Introduction to Quantum Mechanics · Problem 2.42

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · gaussian_wave_packet

Problem

In Problem 2.21 you analyzed the stationary gaussian free particle wave packet. Now solve the same problem for the traveling gaussian wave packet, starting with the initial wave function Ψ(x,0)=Ae−ax2eilx,\Psi(x, 0) = Ae^{-ax^2}e^{ilx}, where ll is a (real) constant. [Suggestion: In going from ϕ(k)\phi(k) to Ψ(x,t)\Psi(x, t), change variables to u≡k−lu \equiv k - l before doing the integral.] Partial answer: Ψ(x,t)=(2aπ)1/41γe−a(x−ℏlt/m)2/γ2eil(x−ℏlt/2m)\Psi(x, t) = \left(\frac{2a}{\pi}\right)^{1/4}\frac{1}{\gamma}e^{-a(x - \hbar l t/m)^2/\gamma^2}e^{il(x - \hbar l t/2m)} where γ≡1+2iaℏt/m\gamma \equiv \sqrt{1 + 2ia\hbar t/m}, as before. Notice that Ψ(x,t)\Psi(x, t) has the structure of a gaussian "envelope" modulating a traveling sinusoidal wave. What is the speed of the envelope? What is the speed of the traveling wave?

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