PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch The oscillator as an algebra
Problem
An oscillator has mass m=2.0×10−26kg and angular frequency ω=5.0×1013rads−1, and is in the number state ∣n⟩ with n=3.
((a)) Using the counting results ⟨n∣x^∣n⟩=⟨n∣p^∣n⟩=0 (from the selection rule that x^ and p^ connect ∣n⟩ only to ∣n±1⟩) and ⟨n∣x^2∣n⟩=(n+21)ℏ/mω, ⟨n∣p^2∣n⟩=(n+21)mℏω, compute Δx and Δp numerically for ∣3⟩.
((b)) Compute the product ΔxΔp and compare it with the generalised uncertainty relation ΔxΔp≥ℏ/2. State for which n equality holds, and connect that state to the Gaussian ground-state wave function produced from the lowering condition a^∣0⟩=0.