Problem Set

1. (Photoelectric effect) A sodium surface has work function $\phi = 2.28$ eV. When illuminated With light of wavelength $\lambda = 400$ nm, find (a) the maximum kinetic energy of emitted Electrons, and (b) the stopping potential.

2. (Compton scattering) X-rays of wavelength $0.071$ nm are scattered at $\theta = 45°$ from a Carbon target. Find (a) the wavelength of the scattered photons, and (b) the kinetic energy of The recoil electrons.

3. (de Broglie wavelength) Electrons are accelerated through a potential difference of $200$ V. Calculate their de Broglie wavelength. If these electrons pass through a double slit with slit Separation $d = 100$ nm, find the angular position of the first diffraction maximum. Hint: use the Small-angle approximation $\sin\theta \approx \theta$ for the double-slit formula $d\sin\theta = \lambda$.

4. (Postulates) Explain why the state space of quantum mechanics must be a complex vector space Rather than a real vector space. Give a physical example that demonstrates the necessity of Complex amplitudes. Hint: consider the Mach-Zehnder interferometer with phase shifters.

5. (Continuity equation) Starting from the Schrodinger equation, derive the continuity equation $\partial|\psi|^2/\partial t + \nabla\cdot\mathbf{J} = 0$. Show that for a stationary state $\psi(\mathbf{r},t) = \phi(\mathbf{r})e^{-iEt/\hbar}$The probability current is time-independent. What does this imply about the probability distribution?

6. (Normalisation) Normalise the wave function $\psi(x) = N\,x(a-x)$ for $0 \lt x \lt a$ (and zero Otherwise). Find $\langle x \rangle$, $\langle x^2 \rangle$And $\langle p^2 \rangle$.

7. (Hermitian operators) Prove that the momentum operator $\hat{p} = -i\hbar\,d/dx$ is Hermitian On the space of wave functions that vanish at infinity. What boundary conditions are required? Show by counterexample that $\hat{p}$ is not Hermitian if the boundary terms do not vanish.

8. (Uncertainty principle) For the harmonic oscillator ground state $\psi_0(x) = (m\omega/\pi\hbar)^{1/4}e^{-m\omega x^2/(2\hbar)}$Calculate $\langle x \rangle$ $\langle x^2 \rangle$, $\langle p \rangle$, $\langle p^2 \rangle$And verify that $\sigma_x\,\sigma_p = \hbar/2$. Also show that $\langle x \rangle = \langle p \rangle = 0$ by symmetry.

9. (Eigenvalue problem) Find the eigenvalues and normalised eigenvectors of the matrix $\hat{A} = \begin{pmatrix}3 & 1\\1 & 3\end{pmatrix}$. Verify that the eigenvectors are orthogonal And that they form a complete basis for $\mathbb{C}^2$. Generalise: what are the eigenvalues of $\begin{pmatrix}a & b\\b & a\end{pmatrix}$?

10. (Infinite square well) A particle is in the ground state of an infinite square well of Width $L$. Suddenly, the well expands symmetrically to width $2L$ (the centre remains fixed). Find the probability that the particle is found in the ground state of the new well. Also find The probability that it is found in the first excited state.

11. (Harmonic oscillator) Using the ladder operators, compute $\langle x^2 \rangle$ $\langle p^2 \rangle$And $\langle x^4 \rangle$ for the state $|n\rangle$ of the harmonic Oscillator. Express your answers in terms of $n$, $m$, $\omega$And $\hbar$.

12. (Delta potential) A particle of mass $m$ and energy $E \gt 0$ is incident on the potential $V(x) = \alpha[\delta(x+a) + \delta(x-a)]$. Find the transmission coefficient. In the limit $a \to 0$Verify that you recover the single-delta-function result.

13. (Tunnelling) A proton with energy $3$ MeV approaches a rectangular barrier of height $10$ MeV and width $5 \times 10^{-15}$ m. Estimate the transmission coefficient. Compare with The alpha decay of a typical heavy nucleus and comment on the exponential dependence on barrier width.

