A Modern Approach to Quantum Mechanics · Problem 6.13

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · infinite-square-well

Problem

Solve the energy eigenvalue equation in position space for a particle of mass

m

in the potential energy well

V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & \text{elsewhere} \end{cases}

Show that the energy eigenvalues are given by

E_n = \frac{\hbar^2 \pi^2 n^2}{2mL^2} \quad n = 1, 2, 3, \ldots

with the corresponding normalized energy eigenfunctions

\psi_n(x) = \begin{cases} \sqrt{\dfrac{2}{L}}\sin\dfrac{n\pi x}{L} & 0 < x < L \quad n = 1, 2, 3, \ldots \\ 0 & \text{elsewhere} \end{cases}