Chapter 8 Exercises

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 8 Exercises

Notation and Conventions

Unless otherwise stated: $\hbar = 1$ in problems involving spin; the infinite square well has width $a$; the QHO has frequency $\omega$ and mass $m$. All operators are assumed to act on the appropriate Hilbert space. Answers to selected problems appear in Appendix H.

Section A: Kets, Bras, and Inner Products (Problems 1--8)

Problem 1. (Warm-up) A spin-1/2 particle is in the state $|\psi\rangle = \frac{3}{5}|\uparrow\rangle + \frac{4i}{5}|\downarrow\rangle$.

(a) Write the corresponding bra $\langle\psi|$.

(b) Verify that $|\psi\rangle$ is normalized.

(d) Verify that the probabilities sum to 1.

Problem 2. (Inner product properties) Let $|\alpha\rangle = 2i|1\rangle + 3|2\rangle$ and $|\beta\rangle = |1\rangle - i|2\rangle$, where $\{|1\rangle, |2\rangle\}$ is an orthonormal basis.

(a) Compute $\langle\alpha|\beta\rangle$.

(b) Compute $\langle\beta|\alpha\rangle$.

(c) Verify that $\langle\alpha|\beta\rangle = \langle\beta|\alpha\rangle^*$.

(d) Compute $\langle\alpha|\alpha\rangle$ and $\langle\beta|\beta\rangle$.

Problem 3. (Wave function to Dirac) The wave function of a particle in the ground state of an infinite square well of width $a$ is $\psi_1(x) = \sqrt{2/a}\sin(\pi x/a)$.

(a) Write the normalization condition in Dirac notation.

(b) Express $\psi_1(x)$ in terms of the ket $|1\rangle$ and the position eigenstate $|x\rangle$.

(c) If the state is $|\psi\rangle = c_1|1\rangle + c_2|2\rangle$ with $c_1 = 3/5$ and $c_2 = 4/5$, write $\psi(x) = \langle x|\psi\rangle$ explicitly.

Problem 4. (Anti-linearity of the bra) Let $|\chi\rangle = (2+i)|a\rangle + (3-2i)|b\rangle$, where $\{|a\rangle, |b\rangle\}$ is orthonormal.

(a) Write $\langle\chi|$ explicitly.

(b) Compute $\langle\chi|\chi\rangle$.

(c) A student writes $\langle\chi| = (2+i)\langle a| + (3-2i)\langle b|$. What is wrong, and what is the correct expression?

Problem 5. (Schwarz inequality) Prove that $|\langle\phi|\psi\rangle|^2 \leq \langle\phi|\phi\rangle\langle\psi|\psi\rangle$ for any two kets, using the fact that $\langle\chi|\chi\rangle \geq 0$ for $|\chi\rangle = |\psi\rangle - \frac{\langle\phi|\psi\rangle}{\langle\phi|\phi\rangle}|\phi\rangle$. Under what condition does equality hold?

Problem 6. (Orthogonal decomposition) A particle in the infinite square well is in the state $|\psi\rangle = A(3|1\rangle - 4i|2\rangle + 2|3\rangle)$.

(a) Determine the normalization constant $A$.

(b) What is the probability of measuring the ground state energy $E_1$?

(c) What is the expectation value $\langle\hat{H}\rangle$ in terms of the energy eigenvalues $E_n$?

Problem 7. (Position-momentum overlap) Starting from $\langle x|p\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{ipx/\hbar}$:

(a) Show that $\langle p|x\rangle = \frac{1}{\sqrt{2\pi\hbar}}e^{-ipx/\hbar}$.

(b) Verify the orthonormality relation $\langle p|p'\rangle = \delta(p - p')$ by inserting a complete set of position states.

(c) Using the result of (b), confirm that $\int |p\rangle\langle p| \, dp$ acts as the identity on position eigenstates.

(a) Construct an orthonormal pair $\{|e_1\rangle, |e_2\rangle\}$ using Gram-Schmidt: set $|e_1\rangle = |a\rangle$ and $|e_2\rangle = N(|b\rangle - \langle a|b\rangle|a\rangle)$ with appropriate $N$.

(b) Verify that $\langle e_1|e_2\rangle = 0$ and $\langle e_2|e_2\rangle = 1$.

