Chapter 9 Exercises

Notation and Conventions

Unless otherwise stated: $\hbar = 1$ in spin problems; the Pauli matrices are $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$; spin operators are $\hat{S}_i = (\hbar/2)\sigma_i$. The infinite square well has width $a$ and eigenstates $|n\rangle$ with energies $E_n = n^2\pi^2\hbar^2/(2ma^2)$. The QHO has frequency $\omega$ and mass $m$ with eigenstates $|n\rangle$ and energies $E_n = (n + 1/2)\hbar\omega$. Answers to selected problems appear in Appendix H.

Section A: Eigenvalue Equations and Discrete Spectra (Problems 1--8)

Problem 1. (Warm-up: spin-1/2 eigenvalues) Consider the operator $\hat{S}_n = \hat{\mathbf{S}} \cdot \hat{n}$, the spin component along the direction $\hat{n} = \sin\theta\cos\phi\,\hat{x} + \sin\theta\sin\phi\,\hat{y} + \cos\theta\,\hat{z}$.

(a) Write $\hat{S}_n$ as a $2 \times 2$ matrix in the $\hat{S}_z$ eigenbasis.

(b) Find the eigenvalues of $\hat{S}_n$ by computing the characteristic polynomial. Show that the eigenvalues are $\pm\hbar/2$ regardless of the direction $\hat{n}$.

(c) Find the normalized eigenstates $|\pm\rangle_n$ as functions of $\theta$ and $\phi$.

(d) Verify that $|+\rangle_n$ and $|-\rangle_n$ are orthonormal.

(e) Verify the completeness relation $|+\rangle_n\langle+|_n + |-\rangle_n\langle-|_n = \hat{I}$.

Problem 2. (Eigenvalues of a general $2 \times 2$ Hermitian matrix) Let $\hat{A} = \begin{pmatrix} a & b \\ b^* & c \end{pmatrix}$ where $a, c \in \mathbb{R}$ and $b \in \mathbb{C}$.

(a) Show that the eigenvalues are $\lambda_\pm = \frac{1}{2}\left[(a + c) \pm \sqrt{(a - c)^2 + 4|b|^2}\right]$.

(b) Show that both eigenvalues are real.

(c) Under what condition on $a$, $b$, $c$ is the eigenvalue degenerate ($\lambda_+ = \lambda_-$)?

(d) Apply this formula to find the eigenvalues of $\hat{S}_x = \frac{\hbar}{2}\sigma_x$. Verify against Problem 1.

Problem 3. (Eigenvalue problem for $\hat{S}_y$) Starting from the matrix $S_y = \frac{\hbar}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$:

(a) Solve the characteristic equation to find the eigenvalues.

(b) Find the normalized eigenstates $|+\rangle_y$ and $|-\rangle_y$ in the $\hat{S}_z$ basis.

(c) Write the spectral decomposition $\hat{S}_y = \lambda_+|+\rangle_y\langle+|_y + \lambda_-|-\rangle_y\langle-|_y$.

(d) Verify the spectral decomposition by computing the right-hand side as a $2 \times 2$ matrix and checking it equals $S_y$.

(e) A particle is in the state $|\psi\rangle = \cos(\alpha/2)|\uparrow\rangle + e^{i\beta}\sin(\alpha/2)|\downarrow\rangle$. Compute $P(S_y = +\hbar/2)$ and $\langle\hat{S}_y\rangle$ as functions of $\alpha$ and $\beta$.

Problem 4. (QHO eigenvalue algebra) Using only the commutation relation $[\hat{a}, \hat{a}^\dagger] = 1$ and the definition $\hat{N} = \hat{a}^\dagger\hat{a}$:

(a) Prove that if $\hat{N}|n\rangle = n|n\rangle$, then $\hat{N}(\hat{a}^\dagger|n\rangle) = (n+1)(\hat{a}^\dagger|n\rangle)$. Interpret this physically.

(b) Prove that $\hat{N}(\hat{a}|n\rangle) = (n-1)(\hat{a}|n\rangle)$.

(c) Show that $n \geq 0$ for all eigenvalues of $\hat{N}$. (Hint: compute $\langle n|\hat{N}|n\rangle$ using $\hat{N} = \hat{a}^\dagger\hat{a}$, and use the fact that the norm of any ket is non-negative.)

(d) Deduce that the spectrum of $\hat{N}$ is $\{0, 1, 2, 3, \ldots\}$ and the ground state satisfies $\hat{a}|0\rangle = 0$.

