Principles of Quantum Mechanics · Problem 7.4.5

The Path Integral Formulation of Quantum Theory — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-time-evolution

Problem

At t=0t=0 a particle starts out in ∣ψ(0)⟩=1/21/2(∣0⟩+∣1⟩)|\psi(0)\rangle = 1/2^{1/2}(|0\rangle + |1\rangle). (1) Find ∣ψ(t)⟩|\psi(t)\rangle; (2) find ⟨X(0)⟩=⟨ψ(0)∣X∣ψ(0)⟩\langle X(0)\rangle = \langle \psi(0)|X|\psi(0)\rangle, ⟨P(0)⟩\langle P(0)\rangle, ⟨X(t)⟩\langle X(t)\rangle, ⟨P(t)⟩\langle P(t)\rangle; (3) find ⟨X˙(t)⟩\langle \dot{X}(t)\rangle and ⟨P˙(t)⟩\langle \dot{P}(t)\rangle using Ehrenfest's theorem and solve for ⟨X(t)⟩\langle X(t)\rangle and ⟨P(t)⟩\langle P(t)\rangle and compare with part (2).

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