A Modern Approach to Quantum Mechanics · Problem 6.5

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · wave-packets-uncertainty

Problem

Consider a wave packet defined by ⟨p∣ψ⟩={0p<−P/2N−P/2<p<P/20p>P/2\langle p|\psi\rangle = \begin{cases} 0 & p < -P/2 \\ N & -P/2 < p < P/2 \\ 0 & p > P/2 \end{cases} (a) Determine a value for NN such that ⟨ψ∣ψ⟩=1\langle \psi|\psi\rangle = 1 using the momentum-space wave function directly. (b) Determine ⟨x∣ψ⟩=ψ(x)\langle x|\psi\rangle = \psi(x). (c) Sketch ⟨p∣ψ⟩\langle p|\psi\rangle and ⟨x∣ψ⟩\langle x|\psi\rangle. Use reasonable estimates of Δpx\Delta p_x from the form of ⟨p∣ψ⟩\langle p|\psi\rangle and Δx\Delta x from the form of ⟨x∣ψ⟩\langle x|\psi\rangle to estimate the product Δx Δpx\Delta x\, \Delta p_x. Check that your result is independent of the value of PP. Note: Simply estimate rather than actually calculate the uncertainties.

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