Problem 5. (Degeneracy in the 2D harmonic oscillator) The two-dimensional isotropic harmonic oscillator has Hamiltonian $\hat{H} = \hbar\omega(\hat{a}_x^\dagger\hat{a}_x + \hat{a}_y^\dagger\hat{a}_y + 1)$.

(a) What are the energy eigenvalues?

(b) List all eigenstates for $E = \hbar\omega$, $E = 2\hbar\omega$, $E = 3\hbar\omega$, and $E = 4\hbar\omega$.

(c) What is the degeneracy of the level $E = (N+1)\hbar\omega$?

(d) Show that $\hat{L}_z = -i\hbar(\hat{a}_x^\dagger\hat{a}_y - \hat{a}_y^\dagger\hat{a}_x)$ commutes with $\hat{H}$, and explain why this implies that the energy degeneracy is related to a symmetry.

Problem 6. (Eigenvalues of a $3 \times 3$ Hamiltonian) A three-level system has the Hamiltonian:

$$H = \begin{pmatrix} E_0 & V & 0 \\ V & E_0 & V \\ 0 & V & E_0 \end{pmatrix}$$

where $E_0$ and $V$ are real and positive.

(a) Find the three eigenvalues.

(b) Find the three normalized eigenstates.

(c) Write the spectral decomposition of $\hat{H}$.

(d) If the system starts in state $|1\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, what is the probability of measuring each energy eigenvalue?

Problem 7. (Functions of operators via spectral decomposition) For the spin-1/2 operator $\hat{S}_z$:

(a) Compute $e^{i\hat{S}_z\phi/\hbar}$ using the spectral decomposition.

(b) Compute $\hat{S}_z^2$ using the spectral decomposition, and verify by direct matrix multiplication.

(c) Show that $\hat{S}_z^2 = \frac{\hbar^2}{4}\hat{I}$ for any spin-1/2 system. Why does this imply that $\hat{S}_z$ has only two eigenvalues?

(d) Compute $\cos(\hat{S}_z\theta/\hbar)$ and $\sin(\hat{S}_z\theta/\hbar)$ using the spectral decomposition.

Problem 8. (Time evolution via spectral decomposition) A spin-1/2 particle is in a uniform magnetic field $\mathbf{B} = B\hat{z}$, with Hamiltonian $\hat{H} = -\gamma B\hat{S}_z = \omega_0\hat{S}_z$ where $\omega_0 = -\gamma B$.

(a) Write the spectral decomposition of $\hat{H}$.

(b) Compute the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$.

(c) If the initial state is $|\psi(0)\rangle = |+\rangle_x = \frac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle)$, find $|\psi(t)\rangle$.

(d) Compute $\langle\hat{S}_x\rangle(t)$, $\langle\hat{S}_y\rangle(t)$, and $\langle\hat{S}_z\rangle(t)$, and show that the spin precesses about the $z$-axis with angular frequency $\omega_0$.

Section B: Continuous Spectra and the Dirac Delta Function (Problems 9--16)

Problem 9. (Delta function identities) Prove the following identities by applying both sides to a test function $f(x)$ and integrating:

(a) $\delta(-x) = \delta(x)$

(b) $\delta(ax) = \frac{1}{|a|}\delta(x)$ for $a \neq 0$

(c) $x\delta(x) = 0$

(d) $\delta(x^2 - a^2) = \frac{1}{2|a|}[\delta(x - a) + \delta(x + a)]$ for $a > 0$

(e) $f(x)\delta(x - a) = f(a)\delta(x - a)$

Problem 10. (Delta function representations) Consider the sequence of functions $f_n(x) = \frac{n}{\sqrt{\pi}}e^{-n^2 x^2}$.

(a) Show that $\int_{-\infty}^{\infty} f_n(x) \, dx = 1$ for all $n$.

(b) Show that $\lim_{n \to \infty} f_n(x) = 0$ for all $x \neq 0$.

(c) Show that $\lim_{n \to \infty} f_n(0) = \infty$.

(d) Argue that $\lim_{n \to \infty} f_n(x) = \delta(x)$ in the distributional sense: for any smooth, rapidly decreasing test function $g(x)$, $\lim_{n \to \infty} \int f_n(x) g(x) \, dx = g(0)$.