14. (Angular momentum algebra) Using the angular momentum commutation relations and the Raising/lowering operators, prove that $[\hat{L}^2, \hat{L}_\pm] = 0$. Then show that $\hat{L}_+|l,l\rangle = 0$ and hence derive the normalisation constant for $\hat{L}_+|l,m\rangle$.

15. (Hydrogen atom) Calculate $\langle r \rangle$, $\langle r^2 \rangle$And $\langle 1/r \rangle$ for the hydrogen atom ground state $\psi_{100}$. Compare $\langle r \rangle$ With the Bohr radius $a_0$. Use the virial theorem to relate $\langle T \rangle$ and $\langle V \rangle$ for The Coulomb potential.

16. (Spin) An electron is in the spin state $|\psi\rangle = \frac{1}{\sqrt{3}}|\uparrow\rangle + \sqrt{\frac{2}{3}}|\downarrow\rangle$. (a) If $S_z$ is measured, what are the possible outcomes and Their probabilities? (b) If $S_x$ is measured, what are the possible outcomes and their Probabilities? (c) What is $\langle S_x \rangle$? (d) Write the density matrix $\hat{\rho}$ for this state and verify $\mathrm{Tr}(\hat{\rho}) = 1$ and $\hat{\rho}^2 = \hat{\rho}$ (pure state).

17. (Singlet state) Two spin-1/2 particles are prepared in the singlet state. If particle 1 is Measured to have $S_z^{(1)} = +\hbar/2$What is the state of particle 2 immediately after? If Particle 2”s spin is then measured along the $x$-axis, what is the probability of obtaining $+\hbar/2$? Explain how this result is consistent with Bell’s theorem and the no-communication theorem.

18. (Variational method) Use the variational principle with the trial function $\psi(x) = A(a^2 - x^2)$ for $|x| \lt a$ (and zero otherwise) to estimate the ground state Energy of the infinite square well $V(x) = 0$ for $|x| \lt L$ and $V(x) = \infty$ otherwise. Take $a = L$ as a fixed parameter. Compare your result with the exact ground state energy $E_1 = \pi^2\hbar^2/(2mL^2)$ and calculate the percentage error. Comment on why the variational Estimate is higher than the exact result.

19. (Perturbation theory) A particle in an infinite square well of width $L$ experiences a small perturbation $H' = \varepsilon x$ for $0 < x < L$. (a) Compute the first-order correction to the ground state energy. (b) Compute the first-order correction to the ground state wave function. (c) Find the second-order correction to the ground state energy.

20. (Time evolution) A spin-1/2 particle is initially in the state $|\psi(0)\rangle = |+\rangle_x$ (eigenstate of $S_x$ with eigenvalue $+\hbar/2$). The Hamiltonian is $H = \omega S_z$. (a) Find $|\psi(t)\rangle$. (b) Compute $\langle S_x \rangle(t)$, $\langle S_y \rangle(t)$, and $\langle S_z \rangle(t)$. (c) Show that the Bloch vector precesses around the $z$-axis.

21. (Scattering) A plane wave $e^{ikz}$ is incident on a spherically symmetric potential $V(r) = V_0$ for $r < a$ and $V(r) = 0$ for $r > a$. Compute the scattering amplitude $f(\theta)$ and the total cross section in the Born approximation. Under what condition is the Born approximation valid?

22. (Entanglement) Two spin-1/2 particles are in the Bell state $|\Phi^+\rangle = (|00\rangle + |11\rangle)/\sqrt{2}$. (a) Compute the reduced density matrix for particle 1. (b) Show that the von Neumann entropy $S(\rho_1) = 1$ bit. (c) If Alice measures $S_z$ on particle 1 and obtains $+\hbar/2$, what is the state of particle 2? (d) Verify that the Bell state violates the CHSH inequality.