Chapter 6: Correlation Functions and Spectra

A. Initial and final potentials with energy levels depicted
V₀=1/2kx² V=k₁x⁴-k₂x²

B. Eigenvalues as a function of the switching parameter. Note the pairwise coalescence of eigenvalues in the double well system.

C. Eigenfunctions of initial harmonic potential

D. Eigenfunctions of final double well potential

Adiabatic switching methods for calculating eigenstates.
The adiabatic theorem in quantum mechanics states that if The Actual formul

is an eigenfunction sufficiently of H(0), and H(t) is a slowly varying function of time, then The Actual formul

will evolve in such a way as to remain an eigenfunction of H(t) for all time. This property may be exploited to calculate eigenfunctions for complicated potentials, V, by starting in an eigenfunction of a simple potential The Actual formul

and slowly "switching" on the difference potential, The Actual formul

where

varies slowly from 0 to 1. In the example below, The Actual formul

is chosen as the harmonic oscillator potential, The Actual formul

and V is the double well potential, The Actual formul