Thermodynamics, Statistical Physics, and Quantum Mechanics

Second solution: Instead of using operator algebra, we can find a wave

function in the momentum representation, where

The Hamiltonian then is

The Schrödinger equation for becomes

This equation has exactly the same form as the standard oscillator

Schrödingerequation:

We then obtain for the momentum probability distribution for the ground
state:

Therefore

where Using the old “differentiate with respect to an
innocent parameter method” of simplifying an integral, we may rewrite
as

where we substituted (S.5.44.10) into (S.5.44.11) and let Fi-
nally,

QUANTUM MECHANICS 295