Problem 11. (Position and momentum eigenstates) Starting from $\hat{p}|p\rangle = p|p\rangle$ and $\langle x|\hat{p} = -i\hbar\frac{\partial}{\partial x}\langle x|$:

(a) Show that $\langle x|p\rangle$ satisfies the differential equation $-i\hbar\frac{\partial}{\partial x}\langle x|p\rangle = p\langle x|p\rangle$.

(b) Solve this differential equation to obtain $\langle x|p\rangle = Ce^{ipx/\hbar}$.

(c) Determine the constant $C$ by requiring $\langle p|p'\rangle = \delta(p - p')$, using the Fourier representation of the delta function.

(d) Verify the completeness relation $\int |p\rangle\langle p| dp = \hat{I}$ by showing that $\int \langle x|p\rangle\langle p|x'\rangle dp = \delta(x - x')$.

Problem 12. (Expectation values via completeness) A particle is in the state $|\psi\rangle$ with position-space wave function $\psi(x) = \left(\frac{2\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2 + ip_0 x/\hbar}$ (a Gaussian centered at $x = 0$ with mean momentum $p_0$).

(a) Compute $\langle\hat{x}\rangle$ and $\langle\hat{x}^2\rangle$ in the position representation.

(b) Find $\phi(p) = \langle p|\psi\rangle$ by Fourier transform.

(c) Compute $\langle\hat{p}\rangle$ and $\langle\hat{p}^2\rangle$ in the momentum representation.

(d) Verify the uncertainty relation $\Delta x \Delta p \geq \hbar/2$.

Problem 13. (Derivative of the delta function) The distributional derivative $\delta'(x)$ is defined by $\int f(x)\delta'(x - a) \, dx = -f'(a)$.

(a) Show that $x\delta'(x) = -\delta(x)$.

(b) Show that $\int x^2 \delta''(x) \, dx = 2$.

(c) Use the Fourier representation $\delta(x) = \frac{1}{2\pi}\int e^{ikx} dk$ to derive $\delta'(x) = \frac{1}{2\pi}\int ik \, e^{ikx} dk$.

(d) Show that $\langle x|\hat{p}^2|p\rangle = p^2\langle x|p\rangle$ by applying $\hat{p}$ twice in the position representation, and verify consistency.

Problem 14. (Normalization of free-particle states) A free particle ($V = 0$) has energy eigenstates $|E, \pm\rangle$ satisfying $\hat{H}|E, \pm\rangle = E|E, \pm\rangle$ where $E \geq 0$ and $\pm$ labels the direction of propagation.

(a) Show that in the position representation, $\langle x|E, +\rangle = \frac{1}{\sqrt{2\pi\hbar v}}e^{ikx}$ where $k = \sqrt{2mE}/\hbar$ and $v = \hbar k/m$, with the normalization chosen so that $\langle E'|E\rangle = \delta(E - E')$.

(b) Why is the factor of $v$ in the normalization physically reasonable? (Hint: think about probability current.)

(c) Write the completeness relation for free-particle energy eigenstates.

Problem 15. (Three-dimensional delta function) In spherical coordinates, the three-dimensional delta function is:

$$\delta^3(\mathbf{r} - \mathbf{r}') = \frac{1}{r^2}\delta(r - r')\delta(\cos\theta - \cos\theta')\delta(\phi - \phi')$$

(a) Verify that $\int \delta^3(\mathbf{r} - \mathbf{r}') f(\mathbf{r}) \, d^3r = f(\mathbf{r}')$ for any continuous $f$.

(b) Using the completeness of spherical harmonics, show that:

$$\delta(\cos\theta - \cos\theta')\delta(\phi - \phi') = \sum_{l=0}^{\infty}\sum_{m=-l}^{l} Y_l^m(\theta, \phi)Y_l^{m*}(\theta', \phi')$$

(c) How does this relate to the completeness relation for the angular momentum eigenstates $|l, m\rangle$?

Problem 16. (Mixed spectrum) The finite square well of depth $V_0$ and width $a$ has a finite number of bound states (discrete spectrum, $E < 0$) and a continuum of scattering states (continuous spectrum, $E > 0$). Write the completeness relation for this system, carefully distinguishing the discrete sum and continuous integral contributions. Explain why the normalization conventions differ for bound and scattering states.

