A Modern Approach to Quantum Mechanics · Problem 6.15

The One-Dimensional Harmonic Oscillator — John S. Townsend

Physics — Quantum Mechanics · time-evolution-infinite-square-well

Problem

A particle of mass

m

in the one-dimensional potential energy well

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

is at time

t = 0

in the state

\psi(x) = \begin{cases} \left(\dfrac{1+i}{2}\right)\sqrt{\dfrac{2}{L}}\sin\dfrac{\pi x}{L} + \dfrac{1}{\sqrt{2}}\sqrt{\dfrac{2}{L}}\sin\dfrac{2\pi x}{L} & 0 < x < L \\ 0 & \text{elsewhere} \end{cases}

(a) What is

\psi(x, t)

? (b) What is

\langle E\rangle

for this state at time

t

? (c) What is the probability that a measurement of the energy will yield the value

\hbar^2\pi^2/2mL^2

? (d) Without detailed computation, give an argument that

\langle x\rangle

is time dependent.