The wave function of a particle of mass $m$ that moves in a one-dimensional potential well satisfies the Schrödinger equation with a potential that is zero in the region $-a \leq x \leq a$ and infinite elsewhere,

$$V(x)=0 \quad \text { for } \quad|x| \leq a, \quad V(x)=\infty \quad \text { for } \quad|x|>a$$

Determine the complete set of normalised energy eigenfunctions for the particle and show that the energy eigenvalues are

$$E=\frac{\hbar^{2} \pi^{2} n^{2}}{8 m a^{2}}$$

where $n$ is a positive integer.

At time $t=0$ the wave function is

$$\psi(x)=\frac{1}{\sqrt{5 a}} \cos \left(\frac{\pi x}{2 a}\right)+\frac{2}{\sqrt{5 a}} \sin \left(\frac{\pi x}{a}\right)$$

in the region $-a \leq x \leq a$, and zero otherwise. Determine the possible results for a measurement of the energy of the system and the relative probabilities of obtaining these energies.

In an experiment the system is measured to be in its lowest possible energy eigenstate. The width of the well is then doubled while the wave function is unaltered. Calculate the probability that a later measurement will find the particle to be in the lowest energy state of the new potential well.