| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
69 |
IB |
2010 |
Quantum Mechanics |
|
Paper 1, Section II, 15D |
2010 |
A particle of unit mass moves in one dimension in a potential
$$V=\frac{1}{2} \omega^{2} x^{2}$$
Show that the stationary solutions can be written in the form
$$\psi_{n}(x)=f_{n}(x) \exp \left(-\alpha x^{2}\right)$$
You should give the value of $\alpha$ and derive any restrictions on $f_{n}(x)$. Hence determine the possible energy eigenvalues $E_{n}$.
The particle has a wave function $\psi(x, t)$ which is even in $x$ at $t=0$. Write down the general form for $\psi(x, 0)$, using the fact that $f_{n}(x)$ is an even function of $x$ only if $n$ is even. Hence write down $\psi(x, t)$ and show that its probability density is periodic in time with period $\pi / \omega$.