductor 1", "$C_{22}$": "capacitance coefficient for conductor 2"}
|
Electrostatics of Conductors
|
Electrostatic field energy of conductor
|
|
290
|
Magnetism
|
Given two conductors with capacitances $C_{1}$ and $C_{2}$ separated by a distance $r$, where $r$ is much larger than the dimensions of the conductors. Try to determine the coefficient $C_{22}$. You should return your answer as an equation.
|
[] |
Equation
|
{"$C_{1}$": "capacitance of conductor 1", "$C_{2}$": "capacitance of conductor 2", "$r$": "distance between the conductors", "$C_{22}$": "self-capacitance coefficient of conductor 2", "$e_{1}$": "charge on conductor 1", "$\\varphi_{1}$": "potential at conductor 1", "$\\varphi_{2}$": "potential at conductor 2", "$C_{11}$": "self-capacitance coefficient of conductor 1", "$C_{12}$": "mutual capacitance coefficient between conductor 1 and conductor 2"}
|
Electrostatics of Conductors
|
Electrostatic field energy of conductor
|
|
291
|
Magnetism
|
Consider a conductor with a sharp conical tip on its surface.
Using spherical coordinates, place the origin at the vertex of the conical tip, with the cone axis as the polar axis.
Let the cone's opening angle be $2 \theta_{0} \ll 1$, and the polar angle range corresponding to the external region of the conductor is $\theta_{0} \leqslant \theta \leqslant \pi$. Assume that the potential $\varphi$ has the form $\varphi(r, \theta) = r^{n} f(\theta)$.
Based on this setup, and using the boundary condition that the potential on the conductor's surface ($\theta=\theta_0$) is constant (i.e., $f(\theta_0)=0$ for the angular dependence part of the potential), and that for small $\theta_0$ and small $n \ll 1$, the function $f(\theta)$ can be approximately expressed as $f(\theta) = \mathrm{const} \cdot (1+2 n \ln \sin (\theta/2) )$, derive the expression for the exponent $n$.
|
[] |
Expression
|
{"$\\theta_{0}$": "opening angle component of the cone", "$\\theta$": "polar angle", "$\\varphi$": "potential", "$r$": "radial coordinate", "$n$": "exponent in the potential function", "$f(\\theta)$": "angular dependence part of the potential", "$\\psi(\\theta)$": "correction term in the angular dependence"}
|
Electrostatics of Conductors
|
Solutions to electrostatic problems
|
|
292
|
Magnetism
|
Given that the conductor boundary is an infinite plane with a hemispherical protrusion whose radius is $R$, determine the charge distribution on the surface of the conductor at the hemispherical protrusion.
Hint: the potential is
$$
\varphi=-4\pi \sigma_0 \cdot z(1-\frac{R^{3}}{r^{3}})
$$
|
[] |
Expression
|
{"$R$": "radius of the hemispherical protrusion", "$\\varphi$": "electric potential", "$\\sigma_0$": "charge density far from the protruding part", "$r$": "radial distance", "$z$": "vertical coordinate"}
|
Electrostatics of Conductors
|
Solutions to electrostatic problems
|
|
293
|
Magnetism
|
Find the charge distribution on a non-charged conductor disk (with radius $a$) that is parallel to a uniform external electric field ${ }^{(1)}$.
|
[] |
Expression
|
{"$a$": "radius of the conductor disk", "$c$": "short semi-axis of the rotational ellipsoid", "$n^{(z)}$": "depolarization factor along the z-axis", "$n^{(x)}$": "depolarization factor along the x-axis", "$n^{(y)}$": "depolarization factor along the y-axis", "$\\nu_{x}$": "component of the unit vector in the direction of the normal to the rotational ellipsoid surface", "$x$": "coordinate along the x-axis", "$y$": "coordinate along the y-axis", "$z$": "coordinate along the z-axis", "$\\sigma$": "charge density", "$\\mathfrak{C}$": "constant related to the charge density", "$\\rho$": "radial coordinate in polar coordinates", "$\\varphi$": "angular coordinate (angle) in polar coordinates", "$p$": "radial position in the plane of the disk"}
|
Electrostatics of Conductors
|
Conductive ellipsoid
|
|
294
|
Magnetism
|
For a conducting sphere placed in a uniform external electric field $\mathfrak{C}$, determine its relative volume change $\frac{\Delta V}{V}$, the bulk modulus of the material is $K$.
|
[] |
Expression
|
{"$\\mathfrak{C}$": "uniform external electric field", "$\\Delta V$": "change in volume", "$V$": "original volume", "$K$": "bulk modulus", "$\\Delta P$": "change in pressure", "$\\alpha$": "polarizability", "$\\delta_{ik}$": "Kronecker delta"}
|
Electrostatics of Conductors
|
force on a conductor
|
|
295
|
Magnetism
|
For a conducting sphere placed in a uniform external electric field $\mathfrak{C}$, determine its shape deformation. Specifically, find the expression for the quantity $\frac{a-b}{R}$ that describes its deformation, where $R$ is the original radius of the sphere, and $a$ and $b$ are the semi-axes of the ellipsoid along and perpendicular to the field direction, respectively. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mathfrak{C}$": "external electric field", "$R$": "original radius of the sphere", "$a$": "semi-axis of the ellipsoid along the field direction", "$b$": "semi-axis of the ellipsoid perpendicular to the field direction", "$V$": "volume of the conductor", "$n$": "depolarization factor term", "$u_{x x}$": "strain tensor component in the x-direction", "$u_{y y}$": "strain tensor component in the y-direction", "$u_{z z}$": "strain tensor component in the z-direction", "$u_{i i}$": "trace of the strain tensor", "$\\sigma_{i k}$": "elastic stress tensor", "$\\sigma_{x x}$": "elastic stress tensor component in the x-direction", "$\\sigma_{y y}$": "elastic stress tensor component in the y-direction", "$\\mu$": "shear modulus of the material"}
|
Electrostatics of Conductors
|
force on a conductor
|
|
296
|
Magnetism
|
Try to determine the volume change of a dielectric ellipsoid in a uniform electric field, assuming the direction of the electric field is parallel to one of the ellipsoid's axes. Specifically, determine $\frac{V-V_0}{V}$.
