Q.No:1 CSIR Dec-2014
An ideal Bose gas is confined inside a container that is connected to a particle reservoir. Each particle can occupy a discrete set of single-particle quantum states. If the probability that a particular quantum state is unoccupied is \(0.1\), then the average number of bosons in that state is
(1)
\(8\)
(2)
\(9\)
(3)
\(10\)
(4)
\(11\)
Check Answer
Option 2
Q.No:2 CSIR June-2015
An ideal Bose gas in d-dimensions obeys the dispersion relation \(\in (\overrightarrow{k}) = Ak^{s}\) where \(A\) and \(s\) are constants. For Bose-Einstein condensation to occur, the occupancy of excited states
\(N_{e}=c \int_{0}^{\infty} \frac{\epsilon^{(d-s) / s}}{e^{\beta(\epsilon-\mu)}-1} d \epsilon\)
Where \(c\) is a constant, should remain finite even for \(\mu = 0\). This can happen if
(1)
\(\frac{d}{s} < \frac{1}{4}\)
(2)
\(\frac{1}{4} < \frac{d}{s}<\frac{1}{2}\)
(3)
\(\frac{d}{s} > 1\)
(4)
\(\frac{1}{2} < \frac{d}{s}<1\)
Check Answer
Option 3
Q.No:3 CSIR June-2015
The low-energy electronic excitations in a two-dimensional sheet of graphene is given by \(E(\vec{k})=\hbar v k\) where \(v\) is the velocity of the excitations. The density of states is proportional to
(1)
\(E\)
(2)
\(E^{3/2}\)
(3)
\(E^{1/2}\)
(4)
\(E^2\)
Check Answer
Option 1
Q.No:4 CSIR Dec-2015
A thin metal film of dimension \(2 \text{ mm}\times 2 \text{ mm}\) contains \(4\times 10^{12}\) electrons. The magnitude of the Fermi wavevector of the system, in the free electron approximation, is
(1)
\(2\sqrt{\pi}\times 10^7 \text{ cm}^{-1}\)
(2)
\(\sqrt{2\pi}\times 10^7 \text{ cm}^{-1}\)
(3)
\(\sqrt{\pi}\times 10^7 \text{ cm}^{-1}\)
(4)
\(2\pi\times 10^7 \text{ cm}^{-1}\)
Check Answer
Option 2
Q.No:5 CSIR June-2016
Suppose the frequency of phonons in a one-dimensional chain of atoms is proportional to the wavevector. If \(n\) is the number density of atoms and \(c\) is the speed of the phonons, then the Debye frequency is
(1)
\(2\pi cn\)
(2)
\(\sqrt{2}\pi cn\)
(3)
\(\sqrt{3}\pi cn\)
(4)
\(\pi cn/2\)
Check Answer
Option 1
Q.No:6 CSIR June-2016
Consider electrons in graphene, which is a planar monatomic layer of carbon atoms. If the dispersion relation of the electrons is taken to be \(\varepsilon(k)=ck\) (where \(c\) is constant) over the entire \(k\)-space, then the Fermi energy \(\varepsilon_F\) depends on the number density of electrons \(\rho\) as
(1)
\(\varepsilon_F \propto \rho^{1/2}\)
(2)
\(\varepsilon_F \propto \rho\)
(3)
\(\varepsilon_F \propto \rho^{2/3}\)
(4)
\(\varepsilon_F \propto \rho^{1/3}\)
Check Answer
Option 1
Q.No:7 CSIR Dec-2016
The electrons in graphene can be thought of as a two-dimensional gas with a linear energy-momentum relation \(E=|\vec{p}|v\), where \(\vec{p}=(p_x, p_y)\) and \(v\) is a constant. If \(\rho\) is the number of electrons per unit area, the energy per unit area is proportional to
(1)
\(\rho^{3/2}\)
(2)
\(\rho\)
(3)
\(\rho^{1/3}\)
(4)
\(\rho^{2}\)
Check Answer
Option 1
Q.No:8 CSIR June-2017
A gas of photons inside a cavity of volume \(V\) is in equilibrium at temperature \(T\). If the temperature of the cavity is changed to \(2T\), the radiation pressure will change by a factor of
(1)
\(2\)
(2)
\(16\)
(3)
\(8\)
(4)
\(4\)
Check Answer
Option 2