Section C: Spectral Theorem and Fourier Transforms (Problems 17--24)

Problem 17. (Spectral decomposition practice) The operator $\hat{A}$ has the matrix representation $A = \begin{pmatrix} 1 & 1-i \\ 1+i & 0 \end{pmatrix}$ in some orthonormal basis $\{|1\rangle, |2\rangle\}$.

(a) Verify that $\hat{A}$ is Hermitian.

(b) Find the eigenvalues $\lambda_\pm$.

(c) Find the normalized eigenstates $|\lambda_\pm\rangle$ in the $\{|1\rangle, |2\rangle\}$ basis.

(d) Write the spectral decomposition $\hat{A} = \lambda_+|\lambda_+\rangle\langle\lambda_+| + \lambda_-|\lambda_-\rangle\langle\lambda_-|$.

(e) Compute $\hat{A}^3$ using the spectral decomposition and verify by direct matrix multiplication.

Problem 18. (Resolution of the identity from the spectral theorem) Given the spectral decomposition $\hat{A} = \sum_n a_n|a_n\rangle\langle a_n|$, show that:

(a) $\hat{I} = \sum_n |a_n\rangle\langle a_n|$ (completeness follows from the spectral theorem).

(b) $\hat{A}^k = \sum_n a_n^k|a_n\rangle\langle a_n|$ for any positive integer $k$.

(c) If all $a_n \neq 0$, then $\hat{A}^{-1} = \sum_n a_n^{-1}|a_n\rangle\langle a_n|$.

(d) $\text{Tr}(\hat{A}) = \sum_n a_n$ and $\text{Tr}(\hat{A}^2) = \sum_n a_n^2$.

Problem 19. (Projection operators) For each eigenvalue $a_n$ of a non-degenerate observable, define $\hat{P}_n = |a_n\rangle\langle a_n|$.

(a) Show that $\hat{P}_n^2 = \hat{P}_n$ (idempotent).

(b) Show that $\hat{P}_n^\dagger = \hat{P}_n$ (Hermitian).

(c) Show that $\hat{P}_m\hat{P}_n = \delta_{mn}\hat{P}_n$ (orthogonality of projectors).

(d) Show that $\sum_n \hat{P}_n = \hat{I}$ (completeness).

(e) Show that $\hat{A} = \sum_n a_n\hat{P}_n$ and that $\hat{P}_n = \prod_{m \neq n}\frac{\hat{A} - a_m\hat{I}}{a_n - a_m}$.

Problem 20. (Fourier transform of the QHO ground state) The QHO ground state wave function is $\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-m\omega x^2/2\hbar}$.

(a) Compute the momentum-space wave function $\phi_0(p) = \langle p|\psi_0\rangle$ by Fourier transform.

(b) Show that $\phi_0(p)$ is also a Gaussian, and determine its width $\sigma_p$.

(c) Verify that $\Delta x \Delta p = \hbar/2$, confirming the minimum-uncertainty property.

(d) Compute $\langle\hat{p}^2\rangle$ directly from $\phi_0(p)$ and verify that $\langle\hat{T}\rangle = \frac{1}{4}\hbar\omega$ where $\hat{T} = \hat{p}^2/(2m)$.

(e) Explain why the ground state wave function is the same shape in both position and momentum space (up to scaling). What property of the QHO makes this possible?

Problem 21. (Fourier transform of a box function) A particle is in the state:

$$\psi(x) = \begin{cases} 1/\sqrt{a} & \text{if } |x| < a/2 \\ 0 & \text{otherwise} \end{cases}$$

(a) Compute $\phi(p)$ by Fourier transform.

(b) Show that $\phi(p)$ is a sinc function and sketch it.

(c) Compute $\langle\hat{p}^2\rangle$ from $\phi(p)$. (Warning: the integral diverges! Explain why, and what this tells you about the position-space wave function.)

(d) Compute $\Delta x$ and discuss the uncertainty relation for this state.

Problem 22. (Momentum-space Schrodinger equation) A particle moves in the potential $V(x) = -\alpha\delta(x)$.

(a) Write the time-independent Schrodinger equation in momentum space.

(b) For the bound state ($E < 0$), show that the momentum-space wave function is $\phi(p) = N/(p^2 + \kappa^2)$ where $\kappa = \sqrt{2m|E|}$.

(c) Determine $|E|$ by the self-consistency condition (integrate $\phi(p)$ over all $p$).

(d) Determine the normalization constant $N$ by requiring $\int|\phi(p)|^2 dp = 1$.

(e) Compute the momentum-space probability density $|\phi(p)|^2$ and sketch it. What is the most probable momentum?

