Introduction to Quantum Mechanics · Problem 3.49

Formalism — David J. Griffiths

Physics — Quantum Mechanics · momentum_space_schrodinger

Problem

(a) Write down the time-dependent "Schrödinger equation" in momentum space, for a free particle, and solve it. *Answer:* exp⁡(−ip2t/2mℏ)Φ(p,0)\exp\left(-ip^2t/2m\hbar\right) \Phi(p, 0). (b) Find Φ(p,0)\Phi(p, 0) for the traveling gaussian wave packet (Problem 2.42), and construct Φ(p,t)\Phi(p, t) for this case. Also construct ∣Φ(p,t)∣2|\Phi(p, t)|^2, and note that it is independent of time. (c) Calculate ⟨p⟩\langle p \rangle and ⟨p2⟩\langle p^2 \rangle by evaluating the appropriate integrals involving Φ\Phi, and compare your answers to Problem 2.42. (d) Show that ⟨H⟩=⟨p⟩2/2m+⟨H⟩0\langle H \rangle = \langle p \rangle^2/2m + \langle H \rangle_0 (where the subscript 0 denotes the *stationary* gaussian), and comment on this result.

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