The ladder operators of a simple harmonic oscillator which obey(adsbygoogle = window.adsbygoogle || []).push({});

$$[H,a^{\dagger}]=\hbar\omega\ a^{\dagger}$$.

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I would like to see a proof of the relation

$$\exp(-iHt)\exp(a^{\dagger})\exp(iHt)|0\rangle=\exp(a^{\dagger}e^{-i\omega t})|0\rangle\exp(i\omega t/2).$$

Thoughts?

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# I Harmonic oscillator ladder operators

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