| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
35 |
IB |
2001 |
Quantum Mechanics |
|
1.II.18F |
2001 |
Consider a quantum-mechanical particle of mass $m$ moving in a potential well,
$$V(x)= \begin{cases}0, & -a<x<a \ \infty, & \text { elsewhere }\end{cases}$$
(a) Verify that the set of normalised energy eigenfunctions are
$$\psi_{n}(x)=\sqrt{\frac{1}{a}} \sin \left(\frac{n \pi(x+a)}{2 a}\right), \quad n=1,2, \ldots$$
and evaluate the corresponding energy eigenvalues $E_{n}$.
(b) At time $t=0$ the wavefunction for the particle is only nonzero in the positive half of the well,
$$\psi(x)= \begin{cases}\sqrt{\frac{2}{a}} \sin \left(\frac{\pi x}{a}\right), & 0<x<a \ 0, & \text { elsewhere }\end{cases}$$
Evaluate the expectation value of the energy, first using
$$\langle E\rangle=\int_{-a}^{a} \psi H \psi d x$$
and secondly using
$$\langle E\rangle=\sum_{n}\left|a_{n}\right|^{2} E_{n},$$
where the $a_{n}$ are the expansion coefficients in
$$\psi(x)=\sum_{n} a_{n} \psi_{n}(x)$$
Hence, show that
$$1=\frac{1}{2}+\frac{8}{\pi^{2}} \sum_{p=0}^{\infty} \frac{(2 p+1)^{2}}{\left[(2 p+1)^{2}-4\right]^{2}}$$