Phy5646/Non-degenerate Perturbation Theory - Problem 3

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(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.