Problem Sheet: Week 10

Infinite Square Well¶

1. In this question we will find the momentum wavefunctions associated to the energy eigenstates of the infinite square well. Recall that the spatial wavefunctions of the energy eigenstates are given by

   $$u_n(x) = \sqrt{\frac{2}{a}} \sin \left(\frac{n \pi x}{a}\right),$$

   for $0 \leq x \leq a$, and $u_n(x) = 0$ otherwise.

   1. Show that the momentum wavefunction $\tilde{u}_n(p)$ associated to $u_n(x)$ is given by

      $$\tilde{u}_n(p) = \sqrt{\frac{\pi a}{\hbar}}\frac{n}{n^2 \pi^2 - a^2p^2/\hbar^2}(1-(-1)^ne^{-iap/\hbar}).$$

   2. For $n = 1$ and $n = 2$, i.e. for the ground state and first excited state, sketch the real and imaginary parts of the momentum wavefunctions, and also the momentum probability density.

2. Consider a particle confined inside an infinite square well of width $a$. The wavefunction of the particle is

   $$\Psi(x) = \begin{cases} \sqrt{\frac{3}{a^3}}x& \text{if } 0 \leq x < a, \\ 0 &\text{otherwise. } \end{cases}$$
   1. Sketch the wavefunction $\Psi(x)$ and confirm that it is normalised.

   In this Problem we would like to write this wavefunction as a superposition of the energy eigenstates of the infinite square well. That is, we would like to write $\Psi(x)$ as

   $$\Psi(x) = \sum_{n=1}^{\infty} c_n u_n(x),$$

   (10.1)

   where $u_n(x) = \sqrt{\frac{2}{a}}\sin \left(\frac{n\pi x}{a}\right)$ for $0 \leq x \leq a$, and $u_n(x) = 0$ otherwise.

   2. Show that the amplitude $c_n$ is given by

      $$c_n = - \frac{\sqrt{6} (-1)^n}{n\pi}.$$

      (10.2)

   3. What are the probabilities for the particle to have the energies $E_1$, $E_2$ and $E_3$, where $E_n = \hbar^2\pi^2n^2/2ma^2$ are the energy levels of the infinite square well?

   4. What is the probability for the particle to have energy at least $E_4$?

3. In this question we will consider an approximation to the wavefunction from Problem 10.2, and study its evolution. One way to approximate a wavefunction is to consider only the amplitudes which have largest modulus. Therefore, consider the following initial wavefunction for a particle,

   $$\Psi(x,0) = A \frac{\sqrt{6}}{\pi}\left(u_1(x) - \frac{1}{2}u_2(x) + \frac{1}{3}u_3(x) \right),$$

   where $A$ is a normalisation constant.

   1. Find the normalisation constant $A$.
   2. Make a sketch of the wavefunction $\Psi(x,0)$.
   3. Write down $\Psi(x,t)$, the wavefunction of the particle at time $t$.
   4. Show that the wavefunction of the particle returns to the form it has at $t = 0$ whenever the time is a multiple of the period $T = 2\pi/\omega$, where
      $$\omega = \frac{\hbar \pi^2}{2 M a^2}.$$

      (10.3)
      That is, show that $\Psi(x,mT) = \Psi(x,0)$, where $m$ is an integer.