Introduction to Quantum Mechanics · Problem 2.13

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator_superposition

Problem

A particle in the harmonic oscillator potential starts out in the state Ψ(x,0)=A[3ψ0(x)+4ψ1(x)].\Psi(x, 0) = A\left[3\psi_0(x) + 4\psi_1(x)\right]. (a) Find AA. (b) Construct Ψ(x,t)\Psi(x, t) and ∣Ψ(x,t)∣2|\Psi(x, t)|^2. Don't get too excited if ∣Ψ(x,t)∣2|\Psi(x, t)|^2 oscillates at exactly the classical frequency; what would it have been had I specified ψ2(x)\psi_2(x), instead of ψ1(x)\psi_1(x)? (c) Find ⟨x⟩\langle x \rangle and ⟨p⟩\langle p \rangle. Check that Ehrenfest's theorem (Equation 1.38) holds, for this wave function. (d) If you measured the energy of this particle, what values might you get, and with what probabilities?

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