Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 15.1-1

THE CANONICAL FORMALISM; STATISTICAL MECHANICS IN HELMHOLTZ REPRESENTATION

Physics — Statistical Mechanics · microcanonical-harmonic-oscillators

Problem

A system is composed of two harmonic oscillators each of natural frequency

\omega_0

and each having permissible energies

(n + \frac{1}{2})\hbar\omega_0

, where

n

is any non-negative integer. The total energy of the system is

E' = n'\hbar\omega_0

, where

n'

is a positive integer. How many microstates are available to the system? What is the entropy of the system? A second system is also composed of two harmonic oscillators, each of natural frequency

2\omega_0

. The total energy of this system is

E'' = n''\hbar\omega_0

, where

n''

is an even integer. How many microstates are available to this system? What is the entropy of this system? What is the entropy of the system composed of the two preceding subsystems (separated and enclosed by a totally restrictive wall)? Express the entropy as a function of

E'

and

E''

. Answer:

S_{\text{tot}} = k_B \ln\left(\frac{E'E''}{2\hbar^2\omega_0^2}\right)

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