Introduction to Quantum Mechanics · Problem 1.16

The Wave Function — David J. Griffiths

Physics — Quantum Mechanics · expectation_values

Problem

A particle is represented (at time t=0t = 0) by the wave function Ψ(x,0)={A(a2−x2),−a≤x≤+a,0,otherwise.\Psi(x, 0) = \begin{cases} A\left(a^2 - x^2\right), & -a \leq x \leq +a, \\ 0, & \text{otherwise}. \end{cases} (a) Determine the normalization constant AA. (b) What is the expectation value of xx? (c) What is the expectation value of pp? (Note that you cannot get it from ⟨p⟩=md⟨x⟩/dt\langle p \rangle = m d\langle x \rangle/dt. Why not?)

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