- #1
vbrasic
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Homework Statement
Find the first-order corrections to energy and the wavefunction, for a 1D harmonic oscillator which is linearly perturbed by ##H'=ax##.
Homework Equations
First-order correction to the energy is given by, ##E^{(1)}=\langle n|H'|n\rangle##, while first-order correction to the wave-function is, $$|n^{(1)}\rangle=\Sigma_{m\neq n}\frac{\langle m|H'|n\rangle}{E_n-E_m}|m\rangle.$$
The Attempt at a Solution
The energy correction is just ##0## by orthogonality arguments. For the corrections to the wavefunction, I have, $$|n^{(1)}\rangle=\frac{a\sqrt{\frac{\hbar}{2m\omega}}}{\hbar\omega}\Sigma_{m\neq n}\frac{\langle m|a+a^+|n\rangle}{n-m}|m\rangle.$$ The first term within the summation is only exists for ##m=n-1##, such that the term becomes, ##\sqrt{n}|n-1\rangle##. Similarly, the second term in the summation only exists for ##m=n+1##, such that the term becomes,##-\sqrt{n+1}|n+1\rangle##. Hence, the first order correction should be, $$\frac{a\sqrt{\frac{\hbar}{2m\omega}}}{\hbar\omega}(\sqrt{n}|n-1\rangle-\sqrt{n+1}|n+1\rangle).$$ I am not sure however, as I'm still getting my feet wet with perturbation theory and would like to know if I'm on the correct track.