2004-46.md

course	course_year	question_number	tags	title	year
Quantum Mechanics	IB	46	IB 2004 Quantum Mechanics	1.I.8D	2004

From the time-dependent Schrödinger equation for $\psi(x, t)$, derive the equation

$$\frac{\partial \rho}{\partial t}+\frac{\partial j}{\partial x}=0$$

for $\rho(x, t)=\psi^{*}(x, t) \psi(x, t)$ and some suitable $j(x, t)$.

Show that $\psi(x, t)=e^{i(k x-\omega t)}$ is a solution of the time-dependent Schrödinger equation with zero potential for suitable $\omega(k)$ and calculate $\rho$ and $j$. What is the interpretation of this solution?