Section B: Completeness and Resolution of Identity (Problems 9--14)

Problem 9. (Inserting a 1) Starting from $\langle\psi|\hat{A}|\psi\rangle$, insert the energy eigenbasis completeness relation $\sum_n |n\rangle\langle n| = \hat{I}$ on both sides of $\hat{A}$ to show that:

$$\langle\hat{A}\rangle = \sum_{m,n} c_m^* A_{mn} c_n$$

where $c_n = \langle n|\psi\rangle$ and $A_{mn} = \langle m|\hat{A}|n\rangle$.

Problem 10. (Parseval's theorem) Using the completeness relation, prove that:

$$\langle\psi|\psi\rangle = \sum_n |c_n|^2$$

where $c_n = \langle n|\psi\rangle$ for an orthonormal basis $\{|n\rangle\}$. Interpret this result physically.

Problem 11. (Continuous completeness) Use the position-space completeness relation to show that:

$$\langle\phi|\psi\rangle = \int_{-\infty}^{\infty} \phi^*(x)\psi(x) \, dx$$

State each step explicitly, identifying where you use $\langle x|\psi\rangle = \psi(x)$ and $\langle\phi|x\rangle = \phi^*(x)$.

Problem 12. (Double completeness insertion) Starting from $\langle\phi|\hat{A}|\psi\rangle$, insert position completeness relations on both sides of $\hat{A}$ to derive:

$$\langle\phi|\hat{A}|\psi\rangle = \int\!\!\int \phi^*(x) \langle x|\hat{A}|x'\rangle \psi(x') \, dx \, dx'$$

Apply this to the momentum operator $\hat{p}$ using $\langle x|\hat{p}|x'\rangle = -i\hbar\frac{\partial}{\partial x}\delta(x - x')$ and show that you recover the familiar wave-mechanics expression.

Problem 13. (Resolution of identity for spin-1/2) Write out the completeness relation for spin-1/2 in the $S_z$ eigenbasis. Then insert this completeness relation to expand an arbitrary spin state $|\chi\rangle$ in this basis. Show that the expansion coefficients are probabilities for spin measurement outcomes along $z$.

Problem 14. (Mixed basis completeness) Consider a three-state system with orthonormal basis $\{|1\rangle, |2\rangle, |3\rangle\}$.

(a) Write the completeness relation.

(b) A new basis is defined by $|a\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)$, $|b\rangle = \frac{1}{\sqrt{2}}(|1\rangle - |2\rangle)$, $|c\rangle = |3\rangle$. Verify that $|a\rangle\langle a| + |b\rangle\langle b| + |c\rangle\langle c| = \hat{I}$.

(c) Express $|1\rangle$ in the $\{|a\rangle, |b\rangle, |c\rangle\}$ basis using the completeness relation.

Section C: Operators and Matrix Representations (Problems 15--22)

Problem 15. (Outer product operators) Let $|u\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)$ and $|v\rangle = \frac{1}{\sqrt{2}}(|1\rangle - |2\rangle)$.

(a) Compute the operator $\hat{A} = |u\rangle\langle v|$ and write its matrix representation in the $\{|1\rangle, |2\rangle\}$ basis.

(b) Compute $\hat{A}^\dagger$ and its matrix representation.

(c) Is $\hat{A}$ Hermitian? Is $\hat{A} + \hat{A}^\dagger$ Hermitian?

Problem 16. (Projection operators) For the spin-1/2 state $|+x\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle + |\downarrow\rangle)$:

(a) Construct the projection operator $\hat{P}_{+x} = |+x\rangle\langle +x|$ and write its $2 \times 2$ matrix.

(b) Verify that $\hat{P}_{+x}^2 = \hat{P}_{+x}$.

(c) Verify that $\text{Tr}(\hat{P}_{+x}) = 1$.

(d) If the spin state is $|\psi\rangle = \cos\theta|\uparrow\rangle + e^{i\phi}\sin\theta|\downarrow\rangle$, compute $\langle\psi|\hat{P}_{+x}|\psi\rangle$. Interpret the result.

Problem 17. (Matrix representation of the Hamiltonian) A two-level system has Hamiltonian $\hat{H} = E_1|1\rangle\langle 1| + E_2|2\rangle\langle 2| + V(|1\rangle\langle 2| + |2\rangle\langle 1|)$ where $V$ is real.

(a) Write the $2\times 2$ matrix $[H]$ in the $\{|1\rangle, |2\rangle\}$ basis.

(b) Find the eigenvalues of $[H]$.

(c) Find the normalized eigenstates. Express them as kets.

(d) Verify that the eigenstates are orthogonal.

