An oscillator of mass and angular frequency has Heisenberg-picture lowering operator .

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The oscillator as an algebra

Problem

An oscillator of mass m=2.0×10−26 kgm = 2.0\times10^{-26}\,\mathrm{kg} and angular frequency ω=5.0×1014 rad s−1\omega = 5.0\times10^{14}\,\mathrm{rad\,s^{-1}} has Heisenberg-picture lowering operator a^H(t)=a^ e−iωt\hat a_H(t) = \hat a\,e^{-i\omega t}. ((a)) Starting from x^H(t)=ℏ/2mω (a^H(t)+a^H†(t))\hat x_H(t) = \sqrt{\hbar/2m\omega}\,\big(\hat a_H(t) + \hat a_H^\dagger(t)\big), write ⟨n+1∣x^H(t)∣n⟩\langle n+1|\hat x_H(t)|n\rangle as an explicit function of tt, keeping only the surviving ladder term. ((b)) Evaluate its amplitude for n=2n=2.

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