Problem 23. (Parseval's theorem application) Use Parseval's theorem to evaluate the integral $\int_{-\infty}^{\infty}\frac{dx}{(x^2 + a^2)^2}$ for $a > 0$.

(Hint: Recognize $f(x) = 1/(x^2 + a^2)$ as the Fourier transform of something, apply Parseval's theorem, and evaluate the simpler integral.)

Problem 24. (Position operator in momentum space) Starting from $\langle p|\hat{x}|\psi\rangle = i\hbar\frac{\partial}{\partial p}\phi(p)$:

(a) Derive this result by inserting a complete set of position eigenstates.

(b) Verify it for the QHO ground state: compute $\langle p|\hat{x}|\psi_0\rangle$ using the formula, and check that it gives the same result as the Fourier transform of $x\psi_0(x)$.

(c) Show that $[\hat{x}, \hat{p}] = i\hbar$ in the momentum representation.

Section D: Rigged Hilbert Space and Synthesis (Problems 25--28)

Problem 25. (Why $|x\rangle$ is not in the Hilbert space) Consider the would-be "wave function" of a position eigenstate: $\psi_x(x') = \langle x'|x\rangle = \delta(x' - x)$.

(a) Show that $\int |\psi_x(x')|^2 dx'$ is not finite (it equals $\delta(0)$).

(b) Show that $\langle\hat{p}^2\rangle$ for this state is infinite.

(c) Show that $\Delta p = \infty$ and $\Delta x = 0$, formally satisfying $\Delta x \Delta p \geq \hbar/2$.

(d) Explain in your own words why position eigenstates are essential mathematical tools despite being unphysical.

Problem 26. (Approximating a position eigenstate) Consider the normalized Gaussian $\psi_\sigma(x) = (\pi\sigma^2)^{-1/4}e^{-x^2/2\sigma^2}$.

(a) Show that $|\psi_\sigma(x)|^2 \to \delta(x)$ as $\sigma \to 0$ (in the distributional sense).

(b) Compute $\Delta x$ and $\Delta p$ for $\psi_\sigma$.

(c) Show that as $\sigma \to 0$, $\Delta x \to 0$ but $\Delta p \to \infty$.

(d) Explain how this illustrates the relationship between physical states (in the Schwartz space $\Phi$) and generalized eigenstates (in $\Phi'$).

Problem 27. (Eigenvalue problem for the parity operator) The parity operator $\hat{\Pi}$ is defined by $\hat{\Pi}|x\rangle = |-x\rangle$.

(a) Show that $\hat{\Pi}$ is both Hermitian and unitary.

(b) Find the eigenvalues of $\hat{\Pi}$.

(c) Show that $\hat{\Pi}^2 = \hat{I}$, and explain why this is consistent with your answer to (b).

(d) Write the spectral decomposition of $\hat{\Pi}$ for the space of functions on $\mathbb{R}$, identifying the projectors onto even and odd functions.

(e) Show that $[\hat{\Pi}, \hat{H}] = 0$ for any Hamiltonian with a symmetric potential $V(-x) = V(x)$, and explain the physical consequence.

Problem 28. (Capstone: Complete analysis of a three-level system) A quantum system has three orthonormal states $\{|1\rangle, |2\rangle, |3\rangle\}$. The Hamiltonian is:

$$\hat{H} = E_0(|1\rangle\langle 1| + |2\rangle\langle 2| + |3\rangle\langle 3|) + V(|1\rangle\langle 2| + |2\rangle\langle 1|) + V(|2\rangle\langle 3| + |3\rangle\langle 2|)$$

where $E_0, V > 0$.

(a) Write $\hat{H}$ as a $3 \times 3$ matrix.

(b) Find all eigenvalues and normalized eigenstates.

(c) Write the spectral decomposition $\hat{H} = \sum_n E_n|E_n\rangle\langle E_n|$.

(d) Compute the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ using the spectral decomposition.

(e) If the system starts in state $|1\rangle$, compute $|\psi(t)\rangle = \hat{U}(t)|1\rangle$ and find the probability of being in state $|3\rangle$ as a function of time.

(f) At what time $t^*$ is the probability of being in state $|3\rangle$ maximum? What is this maximum probability?

(g) Show that the system exhibits quantum beats — periodic oscillations between $|1\rangle$ and $|3\rangle$ mediated by $|2\rangle$.