Introduction to Quantum Mechanics · Problem 2.21

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · gaussian_wave_packet

Problem

The gaussian wave packet. A free particle has the initial wave function Ψ(x,0)=Ae−ax2,\Psi(x, 0) = Ae^{-ax^2}, where AA and aa are (real and positive) constants. (a) Normalize Ψ(x,0)\Psi(x, 0). (b) Find Ψ(x,t)\Psi(x, t). *Hint:* Integrals of the form ∫−∞+∞e−(ax2+bx) dx\int_{-\infty}^{+\infty} e^{-(ax^2+bx)}\, dx can be handled by "completing the square": Let y≡a[x+(b/2a)]y \equiv \sqrt{a}\left[x + (b/2a)\right], and note that (ax2+bx)=y2−(b2/4a)(ax^2 + bx) = y^2 - (b^2/4a). *Answer:* Ψ(x,t)=(2aπ)1/41γe−ax2/γ2,where γ≡1+(2iℏat/m).(2.111)\Psi(x, t) = \left(\frac{2a}{\pi}\right)^{1/4} \frac{1}{\gamma} e^{-ax^2/\gamma^2}, \quad \text{where } \gamma \equiv \sqrt{1 + (2i\hbar a t/m)}. \tag{2.111} (c) Find ∣Ψ(x,t)∣2|\Psi(x, t)|^2. Express your answer in terms of the quantity w≡a/[1+(2ℏat/m)2].w \equiv \sqrt{a/\left[1 + (2\hbar a t/m)^2\right]}. Sketch ∣Ψ∣2|\Psi|^2 (as a function of xx) at t=0t = 0, and again for some very large tt. Qualitatively, what happens to ∣Ψ∣2|\Psi|^2, as time goes on? (d) Find ⟨x⟩\langle x \rangle, ⟨p⟩\langle p \rangle, ⟨x2⟩\langle x^2 \rangle, ⟨p2⟩\langle p^2 \rangle, σx\sigma_x, and σp\sigma_p. *Partial answer:* ⟨p2⟩=aℏ2\langle p^2 \rangle = a\hbar^2, but it may take some algebra to reduce it to this simple form. (e) Does the uncertainty principle hold? At what time tt does the system come closest to the uncertainty limit?

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