Introduction to Quantum Mechanics · Problem 2.11

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator_expectation_values

Problem

(a) Compute ⟨x⟩\langle x \rangle, ⟨p⟩\langle p \rangle, ⟨x2⟩\langle x^2 \rangle, and ⟨p2⟩\langle p^2 \rangle, for the states ψ0\psi_0 (Equation 2.60) and ψ1\psi_1 (Equation 2.63), by explicit integration. *Comment:* In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable ξ≡mω/ℏ x\xi \equiv \sqrt{m\omega/\hbar}\, x and the constant α≡(mω/πℏ)1/4\alpha \equiv (m\omega/\pi\hbar)^{1/4}. (b) Check the uncertainty principle for these states. (c) Compute ⟨T⟩\langle T \rangle and ⟨V⟩\langle V \rangle for these states. (No new integration allowed!) Is their sum what you would expect?

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