Brahmananda Keshab Chandra College
Department of Physics
M.Sc. End Semester Examinations, 2022-23
1st semester, Physics
Statistical Mechanics: PHSPCOR04T
Time Allotted: 2 Hour.
Answer any four from the followings:
Full Marks: 40
4 × 10 = 40
1. (a) For the oxygen gas in a flask, list three physical quantities which can be measured besides
mass, volume and temperature.
2
(b) What are microscopic and macroscopic systems?
2
(c) Verify the relation
N
X
P (nr ) = 1
nr =1
.
3
(d) A classical harmonic oscillator of mass m and spring constant k is known to have a total
energy of E, but its starting time is completely unknown. Find the probability density function, p(x), where p(x)dx is the probability that the mass would be found in the interval dx at
x.
3
2. (a) Consider a free particle inside a one-dimensional box of length L, having an energy between E1 and E2 (E2 > E1 ). With the help of a diagram, show allowed regions in the phase
space and calculate the number of accessible microstates of the system.
3
(b) Consider a one-dimensional harmonic oscillator of mass m whose total energy E =
p2
+ 12 mω 2 x2 , find the volume in the phase space of the oscillator with restriction that total
2m
energy lies between U and U + ∆U .
3
1
(c) Explain Boltzmann statistics. Prove that < E >= 2 kT for each degrees of freedom. 1+3.
1
3. (a) What is statistical and thermodynamic definitions of entropy? How are they related? 1+2
(b) From the extensiveness property of entropy and concept of probability, prove that S =
kB In(Ω).
3
(c) Consider a system composed of a very large number N of distinguishable atoms, nonmoving and mutually non-interacting, each of which has only two (non-degenerate) energy
levels: 0,ϵ 0. Let E/N be the mean energy per atom in the limit N→ ∞. (i) What is the maximum possible value of E/N if the system is not necessarily in thermodynamic equilibrium?
What is the maximum attainable value of E/N if the system is in equilibrium (at positive
temperature, of course)? (ii) For thermodynamic equilibrium, compute the entropy per atom,
S/N, as a function of E/N.
4
4. (a) If the number of particle distribution {n̄k } provides the maximum for number of permutations w{nk } show that ln w{nk } is also maximized for {n̄k } .
2
(b) Derive the most probable number of particles per energy level for Bosonic and Fermionic
particles in microcanonical ensemble.
3+3
(c) A quantum system is kept in thermal equilibrium with a heat bath at temperature T . Show
that the quantum partition function Z = T r(e−β Ĥ ) is invariant under the representation of
the Hamiltonian Ĥ.
2
5. (a) Show that√for large volume (V ) limit one-particle density of states can be approximated
2m3 ϵ.
2
as g(ϵ) = 4πV
h3
(b) How one can determine the fugacity (z) for given N, V, T for an ideal Bose-gas?
4
(c) N number of noninteracting harmonic oscillators are confined in a large (can be considered as infinity) one dimensional box, which is in thermal equilibrium with a heat bath at
temperature T . Determine the specific heat (Cv ) of the system.
4
6. (a) If a hamiltonian is invariant under permutation of the co-ordinates of the particles, show
that the state of the system is either symmetric or anti-symmetric under permutation.
3
(b) In principle, the concept of identical particles is meaningless for classical statistical mechanics, whereas the fundamental particles (e.g., all electrons) should be identical for quantum
system. Justify.
3
(c) Derive an expression for the isotherms of the ideal Bose gas. Qualitatively plot the
isotherms.
4
2