Riley hobson bence · Problem 19.3

Quantum operators

Mathematics — Methods · quantum-operators-time-dependence-expectation-value

Problem

Given the time-dependent Schr\"odinger equation for ∣ψ⟩|\psi\rangle, find the time dependence of the expectation value ⟨ψ∣A∣ψ⟩\langle \psi | A | \psi \rangle of an operator AA. Show that the rate of change of ⟨ψ∣A∣ψ⟩\langle \psi | A | \psi \rangle is i⟨ψ∣[H,A]∣ψ⟩i\langle \psi | [H, A] | \psi \rangle. Use the results that [H,px]=[V,px][H, p_x] = [V, p_x] and [H,x]=[px2,x]/2m[H, x] = \left[p_x^2, x\right]/2m to evaluate specific commutators.

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