Q.No:9 CSIR June-2017
The single particle energy levels of a non-interacting three-dimensional isotropic system, labelled by momentum \(k\), are proportional to \(k^3\). The ratio \(\bar{P}/\epsilon\) of the average pressure \(\bar{P}\) to the energy density \(\epsilon\) at a fixed temperature, is
(1)
\(1/3\)
(2)
\(2/3\)
(3)
\(1\)
(4)
\(3\)
Check Answer
Option 3
Q.No:10 CSIR Dec-2017
The dispersion relation of a gas of spin-\(\frac{1}{2}\) fermions in two dimensions is \(E=\hbar v|\vec{k}|\), where \(E\) is the energy, \(\vec{k}\) is the wave vector and \(v\) is a constant with the dimension of velocity. If the Fermi energy at zero temperature is \(\epsilon_F\), the number of particles per unit area is
(1)
\(\epsilon_F/(4\pi v\hbar)\)
(2)
\(\epsilon_F^3/(6\pi^2 v^3 \hbar^2)\)
(3)
\(\pi\epsilon_F^{3/2}/(3v^3 \hbar^3)\)
(4)
\(\epsilon_F^2/(2\pi v^2 \hbar^2)\)
Check Answer
Option 4
Q.No:11 CSIR Dec-2017
Consider a quantum system of non-interacting bosons in contact with a particle bath. The probability of finding no particle in a given single particle quantum state is \(10^{-6}\). The average number of particles in that state is of the order of
(1)
\(10^3\)
(2)
\(10^6\)
(3)
\(10^9\)
(4)
\(10^{12}\)
Check Answer
Option 2
Q.No:12 CSIR Dec-2017
A metallic nanowire of length \(l\) is approximated as a one-dimensional lattice of \(N\) atoms with lattice spacing \(a\). If the dispersion of electrons in the lattice is given as \(E(k)=E_0-2t\cos{ka}\), where \(E_0\) and \(t\) are constants, then the density of states inside the nanowire depends on \(E\) as
(1)
\(N^3 \sqrt{\frac{t^2}{E-E_0}}\)
(2)
\(\sqrt{\left(\frac{E-E_0}{2t}\right)^2-1}\)
(3)
\(N^3 \sqrt{\frac{E-E_0}{t^2}}\)
(4)
\(\frac{N}{\sqrt{(2t)^2-(E-E_0)^2}}\)
Check Answer
Option 4
Q.No:13 CSIR June-2018
The number of ways of distributing \(11\) indistinguishable bosons in \(3\) different energy levels is
(1)
\(3^{11}\)
(2)
\(11^3\)
(3)
\(\frac{(13)!}{2!(11)!}\)
(4)
\(\frac{(11)!}{3!8!}\)
Check Answer
Option 3
Q.No:14 CSIR Dec-2018
The heat capacity \(C_V\) at constant volume of a metal, as a function of temperature, is \(\alpha T+\beta T^3\), where \(\alpha\) and \(\beta\) are constants. The temperature dependence of the entropy at constant volume is
(1)
\(\alpha T+\frac{1}{3}\beta T^3\)
(2)
\(\alpha T+\beta T^3\)
(3)
\(\frac{1}{2}\alpha T+\frac{1}{3}\beta T^3\)
(4)
\(\frac{1}{2}\alpha T+\frac{1}{4}\beta T^3\)
Check Answer
Option 1
Q.No:15 CSIR Dec-2018
Consider an ideal Fermi gas in a grand canonical ensemble at a constant chemical potential. The variance of the occupation number of the single particle energy level with mean occupation number \(\bar{n}\) is
(1)
\(\bar{n}(1-\bar{n})\)
(2)
\(\sqrt{\bar{n}}\)
(3)
\(\bar{n}\)
(4)
\(\frac{1}{\sqrt{\bar{n}}}\)
Check Answer
Option 1
Q.No:16 CSIR Dec-2018
At low temperatures, in the Debye approximation, the contribution of the phonons to the heat capacity of a two-dimensional solid is proportional to
(1)
\(T^2\)
(2)
\(T^3\)
(3)
\(T^{1/2}\)
(4)
\(T^{3/2}\)
Check Answer
Option 1
Q.No:17 CSIR Dec-2018
The dispersion relation of optical phonons in a cubic crystal is given by \(\omega(k)=\omega_0-ak^2\) where \(\omega_0\) and \(a\) are positive constants. The contribution to the density of states due to these phonons with frequencies just below \(\omega_0\) is proportional to
(1)
\((\omega_0-\omega)^{1/2}\)
(2)
\((\omega_0-\omega)^{3/2}\)
(3)
\((\omega_0-\omega)^{2}\)
(4)
\((\omega_0-\omega)\)
Check Answer
Option 1
Q.No:18 CSIR Dec-2019
Consider black body radiation in thermal equilibrium contained in a two-dimensional box. The dependence of the energy density on the temperature \(T\) is
(1)
\(T^3\)
(2)
\(T\)
(3)
\(T^2\)
(4)
\(T^4\)
Check Answer
Option 1
Q.No:19 CSIR Dec-2019
For \(T\) much less than the Debye temperature of copper, the temperature dependence of the specific heat at constant volume of copper, is given by (in the following \(a\) and \(b\) are positive constants)
(1)
\(aT^3\)
(2)
\(aT+bT^3\)
(3)
\(aT^2+bT^3\)
(4)
\(\exp{\left(-\frac{a}{k_B T}\right)}\)
Check Answer
Option 2
Q.No:20 Assam CSIR Dec-2019
Two spheres are subjected to black-body radiation at temperatures \(T_1\) and \(T_2\) respectively. For the first sphere only half of its surface is exposed to the radiation, while the full surface of the second sphere is exposed. The ratio of the radii of the spheres are \(R_1/R_2=2\). If the forces due to radiation pressure on the two spheres are the same, the ratio \(T_1/T_2\) is
(1)
\(2^{1/4}\)
(2)
\(1/\sqrt{2}\)
(3)
\(1\)
(4)
\(1/2^{1/4}\)
Check Answer
Option 4
Q.No:21 CSIR June-2020
The temperatures of two perfect black bodies \(A\) and \(B\) are \(400 K\) and \(200 K\), respectively. If the surface area of \(A\) is twice that of \(B\), the ratio of total power emitted by \(A\) to that by \(B\) is
(a)
\(4\)
(b)
\(2\)
(c)
\(32\)
(d)
\(16\)
Check Answer
Option c
Q.No:22 CSIR June-2020
Spin \(\frac{1}{2}\) fermions of mass \(m\) and \(4m\) are in a harmonic potential \(V(x)=\frac{1}{2}kx^2\). Which configuration of \(4\) such particles has the lowest value of the ground state energy?
(a)
\(4\) particles of mass \(m\)
(b)
\(4\) particles of mass \(4m\)
(c)
\(1\) particle of mass \(m\) and \(3\) particles of mass \(4m\)
(d)
\(2\) particles of mass \(m\) and \(2\) particles of mass \(4m\)
Check Answer
Option d
Q.No:23 CSIR Feb-2022
The volume and temperature of a spherical cavity filled with black body radiation are
\(V\) and \(300K\) , respectively. If it expands adiabatically to a volume \(2V\) , its temperature will be
closest to
(1)
\(150\) K
(2)
\(300\) K
(3)
\(250\) K
(4)
\(240\) K
Check Answer
Option 4
Q.No:24 CSIR Feb-2022
The total number of phonon modes in a solid of volume \(V\) is \(\int_0^{\omega_D}g(\omega)d\omega=3N\), is the
number of primitive cells, \(\omega_D\) is the Debye frequency and density of photon modes is
\(g(\omega)=AV\omega^2\) (with A>0 a constant). If the density of the solid doubles in a phase transition,
the Debye temperature \(\Theta_D\) will
(1)
increase by a factor of \(2^{2/3}\)
(2)
increase by a factor of \(2^{1/3}\)
(3)
decrease by a factor of \(2^{2/3}\)
(4)
decrease by a factor of \(2^{1/3}\)
Check Answer
Option 2
Q.No:25 CSIR Feb-2022
The dispersion relation of a gas of non-interacting bosons in d dimensions \(E(k)=ak^s\) where \(a\) and \(s\) are positive constants, Bose-Einstein condensation will occur for all values of
(1)
\(d>s\)
(2)
\(d+2>s>d-2\)
(3)
