Skip to article frontmatter Skip to article content

Site not loading correctly?

This may be due to an incorrect BASE_URL configuration. See the MyST Documentation for reference.

Made with MyST

Feedback

Teaching Materials

Introduction to Quantum Mechanics

Made with MyST

1.1 Examples, exercises and experiments

April 17, 2026

Updated: 18 Apr 2026

Exercises¶

Exercise 1 (Verify the normalisation)

Show algebraically that $\psi_n(x) = \sqrt{2/a}\,\sin(n\pi x/a)$ is normalised, i.e., that $\int_0^a |\psi_n|^2\,dx = 1$ . Use the identity $\sin^2\theta = \frac{1}{2}(1-\cos 2\theta)$ .

Exercise 2 (Energy level spacing)

Show that the spacing between adjacent energy levels is $E_{n+1} - E_n = (2n+1)E_1$ .

Is the spacing constant (equally spaced levels)?
What happens to the fractional spacing $(E_{n+1}-E_n)/E_n$ as $n \to \infty$ ?
Relate your answer to part (2) to the correspondence principle.

Exercise 3 (Zero-point energy and box size)

An electron ( $m_e = 9.109\times10^{-31}\;\mathrm{kg}$ ) is confined to a 1D box.

Calculate the zero-point energy $E_1$ for a box of length $a = 1\;\mathrm{nm}$ (roughly the size of a small molecule). Express your answer in eV ( $1\;\mathrm{eV} = 1.602\times10^{-19}\;\mathrm{J}$ ).
Calculate $E_1$ for a box of length $a = 1\;\mathrm{m}$ (macroscopic). Why is the zero-point energy negligible on a macroscopic scale?
For what box length $a$ does $E_1 = k_BT$ at room temperature ( $T = 300\;\mathrm{K}$ , $k_B = 1.381\times10^{-23}\;\mathrm{J/K}$ )?

Exercise 4 (Probability in a region)

For a particle in a 1D box of length $a$ :

Find the probability of locating the particle in the middle third of the box ( $a/3 \leq x \leq 2a/3$ ) for $n = 1$ and $n = 2$ .
For the $n=1$ state, is the particle more likely to be found in the middle third or the outer thirds? Does this make sense given the shape of $|\psi_1|^2$ ?
For large $n$ , what value would you expect the probability of finding the particle in any third of the box to approach? Explain.

Exercise 5 (Expectation values)

For the $n$ th state of the particle in a 1D box:

Verify analytically that $\langle p \rangle_n = 0$ using $\hat{p} = -i\hbar\,d/dx$ .
Compute $\langle p^2 \rangle_n$ and show that $\langle p^2 \rangle_n / (2m) = E_n$ . Interpret this result physically.
Show that $\Delta x\,\Delta p > \hbar/2$ for all $n \geq 1$ .

Teaching Materials

A Particle in a 1D Box