Problem 18. (Hermitian conjugation practice) For each expression, compute the Hermitian conjugate:

(a) $\hat{A}|n\rangle\langle m|$

(b) $\alpha\hat{A}\hat{B}^\dagger|\psi\rangle$

(c) $\langle\phi|\hat{A}\hat{B}|\psi\rangle$

(d) $e^{i\hat{H}t/\hbar}$

(e) $|n\rangle\langle n|\hat{A}|m\rangle\langle m|$

Problem 19. (QHO matrix elements) Using the ladder-operator expressions $\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^\dagger)$ and $\hat{p} = i\sqrt{\frac{m\omega\hbar}{2}}(\hat{a}^\dagger - \hat{a})$:

(a) Compute $\langle m|\hat{x}|n\rangle$ for general $m, n$.

(b) Compute $\langle m|\hat{p}|n\rangle$ for general $m, n$.

(c) Write the $4\times 4$ truncated matrices $[x]$ and $[p]$ using $\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\}$.

(d) Compute $[x][p] - [p][x]$ and verify you get $i\hbar [I]$ (up to truncation effects in the corners).

Problem 20. (Commutator in Dirac notation) Using the completeness relation and the matrix elements from Problem 19, verify the canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$ by computing $\langle n|[\hat{x}, \hat{p}]|n\rangle$ directly.

Problem 21. (Spectral decomposition) The operator $\hat{A}$ has eigenvalues $+1$ and $-1$ with corresponding eigenstates $|+\rangle = \cos\theta|\uparrow\rangle + \sin\theta|\downarrow\rangle$ and $|-\rangle = -\sin\theta|\uparrow\rangle + \cos\theta|\downarrow\rangle$.

(a) Write the spectral decomposition $\hat{A} = \sum_n a_n |a_n\rangle\langle a_n|$.

(b) Compute the matrix of $\hat{A}$ in the $\{|\uparrow\rangle, |\downarrow\rangle\}$ basis. For what value of $\theta$ do you get $\sigma_x$? For what value do you get $\sigma_z$?

(c) Compute $\hat{A}^2$ using the spectral decomposition. What do you get?

Problem 22. (Expectation value two ways) A spin-1/2 particle is in the state $|\psi\rangle = \cos(\pi/8)|\uparrow\rangle + \sin(\pi/8)|\downarrow\rangle$.

(a) Compute $\langle\hat{S}_z\rangle$ using $\langle\psi|\hat{S}_z|\psi\rangle$.

(b) Compute $\langle\hat{S}_z\rangle$ using $\langle\hat{S}_z\rangle = \sum_n s_n |\langle s_n|\psi\rangle|^2$.

(c) Verify both methods give the same answer.

(d) Compute $\langle\hat{S}_x\rangle$ and $\langle\hat{S}_y\rangle$.

Section D: Change of Basis and Unitary Transformations (Problems 23--28)

Problem 23. (Unitary matrix construction) For spin-1/2, the $S_y$ eigenstates are $|+y\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle + i|\downarrow\rangle)$ and $|-y\rangle = \frac{1}{\sqrt{2}}(|\uparrow\rangle - i|\downarrow\rangle)$.

(a) Construct the unitary matrix $[U]$ that transforms from the $S_z$ basis to the $S_y$ basis: $U_{mn} = \langle m_y|n_z\rangle$.

(b) Verify $[U][U]^\dagger = [I]$.

(c) Compute $[S_z]' = [U]^\dagger [S_z] [U]$ in the $S_y$ basis.

(d) Verify that $\text{Tr}([S_z]') = \text{Tr}([S_z])$.

Problem 24. (Basis change for QHO) A particle in a harmonic oscillator is in the state $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$.

(a) Write the "state vector" (column vector) in the energy basis $\{|0\rangle, |1\rangle, |2\rangle, \ldots\}$.

(b) Using the position completeness relation, write $\psi(x) = \langle x|\psi\rangle$ explicitly in terms of the QHO eigenfunctions $\psi_0(x)$ and $\psi_1(x)$.

(c) Compute $\langle\hat{x}\rangle$ using the Dirac method (ladder operators). Then verify by integration if you wish.

Problem 25. (Unitary time evolution) Show that the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$ is unitary by direct computation of $\hat{U}^\dagger(t)\hat{U}(t)$ using the spectral decomposition of $\hat{H}$.

Problem 26. (Rotation of spin states) The operator $\hat{R}_y(\theta) = e^{-i\theta\hat{S}_y/\hbar}$ rotates a spin-1/2 state by angle $\theta$ about the $y$-axis. In the $S_z$ basis:

$$[R_y(\theta)] = \begin{pmatrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{pmatrix}$$

(a) Verify that $[R_y]$ is unitary.

