A Modern Approach to Quantum Mechanics · Problem 6.17

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · energy-measurement-non-eigenstate

Problem

A particle in the potential energy well V(x)={00<x<L∞elsewhereV(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{elsewhere} \end{cases} is in the state ψ(x)={Nx(x−L)0<x<L0elsewhere\psi(x) = \begin{cases} Nx(x-L) & 0 < x < L \\ 0 & \text{elsewhere} \end{cases} (a) Determine the value of NN so that the state is properly normalized. (b) What is the probability that a measurement of the energy yields the ground-state energy of the well? (c) What is ⟨E⟩\langle E\rangle for this state?

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