2017-68.md

course	course_year	question_number	tags	title	year
Quantum Mechanics	IB	68	IB 2017 Quantum Mechanics	Paper 3, Section I, B	2017

A particle of mass $m$ is confined to a one-dimensional box $0 \leqslant x \leqslant a$. The potential $V(x)$ is zero inside the box and infinite outside.

(a) Find the allowed energies of the particle and the normalised energy eigenstates.

(b) At time $t=0$ the particle has wavefunction $\psi_{0}$ that is uniform in the left half of the box i.e. $\psi_{0}(x)=\sqrt{\frac{2}{a}}$ for $0<x<a / 2$ and $\psi_{0}(x)=0$ for $a / 2<x<a$. Find the probability that a measurement of energy at time $t=0$ will yield a value less than $5 \hbar^{2} \pi^{2} /\left(2 m a^{2}\right)$.