A Modern Approach to Quantum Mechanics · Problem 7.20

Path Integrals — John S. Townsend

Physics — Quantum Mechanics · coherent-states-displacement-operator

Problem

Show that D^(α)∣0⟩=eαa^†e−α∗a^e[αa^†,α∗a^]/2∣0⟩=e−∣α∣2/2∑n=0∞αnn!∣n⟩=∣α⟩\hat{D}(\alpha)|0\rangle = e^{\alpha\hat{a}^\dagger}e^{-\alpha^*\hat{a}}e^{[\alpha\hat{a}^\dagger,\alpha^*\hat{a}]/2}|0\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle = |\alpha\rangle

Related theory

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