Problem Set-3: Canonical Partition function
CHM322: Dr. Arnab Ghosh
February 28, 2025
P
1. Use the Gibbs expression for entropy S = −kB i Pi ln Pi to derive the
formula for the entropy of mixing.
P P
2. Maximize
P S = −kB i Pi ln Pi subject to the constraints that i Pi = 1
and i Pi Ei = U . Under this condition, show that the entropy can be
written as S/kB = ln Z + βU .
P
3. Maximize the Gibbs-Shannon
P P−kB i Pi ln Pi subject to the
entropy S =
constraints that i Pi = 1 and hf (x)i = i Pi f (xi ), and show that
e−βf (xi ) X d
Pi = ; Z(β) = e−βf (xi ) ; hf (x)i = − ln Z(β)
Z(β) i
dβ
4. Derive an expression for variance in energy for a two-state system under
canonical ensemble.
5. Find the average energy under canonical ensemble. for (a) An n-state
system with energy 0, ε, 2ε, ..., nε. and (b) A harmonic oscillator with
energy 0, ε, 2ε, ...
6. For a two-dimensional HO, n = nx + ny , where
1 1
E nx = n x + ~ωx ; Eny = ny + ~ωy
2 2
nx = 0, 1, 2, ... and ny = 0, 1, 2, .... What is the canonical partition func-
tion at a given temperature? Reduce it to the degenerate case ωx = ωy .
7. Show that at high temperature, such that kB T ~ω, the partition func-
tion of the simple harmonic oscillator is approximately Z ≈ (β~ω)−1 .
Hence find U , Cv , F , and S at high temperature. Repeat the problem for
the high temperature limit of the rotational energy levels of the diatomic
molecule for which Z ≈ (β~2 /2I)−1 .
1
8. Consider a collection of N two level systems (TLS) in thermal equilibrium
at temperature T . Each system has only two states: a ground state of
energy 0 and an excited state of energy ε. Find each of the following
quantities and make a sketch of the temperature dependence. (a) The
probability that the above system will be found in the exited state. (b)
The entropy of the entire system.
9. Show that for N non-interacting spin-1/2 particles in a magnetic field B
the internal energy U is given by
µB B
U = −N µB B tanh
kB T
10. The internal levels of an isolated hydrogen atom are given by E = −RH /n2
where RH = 13.6 eV. The degeneracy of each level is given by 2n2 . Argue
that the expression of Z
∞
X
2 RH
Z= 2n exp
n=1
n2 kB T
diverges for T 6= 0. Give argument that if the hydrogen atom is confined
in a box of finite size, which would cut off highly excited states n ≥ 3, Z
would not then diverge. Estimate mean energy of the H-atom under this
condition at 300 K.
11. ∗ The energy of classical rotor is given by
!
1 p2φ
E= p2θ +
2I sin2 θ
Using classical statistical mechanics, calculate Cv of a heteronuclear di-
atomic molecule with moment of inertia I at temperature T .
12. In QM, energy of the rigid rotor is given by
~2
Ej = j(j + 1), j = 0, 1, 2, 3...
2I
where each j is (2j+1) fold degenerate. Find the expression of the partition
function and average energy of the system as a function temperature.
Derive an expression for the specific heat at high and low temperatures
and specify the range of temperatures where your expressions are valid.
13. Show that the single-partition function Z1 of a two-dimensional gas con-
fined in an area A is given by
A
Z1 =
λ2th
√
where λth = ~/ 2πmkB T
2
14. Show that S given by Sackur-Tetrode equation
5
S = N kB − ln(nλ3th )
2
is an extensive quantity. Use this to show that the entropy of a gas of
distinguishable particles is given by
3 2
S = N kB − ln(λth /V )
2
which is not extensive. [This non-extensivity of entropy is the orig-
inal version of Gibbs paradox]
15. An atom in a solid has two energy levels: a ground state of degeneracy g1
and an excited state of degeneracy g2 at an energy ∆ above the ground
state. Show that the partition function Zatom is
Zatom = g1 + g2 e−β∆
Show that the heat capacity of the atom is given by
g1 g2 ∆2 e−β∆
Cv =
kB T 2 (g1 + g2 e−β∆ )2
A mono-atomic gas of such atoms has a partition function given by Z =
Zatom ZN , where ZN is the partition function due to the translational
motion of the gas atoms and is given by ZN = (1/N !)[V /λ3th ]N . Show
that Cv of such gas is
g1 g2 ∆2 e−β∆
3
Cv = N kB T +
2 kB T 2 (g1 + g2 e−β∆ )2
16. The entropy of an ideal paramagnet in a magnetic field is given approxi-
mately by S = S0 −CU 2 , where C is a constant. (a) Using the fundamental
definition of temperature, determine the internal energy of the system U
and sketch a graph of U vs T for −∞ < T < +∞. Briefly explain the
physical significance of the negative temperature in the above plot.
17. Show that the single particle partition function Z1 of a gas of H-atoms is
approximately given by
V eβRH
Z1 =
λ3th
where RH = 13.6 eV and the contribution due to excited states are ne-
glected.
18. ∗ A certain magnetic system contains n independent molecules per unit
volume, each has four energy levels: 0, ∆ − gµB B, ∆, ∆ + gµB B (g is
3
constant). Write down canonical partition function, compute Helmholtz
free energy and finally magnetization M . Hence show that the magnetic
susceptibility χ is given by
µ0 M 2nµ0 (gµB )2
χ = lim =
B→0 B kB T (3 + e∆/kB T )
19. ∗ A material consists of n independent particles and is in a weak external
magnetic field B. Each particle can have a magnetic moment mµB along
the magnetic field, where m = J, J − 1, ..., −J + 1, −J, J being an integer
and µ is a constant. The system is at temperature T . (a) Find the
partition function. (b) Estimate the average magnetization of the material
at high temperature (kB T µB B).
20. ∗ The energy of a paramagnet can be written as U = −m.B. Show that
if B is varied isothermally then T δS = −B.δm.
21. ∗ N weakly coupled particles obeying MB statistics may each exist in one
of the 3 non-degenerate energy levels of energies −E, 0, +E. The system
is in contact with a thermal reservoir at temperature T . (a) What is the
entropy of the system at T = 0K? (b) What is maximum possible entropy
of the system. (c) If C(T ) is the heat capacity of the system, evaluate the
R∞
value of the integral 0 C(T )
T dT ?
