Introduction to Quantum Mechanics · Problem 3.36

Formalism — David J. Griffiths

Physics — Quantum Mechanics · uncertainty_principle

Problem

Suppose Ψ(x,0)=Ax2+a2,(−∞<x<∞)\Psi(x, 0) = \frac{A}{x^2 + a^2}, \quad (-\infty < x < \infty) for constants AA and aa. (a) Determine AA, by normalizing Ψ(x,0)\Psi(x, 0). (b) Find ⟨x⟩\langle x \rangle, ⟨x2⟩\langle x^2 \rangle, and σx\sigma_x (at time t=0t = 0). (c) Find the momentum space wave function Φ(p,0)\Phi(p, 0), and check that it is normalized. (d) Use Φ(p,0)\Phi(p, 0) to calculate ⟨p⟩\langle p \rangle, ⟨p2⟩\langle p^2 \rangle, and σp\sigma_p (at time t=0t = 0). (e) Check the Heisenberg uncertainty principle for this state.

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