Second order correction to the wavefunction

robbo96 · Oct 18, 2010

Hi all,

I've been doing a lot of thinking and I was wondering precisely how the 2nd order correction to the wave function from perturbation theory is derived:

$Equation for the second-order energy correction term in perturbation theory: -λ² Σ_{k≠n} |k⁽⁰⁾⟩⟨n⁽⁰⁾|V|n⁽⁰⁾⟩⟨k⁽⁰⁾|V|n⁽⁰⁾⟩/(E_n⁽⁰⁾ - E_k⁽⁰⁾)² - ½λ²|n⁽⁰⁾⟩ Σ_{k≠n} ⟨n⁽⁰⁾|V|k⁽⁰⁾⟩⟨k⁽⁰⁾|V|n⁽⁰⁾⟩/(E_n⁽⁰⁾ - E_k⁽⁰⁾)² + O(λ³).$

I mean, I can see where bits and pieces come from and I've tried to work through it as an exercise. Does anyone have a reference text on this that they can point me in the direction of? I've exhausted myself looking.

thanks!

Robert

Avodyne · Oct 18, 2010

http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/time-independent_perturbation_theory

Note that this calculation uses the normalization [itex]\langle n|n^{(0)}\rangle=1[/itex], and the result is missing your last term. This comes from changing the normalization to [itex]\langle n|n\rangle=1[/itex].

Second order correction to the wavefunction

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