| course |
course_year |
question_number |
tags |
title |
year |
Quantum Mechanics |
IB |
68 |
IB |
2009 |
Quantum Mechanics |
|
Paper 1, Section II, B |
2009 |
A particle of mass $m$ moves in one dimension in a potential $V(x)$ which satisfies $V(x)=V(-x)$. Show that the eigenstates of the Hamiltonian $H$ can be chosen so that they are also eigenstates of the parity operator $P$. For eigenstates with odd parity $\psi^{\text {odd }}(x)$, show that $\psi^{o d d}(0)=0$.
A potential $V(x)$ is given by
$$V(x)= \begin{cases}\kappa \delta(x) & |x|<a \ \infty & |x|>a\end{cases}$$
State the boundary conditions satisfied by $\psi(x)$ at $|x|=a$, and show also that
$$\frac{\hbar^{2}}{2 m} \lim {\epsilon \rightarrow 0}\left[\left.\frac{d \psi}{d x}\right|{\epsilon}-\left.\frac{d \psi}{d x}\right|_{-\epsilon}\right]=\kappa \psi(0)$$
Let the energy eigenstates of even parity be given by
$$\psi^{\text {even }}(x)=\left{\begin{array}{lc}
A \cos \lambda x+B \sin \lambda x & -a<x<0 \\
A \cos \lambda x-B \sin \lambda x & 0<x<a \\
0 & \text { otherwise }
\end{array}\right.$$
Verify that $\psi^{\text {even }}(x)$ satisfies
$$P \psi^{\text {even }}(x)=\psi^{\text {even }}(x)$$
By demanding that $\psi^{e v e n}(x)$ satisfy the relevant boundary conditions show that
$$\tan \lambda a=-\frac{\hbar^{2}}{m} \frac{\lambda}{\kappa}$$
For $\kappa>0$ show that the energy eigenvalues $E_{n}^{\text {even }}, n=0,1,2, \ldots$, with $E_{n}^{\text {even }}<E_{n+1}^{\text {even }}$, satisfy
$$\eta_{n}=E_{n}^{e v e n}-\frac{1}{2 m}\left[\frac{(2 n+1) \hbar \pi}{2 a}\right]^{2}>0$$
Show also that
$$\lim {n \rightarrow \infty} \eta{n}=0,$$
and give a physical explanation of this result.
Show that the energy eigenstates with odd parity and their energy eigenvalues do not depend on $\kappa$.