| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
36 |
IB |
2001 |
Quantum Mechanics |
|
2.I $. 9 \mathrm{~F} \quad$ |
2001 |
Consider a solution $\psi(x, t)$ of the time-dependent Schrödinger equation for a particle of mass $m$ in a potential $V(x)$. The expectation value of an operator $\mathcal{O}$ is defined as
$$\langle\mathcal{O}\rangle=\int d x \psi^{*}(x, t) \mathcal{O} \psi(x, t)$$
Show that
$$\frac{d}{d t}\langle x\rangle=\frac{\langle p\rangle}{m},$$
where
$$p=\frac{\hbar}{i} \frac{\partial}{\partial x},$$
and that
$$\frac{d}{d t}\langle p\rangle=\left\langle-\frac{\partial V}{\partial x}(x)\right\rangle$$
[You may assume that $\psi(x, t)$ vanishes as $x \rightarrow \pm \infty .]$