|
[] |
Expression
|
{"$V$": "volume of the dielectric ellipsoid", "$V_0$": "initial volume of the dielectric ellipsoid", "$\\varepsilon$": "dielectric permittivity", "$n$": "form factor of the ellipsoid, related to its shape", "$\\mathfrak{E}$": "magnitude of the electric field", "$K$": "compressibility coefficient", "$P$": "pressure", "$T$": "temperature"}
|
Electrostatics of Dielectrics
|
Electrostriction of Isotropic Dielectrics
|
|
297
|
Magnetism
|
Determine the electrothermal effect of a dielectric ellipsoid in a uniform electric field, assuming the direction of the field is parallel to one of the axes of the ellipsoid.
\footnotetext{
(1) If the object is thermally insulated, the application of an electric field will cause a temperature change of $\Delta T=-Q / \mathscr{C}_{P}$, where $\mathscr{C}_{P}$ is the constant pressure heat capacity of the object.
}
|
[] |
Expression
|
{"$Q$": "electrothermal effect", "$T$": "temperature", "$V$": "volume", "$\\mathfrak{C}$": "electric field strength constant", "$\\alpha$": "coefficient of thermal expansion", "$\\varepsilon$": "dielectric constant", "$n$": "depicts orientation related to field or geometry", "$\\mathscr{C}_{P}$": "constant pressure heat capacity"}
|
Electrostatics of Dielectrics
|
Electrostriction of Isotropic Dielectrics
|
|
298
|
Magnetism
|
Assume the parallel plane plates are perpendicular to the electric field. Try to determine the difference between the heat capacity $\mathscr{C}_{\varphi}$ when the potential difference between the plates remains constant and the heat capacity $\mathscr{C}_{D}$ when the electric displacement remains constant, while the external pressure is also maintained constant in both situations.
|
[] |
Expression
|
{"$\\mathscr{C}_{\\varphi}$": "heat capacity when the potential difference remains constant", "$\\mathscr{C}_{D}$": "heat capacity when the electric displacement remains constant", "$D$": "electric displacement", "$\\mathbb{C}$": "external electric field", "$\\varphi$": "potential difference between plates", "$E$": "electric field", "$l$": "thickness of the plates", "$V$": "volume of the plates", "$\\varepsilon$": "dielectric constant", "$\\alpha$": "thermal expansion coefficient", "$T$": "temperature", "$P$": "pressure", "$\\mathscr{S}$": "entropy of the plates", "$\\mathscr{S}_{0}$": "reference entropy of the plates", "$\\mathfrak{C}$": "constant related to the electric displacement"}
|
Electrostatics of Dielectrics
|
Electrostriction of Isotropic Dielectrics
|
|
299
|
Magnetism
|
Under the same conditions as the previous sub-question (the total volume of the parallel plane panel remains constant, $\mathscr{C}_{\varphi} \equiv \mathscr{C}_{E}$), consider representing the difference $\mathscr{C}_{\varphi}-\mathscr{C}_{D}$ (i.e., $\mathscr{C}_{E}-\mathscr{C}_{D}$) using the external field $\mathfrak{C}$.
Determine the corresponding mathematical expression for this difference.
|
[] |
Expression
|
{"$\\mathscr{C}_{\\varphi}$": "specific capacitance under condition \\(\\varphi\\)", "$\\mathscr{C}_{E}$": "specific capacitance under electric field", "$\\mathscr{C}_{D}$": "initial specific capacitance (reference)", "$\\mathfrak{C}$": "external field", "$E$": "electric field intensity", "$T$": "temperature", "$V$": "volume", "$\\varepsilon$": "permittivity", "$\\mathbb{C}$": "product of permittivity and electric field", "$\\rho$": "constant density or charge density"}
|
Electrostatics of Dielectrics
|
Electrostriction of Isotropic Dielectrics
|
|
300
|
Magnetism
|
In an infinite anisotropic medium, there is a spherical cavity, and the uniform electric field far from the cavity within the medium is known to be $E^{(e)}$. Find the $x$ component of the electric field inside the cavity $E_x^{(i)}$, expressed in terms of $E_x^{(e)}$ and the medium and geometric parameters.
|
[] |
Expression
|
{"$E^{(e)}$": "uniform electric field far from the cavity within the medium", "$E_x^{(i)}$": "x component of the electric field inside the cavity", "$E_x^{(e)}$": "x component of the electric field far from the cavity", "$\\varepsilon^{(x)}$": "dielectric constant in the x direction", "$\\varepsilon^{(y)}$": "dielectric constant in the y direction", "$\\varepsilon^{(z)}$": "dielectric constant in the z direction", "$a$": "radius of the sphere", "$n^{(x)}$": "depolarization coefficient of the ellipsoid in the x direction", "$n^{(y)}$": "depolarization coefficient of the ellipsoid in the y direction", "$n^{(z)}$": "depolarization coefficient of the ellipsoid in the z direction", "$\\varphi^{(i)}$": "potential inside the cavity", "$\\varphi^{(e)}$": "potential outside the cavity"}
|
Electrostatics of Dielectrics
|
Dielectric properties of crystals
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.