It is difficult to calculate second order energy shift using standard perturbati

Question

It is difficult to calculate second order energy shift using standard perturbation theory procedure, however, note that classical Hamiltonian of this oscillator can be represented as Using this representation you can introduce effective angular frequency omega*. Find what this effective angular frequency is. Then use this effective frequency for the energy spectrum of quantum harmonic oscillator En = h omega*(n + 1/2). Expand it in series with respect to the small parameter lambda and find the first and the second order energy shifts. If you do it right then the first order energy shift should match your answer from part a).

Explanation / Answer

H = p2/2m + m/2 (2 + 2/m) x2

>>>> * = (2 + 2/m)

             = (1+2/(m2))0.5

             = (1 + /(m2) + ...)

>>>> first order: /(m)

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