Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 257

We equate terms of like powers of

The last equation defines The middle equation defines once is known.

The top equation gives the eigenvalue:

Harmonic Oscillator

5.14 Given a Gaussian (MIT)

Denote the eigenfunctions of the harmonic oscillator as with eigen-
value They are a complete set of states, and we can expand any
function in thisset. In particular, we expand our function in terms
of coefficients

The expectation value of the energy is the integral of the Hamiltonian H
for the harmonic oscillator: