TUTORIAL PROBLEMS 1
1.1 Consider a system in which the allowed (nondegenerate) states have energies
0, , 2, 3,. . The system has 4 distinguishable (localized) particles and total energy U = 6 .
a) Tabulate the nine possible distributions of the four parti
The Classical Statistical Treatment of an Ideal Gas
3.1 THERMODYNAMICS FROM THE PARTITION FUNCTION
The power of statistical thermodynamics is beautifully demonstrated in the application of the
theory to an ideal gas. We begin by showing that all the therm
APPENDIX:
Integrals
1] The factorial integral
n!=
0
x n e x dx
this is easy to do as integration by parts.
2] The gamma function extends the concept of factorial to non-integer values
(z) =
0
x z 1e x dx .
When z = n is an integer then (n) = ( n 1)! . In
Mid-Session Test 2013 Worked Solution
QU 1
a) g j > N j
w FD =
g j!
(
w BE =
and
)
N j! g j N j !
(N
)
N !( g 1)!
j
+ g j 1 !
j
j
N j terms
BE:
(N + g 1)! = (N g) (g 1)! g
g 1)(N g 2).(N
N
(g 1)!
(g 1)!
j
j
j
J
j
J
j
J
j
j
j
FD:
j
N terms
j g j N j !
N
PROBLEMS
6.1 The partition function of an Einstein solid is
Z=
e"# E 2T
1" e"# E T
where " E is the Einstein temperature. Treat the crystalline lattice as an assembly of 3N
distinguishable oscillators.
(a) Calculate the Helmholtz function F
(b) Calculate
The Thermodynamics of Magnetism
7.1 INTRODUCTION
All magnetic fields are due to electric charges in motion. Individual atoms can give rise to
magnetic fields when their electrons have a net magnetic moment as a result of their orbital
and/or spin angular
PROBLEMS
5.1 (a) Calculate the fractional number N j N of oscillators in the three lowest quantum states
( j = 0,1,2 ) for T = " 4 and for T = " . (b) Sketch N j N versus j for the two temperatures.
(0.982, 0.0180, 0.00033 and 0.632, 0.232, 0.085)
5.2 (a)
Statistical Thermodynamics
1.1 INTRODUCTION
The combination of energy and entropy concepts led to a science of thermodynamics that
provides great generality and reliability of prediction. Classical thermodynamics is a
macroscopic description which provide
The Heat Capacity of a Diatomic Gas
5.1 INTRODUCTION
We have seen how statistical thermodynamics provides deep insight into the classical
description of a monatomic ideal gas. We might have reason to hope, therefore, that the
statistical model can resolve
The Heat Capacity of a Solid
6.1 INTRODUCTION
The investigation of the heat capacity of solids is important in the study of condensed matter.
This example, like the diatomic case, illustrates the shortcomings of classical kinetic theory
and the need for s
Classical and Quantum Statistics
2.1 BOLTZMANN STATISTICS
The aim is to determine the equilibrium configuration for a system of N distinguishable
noninteracting particles subject to the constraints
n
N
j
= N,
(2.1)
j=1
n
N
j
j
=U.
(2.2)
j=1
Here N j is t
PROBLEMS
7.1 Show that the Bohr magneton is given by B = eh 2me . Assume that the electron in a
hydrogen atom moves in a circular orbit of radius a about the proton and that its angular
momentum is h . The magnetic moment is the product of the electron cu
Classical Ensemble Theory
4.1 Discrete Statistical Approach
4.1.1 Canonical Ensemble
The simplest physical argument to obtain the canonical ensemble begins with the quantum
viewpoint where there are discrete energy levels rather than a continuum of levels
PROBLEMS
3.1 (a) Calculate the entropy S and the Helmholtz function F for an assembly of
distinguishable particles. ( U T + Nk ln Z , "NkT ln Z )
(b) Show that the total energy U and the pressure P are the same for distinguishable particles
as for molecul
PROBLEMS
9.1 Assume that for T = 3TF the value of the chemical potential is 5.6 F . Calculate the value
of the Fermi function f ( ) at the temperature T for values of
and (d) 2.0.
F
of (a) 0, (b) 0.5, (c) 1.0,
(0.134; 0.116; 0.0997; 0.0735)
9.2 Consider a
FORMULA SHEET
Boltzmann Entropy
S = k lnW
Statistics and Distributions
N
n
N j N j
= e
gj Z
gj j
W = N!
j=1 N j !
Boltzmann
n
N
Maxwell-Boltzmann
Fermi-Dirac
W =
N j N j
= e
gj Z
gj j
W =
j=1 N j !
n
j=1
n
Bose-Einstein
W =
j=1
n
kT
Z = g je
j=1
n
kT
g j
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF PHYSICS
FINAL EXAMINATION
JUNE 2008
PHYS3020
Statistical Physics
Time Allowed 2 hours
Total number of questions - 5
Answer ALL questions
All questions ARE of equal value
Candidates may not bring their own calcul
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF PHYSICS
FINAL EXAMINATION
JUNE/JULY 2006
PHYS3020
Statistical Physics
Time Allowed 2 hours
Total number of questions - 5
Answer ALL questions
All questions ARE of equal value
Candidates may not bring their own c
FORMULA SHEET
Boltzmann Entropy
S = k lnW
Statistics and Distributions
N
n
N j N j
= e
gj Z
gj j
W B = N!
j=1 N j !
Boltzmann
N
n
Maxwell-Boltzmann
W MB
Fermi-Dirac
W FD =
n
j=1
n
W BE =
Bose-Einstein
N j N j
= e
gj Z
gj j
=
j=1 N j !
n
kT
Z = g je
j=1
Fermi-Dirac Gases
9.1 THE FERMI ENERGY
The behavior of fermions, indistinguishable particles of half-integer spin, is governed by
Fermi-Dirac statistics. Fermions obey the Pauli exclusion principle, which prohibits the
occupation of any quantum state by m
PROBLEMS
8.1 (a) Calculate the total electromagnetic energy inside an oven of volume 1 m3 heated to a
temperature of 400F. ( 3.89 10 5 J )
(b) Show that the thermal energy of the air in the oven is a factor of approximately 1010
larger than the electromag
Bose-Einstein Gases
8.1 BLACKBODY RADIATION
We can apply statistical physics to radiant energy (as photons) as well as material particles. A
hot body loses heat by radiation and that energy loss is due to the emission of electromagnetic
waves (or the loss