\(s>2\) independent of \(d\)
(4)
\(d>2\) independent of \(s\)
Check Answer
Option 1
Q.No:26 CSIR Sep-2022
The energy levels of a system, which is in equilibrium at temperature \(T=1/k_B \beta\), are 0, \(\epsilon\), and \(2\epsilon\). If two identical bosons occupy these energy levels, the probability of the total energy being \(3\epsilon\) is
(1)
\(\frac{e^{-3\beta\epsilon}}{1+e^{-\beta\epsilon}+e^{-2\beta\epsilon}+e^{-3\beta\epsilon}+e^{-4\beta\epsilon}}\)
(2)
\(\frac{e^{-3\beta\epsilon}}{1+2e^{-\beta\epsilon}+2e^{-2\beta\epsilon}+e^{-3\beta\epsilon}+e^{-4\beta\epsilon}}\)
(3)
\(\frac{e^{-3\beta\epsilon}}{e^{-\beta\epsilon}+2e^{-2\beta\epsilon}+e^{-3\beta\epsilon}+e^{-4\beta\epsilon}}\)
(4)
\(\frac{e^{-3\beta\epsilon}}{1+e^{-\beta\epsilon}+2e^{-2\beta\epsilon}+e^{-3\beta\epsilon}+e^{-4\beta\epsilon}}\)
Check Answer
Option 4
Q.No:27 CSIR June-2023
The single particle energies of a system of \(N\) non-interacting fermions of spin \(s\) (at \(T=0\)) are \(E_n=n^2E_0\), \(n=1,2,3...\). The ratio \(\epsilon_F(\frac{3}{2})/\epsilon_F(\frac{1}{2})\) of the Fermi energies for fermions of spin \(3/2\) and spin \(1/2\), is
1) \(1/2\)
2) \(1/4\)
3) \(2\)
4) \(1\)
Check Answer
Option 2
Q.No:28 CSIR June-2023
The energy levels available to each electron in a system of \(N\) non-interacting electrons are \(E_n=nE_0\) , \(n=0,1,2,...\). A magnetic field, which does not affect the energy spectrum, but completely polarizes the electron spins, is applied to the system. The change in the ground state energy of the system is
1) \(\frac{1}{2}N^2E_0\)
2) \(N^2E_0\)
3) \(\frac{1}{8}N^2E_0\)
4) \(\frac{1}{4}N^2E_0\)
Check Answer
Option 4
Q.No:29 CSIR June-2023
The dispersion relation of a gas of non-interacting bosons in two dimensions is \(E(k)=c\sqrt{|k|}\), where \(c\) is a positive constant. At low temperatures, the leading dependence of the specific heat on temperature \(T\), is
1) \(T^4\)
2) \(T^3\)
3) \(T^2\)
4) \(T^{3/2}\)
Check Answer
Option 1
Q.No:30 CSIR June-2023
The dispersion relation of electrons in three dimensions is \(\epsilon(k)=\hbar v_Fk\), where \(v_F\) is the Fermi velocity. If at low temperatures (\(T<<T_F\)) the Fermi energy \(\epsilon_F\) depends on the number density \(n\) as \(\epsilon_F(n)\sim n^\alpha\), the value of \(\alpha\) is
1) \(1/3\)
2) \(2/3\)
3) \(1\)
4) \(3/5\)
Check Answer
Option 1
Q.No: 31 CSIR june-2023
Two electrons in thermal equilibrium at temperature \(T=k_B/\beta\) can occupy two sites. The energy of the configuration in which they occupy the different sites is \(J\textbf{S}_1\cdot\textbf{S}_2\) (where \(J>0\) is a constant and \(\textbf{S}\) denotes the spin of an electron), while it is \(U\) if they are at the same site. If \(U=10J\), the probability for the system to be in the first excited state is
1) \(e^{-3\beta J/4}/(3e^{\beta J/4}+e^{-3\beta J/4}+2e^{-10\beta J})\)
2) \(3e^{-\beta J/4}/(3e^{-\beta J/4}+e^{3\beta J/4}+2e^{-10\beta J})\)
3) \(e^{-\beta J/4}/(2e^{-\beta J/4}+3e^{3\beta J/4}+2e^{-10\beta J})\)
4) \(3e^{-3\beta J/4}/(2e^{\beta J/4}+3e^{-3\beta J/4}+2e^{-10\beta J})\)
Check Answer
Option 2
Q.No: 32 CSIR Dce-2023
