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Physics LibreTexts

2.6: Example

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Consider a simple harmonic oscillator in its ground state, to which we apply a perturbation ˆV=λx2. We know the unperturbed wavefunction |n0=[mω0/π]14 exp{mω0x2/2}, so we can evaluate the first order shift in energy according to the perturbation theory:

ΔE0=n0|λx2|n0=λmω0/πx2 exp{mω0x2/}dx=λ2mω0

In this case we know the exact shift, since the perturbation is simply an additional harmonic potential, giving a total k=mω20+2λ and an exact ground state energy of 12ω20+2λ/m. It is easy to verify that to first order in λ these expressions are identical.

To determine the amount of mixing of states, we need to evaluate matrix elements like n0|λx2|ni. We won’t evaluate these here, but we will note that for odd i the integral is zero - the symmetric perturbation only mixes in symmetric excited states.


This page titled 2.6: Example is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform.

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