(b) Apply $\hat{R}_y(\pi/2)$ to $|\uparrow\rangle$. What state do you get? Express it in terms of $|+x\rangle$ and $|-x\rangle$.

(c) Show that $\hat{R}_y(\pi)|\uparrow\rangle = |\downarrow\rangle$ (up to a phase).

(d) Show that $\hat{R}_y(2\pi)|\uparrow\rangle = -|\uparrow\rangle$. This sign change under $2\pi$ rotation is unique to half-integer spin. (We will explore this in Chapter 13.)

Problem 27. (Position-momentum duality as basis change) Define the "position representation" of operator $\hat{A}$ as $A(x, x') = \langle x|\hat{A}|x'\rangle$ and the "momentum representation" as $\tilde{A}(p, p') = \langle p|\hat{A}|p'\rangle$.

(a) Express $\tilde{A}(p, p')$ in terms of $A(x, x')$ by inserting two complete sets of position states.

(b) Evaluate this explicitly for $\hat{A} = \hat{x}$ (position operator). Show that $\langle p|\hat{x}|p'\rangle = i\hbar\frac{\partial}{\partial p}\delta(p - p')$.

(c) Compare with $\langle x|\hat{p}|x'\rangle = -i\hbar\frac{\partial}{\partial x}\delta(x - x')$. Note the beautiful symmetry.

Problem 28. (Active vs. passive transformations) Let $\hat{U}$ be a unitary operator.

(a) Show that the "active" transformation $|\psi\rangle \to \hat{U}|\psi\rangle$ and the "passive" transformation $\hat{A} \to \hat{U}^\dagger\hat{A}\hat{U}$ give the same expectation values: $\langle\psi|\hat{U}^\dagger\hat{A}\hat{U}|\psi\rangle = \langle\psi'|\hat{A}|\psi'\rangle$ where $|\psi'\rangle = \hat{U}|\psi\rangle$.

(b) Explain why this means you can either rotate the state or rotate the operator — the physics is the same.

Section E: The Trace and Functions of Operators (Problems 29--32)

Problem 29. (Trace computations) For the spin-1/2 Pauli matrices:

(a) Compute $\text{Tr}(\sigma_x)$, $\text{Tr}(\sigma_y)$, $\text{Tr}(\sigma_z)$.

(b) Compute $\text{Tr}(\sigma_x \sigma_y)$, $\text{Tr}(\sigma_y \sigma_z)$, $\text{Tr}(\sigma_x \sigma_z)$.

(c) Compute $\text{Tr}(\sigma_x^2)$, $\text{Tr}(\sigma_y^2)$, $\text{Tr}(\sigma_z^2)$.

(d) Show that $\text{Tr}(\sigma_i \sigma_j) = 2\delta_{ij}$.

Problem 30. (Trace and expectation values) Show that the expectation value of $\hat{A}$ in a pure state $|\psi\rangle$ can be written as $\langle\hat{A}\rangle = \text{Tr}(\hat{A}|\psi\rangle\langle\psi|)$. Verify this for $\hat{A} = \hat{S}_z$ and $|\psi\rangle = |\uparrow\rangle$.

Problem 31. (Functions of operators) The Hamiltonian of a spin-1/2 particle in a magnetic field along $z$ is $\hat{H} = -\gamma B \hat{S}_z = \omega_0 \hat{S}_z$ (with $\omega_0 = -\gamma B$).

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

(b) Compute $e^{-i\hat{H}t/\hbar}$ using the spectral decomposition.

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

(d) Show that $\langle\hat{S}_x\rangle(t) = \frac{\hbar}{2}\cos(\omega_0 t)$. This is Larmor precession.

Problem 32. (Cyclic property of the trace) Prove the cyclic property $\text{Tr}(\hat{A}\hat{B}) = \text{Tr}(\hat{B}\hat{A})$ by inserting a complete set of states. Then show by explicit example that $\hat{A}\hat{B} \neq \hat{B}\hat{A}$ in general (the trace is cyclic, but the operators themselves need not commute).

