| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
69 |
IB |
2015 |
Quantum Mechanics |
|
Paper 1, Section II, D |
2015 |
Write down expressions for the probability density $\rho(x, t)$ and the probability current $j(x, t)$ for a particle in one dimension with wavefunction $\Psi(x, t)$. If $\Psi(x, t)$ obeys the timedependent Schrödinger equation with a real potential, show that
$$\frac{\partial j}{\partial x}+\frac{\partial \rho}{\partial t}=0$$
Consider a stationary state, $\Psi(x, t)=\psi(x) e^{-i E t / \hbar}$, with
$$\psi(x) \sim \begin{cases}e^{i k_{1} x}+R e^{-i k_{1} x} & x \rightarrow-\infty \ T e^{i k_{2} x} & x \rightarrow+\infty\end{cases}$$
where $E, k_{1}, k_{2}$ are real. Evaluate $j(x, t)$ for this state in the regimes $x \rightarrow+\infty$ and $x \rightarrow-\infty$.
Consider a real potential,
$$V(x)=-\alpha \delta(x)+U(x), \quad U(x)= \begin{cases}0 & x<0 \ V_{0} & x>0\end{cases}$$
where $\delta(x)$ is the Dirac delta function, $V_{0}>0$ and $\alpha>0$. Assuming that $\psi(x)$ is continuous at $x=0$, derive an expression for
$$\lim _{\epsilon \rightarrow 0}\left[\psi^{\prime}(\epsilon)-\psi^{\prime}(-\epsilon)\right]$$
Hence calculate the reflection and transmission probabilities for a particle incident from $x=-\infty$ with energy $E>V_{0} .$