Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 253

uvolume” (length) specific heat.

(a) Compute c, for the regime T + 00.

(b) For T + 0

c, -+ Aw-“Tr ,

where A is a constant that you need not compute. Compute the exponents
(Y and 7.
The problem is to be treated quantum mechanically.
(Prince ton)
Solution:
T 1

= (^1) ehw,/kT - 1 ‘
n=- iY-
When kT >> hw,

• NkT.

kT

u c5 Chwn. - -

hwn

= Nk.

dU

Hence c, = -

dT

(b) When kT << hw, we have

Y 1 N/2

= c hwn ehw,/kT - 1 c5^2 C tiune-hWJkT.

n=-+ n=O

so

Notice that as N >> 1 we have approximately

N/2 sin2 Ge-(hw/rrkT) sin(nz/N) N

cos N

N. -d (sin y)

TX T