Each allowed energy level of a system of non-interacting fermions has a degeneracy \( M \). If there are \( N \) fermions and \( R \) is the remainder upon dividing \( N \) by \( M \), then the degeneracy of the ground state is
1) \( R^M \)
2) 1
3) \( M \)
4) \( ^MC_R \)
Check Answer
Option 4
Q.No: 33 CSIR Dce-2023
A system of non-relativistic and non-interacting bosons of mass \(m\) in two dimensions has a density \(n\). The Bose-Einstein condensation temperature \(T_c\) is
1) \(\frac{12\hbar^2n}{\pi mk_B}\)
2) \(\frac{3\hbar^2n}{\pi mk_B}\)
3) \(\frac{6\hbar^2n}{\pi mk_B}\)
4) \(0\)
Check Answer
Option 4
Q.No: 34 CSIR Dce-2023
Four distinguishable particles fill up energy levels \(0, \varepsilon, 2\varepsilon\). The number of available microstates for the total energy \(4\varepsilon\) is
1) 20
2) 24
3) 11
4) 19
Check Answer
Option 4
Q.No: 35 CSIR June-2024
The Debye temperature of a two-dimensional insulator is \(150 \, K\). The ratio of the heat required to raise its temperature from \(1 \, K\) to \(2 \, K\) and from \(2 \, K\) to \(3 \, K\) is
1) \(7:19\)
2) \(3:13\)
3) \(1:1\)
4) \(3:5\)
Check Answer
Option 1
Q.No: 36 CSIR June-2024
The Debye temperature of a two-dimensional insulator is \(150 \, K\). The ratio of the heat required to raise its temperature from \(1 \, K\) to \(2 \, K\) and from \(2 \, K\) to \(3 \, K\) is
1) \(7:19\)
2) \(3:13\)
3) \(1:1\)
4) \(3:5\)
Check Answer
Option 1
Q.No: 37 CSIR Dec-2024
For an ideal Bose gas, the density of states is given by
\(\rho(E) = C E^{2}\), where \(C\) is a positive constant.
Assume that the number of bosons is not conserved.
The variation of the specific heat of the gas with temperature \(T\) is closest to
1) \(T^{2}\)
2) \(T^{3}\)
3) \(T\)
4) \(T^{4}\)
Check Answer
Option 2
Q.No: 38 CSIR Dec-2024
A spherical cavity of volume \(V\) is filled with thermal radiation at temperature \(T\).
The cavity expands adiabatically to 8 times its initial volume.
If \(\sigma\) is Stefan’s constant and \(c\) is the speed of light in vacuum,
what is the closest value of the work done in the process?
1) \(8 \left(\frac{\sigma T^{4} V}{c}\right)\)
2) \(4 \left(\frac{\sigma T^{4} V}{c}\right)\)
3) \(\frac{1}{2} \left(\frac{\sigma T^{4} V}{c}\right)\)
4) \(2 \left(\frac{\sigma T^{4} V}{c}\right)\)
Check Answer
Option 4
Q.No: 39 CSIR Dec-2024
Consider a free fermion gas in a hypercubic infinite potential well in hypothetical 4-dimensional space.
What is the expression for the ground state energy per particle in terms of the Fermi energy \(E_{F}\)?
(Ignore spin degeneracy of the fermions)
1) \(\frac{4}{5} E_{F}\)
2) \(\frac{2}{3} E_{F}\)
3) \(\frac{1}{3} E_{F}\)
4) \(\frac{2}{5} E_{F}\)
Check Answer
Option 2
Q.No: 40 CSIR Dec-2024
Bose condensation experiments are carried out on two samples A and B of an ideal Bose gas.
The same gas species is used in both. The condensate densities achieved at a given temperature
below the critical temperature are \(n_{A} = 1.80\times 10^{14}\,\text{cm}^{-3}\) and
\(n_{B} = 1.44\times 10^{15}\,\text{cm}^{-3}\), respectively. If \(P_{A}\) and \(P_{B}\) are
the pressures of the two gas samples, the ratio \(\frac{P_{A}}{P_{B}}\) is
1) \(1\)
2) \(\left(\frac{1}{8}\right)^{3/2}\)
3) \(\left(\frac{1}{8}\right)^{2/3}\)
4) \(8\)
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