Thermodynamics, Statistical Physics, and Quantum Mechanics

224 SOLUTIONS

where we set So,

Note that, in this approximation, the nextterm in the potential

would not have introduced any additional shift (only antisymmetric terms

do).

Second solution: (see Problem 1.37, Part I) We can solve the equation of

motion for the nonlinear harmonic oscillator corresponding to the potential

where is the principal frequency. The solution (see
(S.1.37.10) of Part I) gives

where is defined from the initial conditions and A is the amplitude of

oscillations of the linear equation. The average over a period

is

We need to calculate the thermodynamic average of

Substituting we obtain

the same as before.