Section F: Rosetta Stone Translations (Problems 33--35)

Problem 33. (Complete translation) Translate each of the following wave mechanics expressions into Dirac notation:

(a) $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$

(b) $\int \psi_m^*(x) \hat{H} \psi_n(x) \, dx = E_n \delta_{mn}$

(c) $\langle\hat{p}^2\rangle = -\hbar^2 \int \psi^*(x) \frac{d^2\psi}{dx^2} dx$

(d) $\psi(x) = \sum_{n=1}^{\infty} c_n \sqrt{\frac{2}{a}} \sin\!\left(\frac{n\pi x}{a}\right)$

(e) $\frac{d\langle\hat{x}\rangle}{dt} = \frac{\langle\hat{p}\rangle}{m}$ (Ehrenfest's theorem)

Problem 34. (Reverse translation) Translate each Dirac expression into wave mechanics:

(a) $\langle\phi|\hat{x}^2|\psi\rangle$

(b) $\sum_n E_n |n\rangle\langle n| = \hat{H}$

(c) $\langle p|\hat{x}|\psi\rangle = i\hbar \frac{\partial}{\partial p} \phi(p)$

(d) $\hat{U}(t)|\psi(0)\rangle = e^{-i\hat{H}t/\hbar}|\psi(0)\rangle$

Problem 35. (The same calculation, two languages) A particle is in the first excited state $|2\rangle$ of an infinite square well of width $a$.

(a) Using wave mechanics, compute $\langle\hat{x}\rangle = \int_0^a \psi_2^*(x) \, x \, \psi_2(x) \, dx$.

(b) Using Dirac notation with completeness insertion, compute $\langle 2|\hat{x}|2\rangle = \sum_{m} \langle 2|\hat{x}|m\rangle \langle m|2\rangle$ by first computing the matrix elements $\langle m|\hat{x}|n\rangle = \int_0^a \psi_m^*(x) \, x \, \psi_n(x) \, dx$ for the relevant values of $m$.

(c) Verify both methods agree. Which was more work for this particular problem?

(d) Now compute $\langle 2|\hat{x}^2|2\rangle$ using whichever method you prefer. Combine with part (a) to find $\sigma_x = \sqrt{\langle\hat{x}^2\rangle - \langle\hat{x}\rangle^2}$.

Challenge Problems

Problem C1. (Generalized uncertainty relation in Dirac notation) Derive the generalized uncertainty relation $\sigma_A \sigma_B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$ entirely in Dirac notation, following these steps:

(a) Define $|f\rangle = (\hat{A} - \langle\hat{A}\rangle)|\psi\rangle$ and $|g\rangle = (\hat{B} - \langle\hat{B}\rangle)|\psi\rangle$.

(b) Show that $\sigma_A^2 = \langle f|f\rangle$ and $\sigma_B^2 = \langle g|g\rangle$.

(d) Show that $\langle f|g\rangle = \frac{1}{2}\langle[\hat{A}, \hat{B}]\rangle + \frac{1}{2}\langle\{\hat{A} - \langle\hat{A}\rangle, \hat{B} - \langle\hat{B}\rangle\}\rangle$.

(e) Use $|\langle f|g\rangle| \geq |\text{Im}\langle f|g\rangle|$ to complete the proof.

Problem C2. (No-cloning theorem — preview of Ch 25) Suppose there exists a unitary operator $\hat{U}$ that "clones" an arbitrary quantum state: $\hat{U}(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle$ for all $|\psi\rangle$.

(a) Apply this to two different states $|\psi\rangle$ and $|\phi\rangle$ and compute $\langle\phi|\psi\rangle$ before and after cloning.

(b) Use the unitarity of $\hat{U}$ (which preserves inner products) to show that $\langle\phi|\psi\rangle = \langle\phi|\psi\rangle^2$.

(c) Show that this implies $\langle\phi|\psi\rangle = 0$ or $\langle\phi|\psi\rangle = 1$ — i.e., the states must be either identical or orthogonal. Conclude that a universal quantum cloner cannot exist.

Problem C3. (Trace of the density operator — preview of Ch 23) Define $\hat{\rho} = |\psi\rangle\langle\psi|$ for a normalized state.

(a) Show that $\hat{\rho}^2 = \hat{\rho}$ and $\text{Tr}(\hat{\rho}) = 1$.

(b) Show that $\langle\hat{A}\rangle = \text{Tr}(\hat{\rho}\hat{A})$.

(c) If $\hat{\rho} = \frac{1}{2}\hat{I}$ (the "maximally mixed state" for spin-1/2), compute $\langle\hat{S}_z\rangle$, $\langle\hat{S}_x\rangle$, and $\langle\hat{S}_y\rangle$. Is $\hat{\rho}^2 = \hat{\rho}$ in this case?

(d) Discuss: what is the physical difference between a pure state and a mixed state?