Physics 561 - Problem Set 1 - due Thursday September 6 2001
John Marko, SES 2374, 996-6064, jmarko@uic.edu

1. Method of Lagrange Multipliers Find the stationary point of the function

f(x,y) = x2 - y2 + y4
along the curve y2 = 5x by

(a) directly eliminating y by plugging the constraint equation into f and then finding the minimum with respect to x.

(b) using the method of Lagrange multipliers applied to the new function

f(x,y) - l[ y2 - 5x]
Note that you will have to determine the appropriate multiplier l to satisfy the constraint equation.

In each case check whether the stationary point is a minimum or a maximum.

(c) Explain why the method works (i.e. finds a stationary point subject to a constraint).

2. Number of States for a Classical Ideal Gas

(a) Up to a numerical constant, write down a formula for the surface area of a sphere in d dimensions as a function of its radius, for d >> 1 (i.e. what is the `scaling' of the area with the sphere radius - this is really an exercise in dimensional analysis)

(b) Now consider a classical-mechanical `ideal gas' of N particles with (kinetic) energy

E = N
å
 i = 1 
 pi2
2 m
If energy E is fixed, find the surface area of the 3N-dimensional sphere in momentum space that the system is required to be located on.

(c) Use (b) and Boltzmann's formula to find the entropy of the gas (don't worry about non-extensive contributions)

(d) Find the temperature T(N,E), and the pressure P(N,T).


3. Helmholtz Free Energy in Canonical Ensemble: A physical system in equilibrium at temperature T has a set of states { s } with energy spectrum Es. The canonical partition function is Z = ås e-bEs, where as usual, b = (kB T)-1. Show that

-kB T lnZ = < E > - ST

Note that in classical thermodynamics E - ST is the Helmholtz free energy (F in this class, but in some books it is denoted A, and is also sometimes called the `Helmholtz potential').

4. Specific Heat in Canonical Ensemble: Using canonical statistical mechanics (i.e. Z = åe-beta E, < E > = -b lnZ etc.) show that the specific heat at constant volume,

CV º æ
ç
è 		< E >
T
ö
÷
ø

V 
is proportional to the mean-squared-fluctuation of the energy:
< E2 > - < E > 2
and find the proportionality constant.

5. Harmonic Oscillator in Thermal Equilibrium: (a) Consider a harmonic oscillator with energy spectrum (h/2p) w0 (n + 1/2) at temperature T. Find the canonical partition function, Helmholtz free energy, average energy < E > , entropy S and specific heat (C = < E > /T). Plot S/kB and C/kB as a function of the dimensionless quantity kB T / ((h/2p) w0).

(b) For high temperatures, show that the `classical' partition function

Z(T) = ó
õ 		dx dp
h
 exp é
ê
ë 		- p2/m + kx2
2 k B T
 ù
ú
û
has the same thermodynamics as the quantum oscillator. At what characteristic temperature does the quantum oscillator become well described by the `classical' expression?

6. Two-level System: Consider a system with two energy levels of energy 0 and e. Find the canonical partition function, Helmholtz free energy, average energy, and specific heat. Plot S/kB and C/kB as as a function of kB T / e.

Note that your results for 4. and 5. represent either the time-averaged properties of one oscillator, or equivalently the properties per oscillator for a large system of identical and independent oscillators (it is a good exercise to explicitly show this).

Also note that 5. and 6. can also be interpreted as the partition function for occupation of a single momentum-spin state by bosonic or fermionic particles (much more on this coming soon).


File translated from TEX by TTH, version 2.53.
On 30 Aug 2001, 13:57.