Problem Class Week 15 Problems

Updated: 08 Feb 2026

These are the problems that we will solve during the problem class in week 15. You are not expected to attempt these questions in advance of the problem class. We will work through them in sequence, going through the solutions after everybody has time to attempt the questions.

I have included additional questions for you to attempt / discuss if you complete the questions before we go over the solution. If you also complete these, please work through Problem Sheet: Week 15

Problems¶

Problem 1

Consider a particle of mass $M$ confined into an infinite square well with walls at $x = 0$ and $x = L$ . The initial state of the particle is the ground state $\ket{\psi_\mathrm{in}} = \ket{E_1}$ , with corresponding initial wavefunction $\psi_\mathrm{in}(x) = u_1(x) = \sqrt{\frac{2}{L}}\sin \frac{\pi x}{L}$ (inside the well, and vanishing outside).

Write down the state $\ket{\psi(t)}$ of the particle at time $t$ , assuming that the state at $t = 0$ is $\ket{\psi_\mathrm{in}}$ .
Write down the wavefunction $\psi(x,t)$ of the particle at time $t$ , as well as the associated probability density $\pd(x,t)$ .
Describe qualitatively how the particle evolves in time.

Consider now that the well is switched off at time $t = t_\mathrm{off}$ . That is, $V(x) = 0$ from $t = t_\mathrm{off}$ onwards, so that the particle becomes free.

Write down (as an integral) the state of the particle at a time $t > t_\mathrm{off}$ . You may use the result of Exercise 7.4. Namely, that the momentum wavefunction corresponding to $u_1(x)$ is
$\tilde{u}_1(p) = \sqrt{\frac{\pi L}{\hbar}}\frac{1}{\pi^2 - p^2 L^2/\hbar^2}(1+e^{-ipL/\hbar}).$
(7.26)

Problem 2

The first two energy eigenfunctions of the QHO are

\begin{align} u_0(x) &= \sqrt{\frac{1}{\ell\sqrt{\pi}}} e^{-x^2/2\ell^2}, & u_1(x) &= \sqrt{\frac{2}{\ell\sqrt{\pi}}}\frac{x}{\ell} e^{-x^2/2\ell^2}, \end{align}

where $\ell = \sqrt{\frac{\hbar}{M\omega}}$ , $M$ is the mass of the particle, and $\omega$ is the (classical) frequency of the oscillator.

Consider the initial wavefunction

\psi_\mathrm{in}(x) = \sqrt{\frac{2}{3\ell \sqrt{\pi}}}\left(1+\frac{x}{\ell}\right)e^{-x^2/2\ell^2}.

By inspection (i.e. by comparing coefficients), write $\psi_\mathrm{in}(x)$ in the form $\psi_\mathrm{in}(x) = \alpha u_0(x) + \beta u_1(x)$ , for suitable constants $\alpha$ and $\beta$ .
Use part 1 to write down the quantum state $\ket{\psi_\mathrm{in}}$ of the particle, as a superposition of energy eigenstates $\ket{E_0}$ and $\ket{E_1}$ .
If the energy of the particle is measured, what are the possible outcomes, and what are their corresponding probabilities?
If the quantum state of the particle at $t = 0$ is $\ket{\psi_\mathrm{in}}$ , use the superposition principle to write down the quantum state $\ket{\psi(t)}$ of the particle at time $t$ .

Problem 3

Consider two distinct particles, one confined in an infinite square well of width $L$ (with walls at $x = 0$ and $x = L$ ), and the second confined in an well of width $\mathcal{L} = L/2$ (with walls at $x = 0$ and $x = \mathcal{L}$ ).

In each case, we will consider a particle prepared in the $n^\mathrm{th}$ energy eigenstate of the well $\ket{E_n}$ .

What is the energy of each particle?
How many times bigger is the energy of the particle in the well of width $\mathcal{L}$ ?
For each well, write down an expression (as an integral) for the average squared-position of the particle $\langle x^2 \rangle$ (you do not need to solve this integral!)
How many times larger is $\langle x^2 \rangle$ for the particle in the well of width $L$ compared to the particle in the well of width $\mathcal{L} = L/2$ .

Solutions¶

Solutions

Tuesday:

Thursday: