Phy5646/Non-degenerate Perturbation Theory - Problem 3
(Submitted by Team 1)
This example taken from "Quantum Physics" 3rd ed., Stephen Gasiorowicz, p. 177.
Problem:
A charged particle in a simple harmonic oscillator, for which , subject to a constant electric field so that . Calculate the energy shift for the level to first and second order in . (Hint: Use the operators and for the evaluation of the matrix elements).
Solution:
(a) To first order we need to calculate . It is easy to show that . One way is to use the relation
and since and we see that .
(b) The second-order term involves
The only contributions come from and , so that
and thus
The result is independent of . We can check for its correctness by noting that the total potential energy is
Thus the perturbation shifts the center of the potential by and lowers the energy by , which agrees with our second-order result.