Problems and Solutions on Thermodynamics and Statistical Mechanics

194 Problems d Sdutio~ on Thermodynamics €4 Statistical Mechanics

and the internal energy is

The molar specific heat is

hwo 6

x=---=-

dU x2 ex

C =-=R

’ dT (e5 - 1)2 ’ kT T‘

(a) In the limit of high temperatures, T >> 0, or z << 1, we have

C,=R.

(b) In the limit of low temperatures, T << 6, or x >> 1, we have

C, w R(6/T)2 exp(-B/T).

2032
A one-dimensional quantum harmonic oscillator (whose ground state
energy is hw/2) is in thermal equilibrium with a heat bath at temperature
T.
(a) What is the mean value of the oscillator’s energy, (E), as a function

(b) What is the value of AE, the root-mean-square fluctuation in en-

(c) How do (E) and AE behave in the limits kT << hw and kT >> hw

of T?

ergy about (E)?

(MIT)

Solution:

The partition function is

(a) The mean energy is

(E) = kT2- a Inz = - hw coth (g)

aT 2