A Modern Approach to Quantum Mechanics · Problem 7.9

Path Integrals — John S. Townsend

Physics — Quantum Mechanics · harmonic-oscillator-ehrenfest-theorem

Problem

Show that in the superposition of adjacent energy states (7.63) the average value of the position of the particle is given by ⟨x⟩=⟨ψ∣x^∣ψ⟩=Acos⁡(ωt+δ)\langle x\rangle = \langle\psi|\hat{x}|\psi\rangle = A\cos(\omega t + \delta) and the average value of the momentum is given by ⟨px⟩=⟨ψ∣p^x∣ψ⟩=−mωAsin⁡(ωt+δ)\langle p_x\rangle = \langle\psi|\hat{p}_x|\psi\rangle = -m\omega A\sin(\omega t + \delta) in accord with Ehrenfest's theorem, (6.33) and (6.34).

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