Chapter 10 Exercises

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 10 Exercises

Notation and Conventions

Unless otherwise stated: $\hbar = 1$ in algebraic problems. Operators carry hats ($\hat{A}$). The canonical commutation relations are $[\hat{x}_i, \hat{p}_j] = i\hbar\delta_{ij}$. The Levi-Civita symbol is $\epsilon_{ijk}$. Answers to selected problems appear in Appendix G.

Section A: Symmetry Transformations and Unitary Operators (Problems 1--7)

Problem 1. (Warm-up: unitarity verification) Consider the translation operator $\hat{T}(a) = e^{-i\hat{p}a/\hbar}$ for a one-dimensional system.

(a) Show that $\hat{T}(a)$ is unitary by verifying $\hat{T}^\dagger(a)\hat{T}(a) = \hat{I}$. Use the Hermiticity of $\hat{p}$.

(b) Verify the group property: $\hat{T}(a)\hat{T}(b) = \hat{T}(a + b)$.

(c) Find $\hat{T}^{-1}(a)$ and verify that $\hat{T}^{-1}(a) = \hat{T}(-a)$.

(d) In the position basis, show that $\langle x|\hat{T}(a)|\psi\rangle = \psi(x - a)$ by inserting a completeness relation and using $\hat{T}(a)|x\rangle = |x + a\rangle$.

Problem 2. (Transformation of operators) If $\hat{U}$ is a unitary symmetry transformation, the transformed operator is $\hat{A}' = \hat{U}\hat{A}\hat{U}^\dagger$.

(a) Show that the eigenvalues of $\hat{A}$ and $\hat{A}'$ are the same. (Hint: if $\hat{A}|a\rangle = a|a\rangle$, what is $\hat{A}'\hat{U}|a\rangle$?)

(b) Show that $[\hat{A}', \hat{B}'] = \hat{U}[\hat{A}, \hat{B}]\hat{U}^\dagger$. Conclude that commutation relations are preserved under unitary transformations.

(c) Using the result of (b) and $[\hat{x}, \hat{p}] = i\hbar$, verify that $[\hat{x}', \hat{p}'] = i\hbar$ for any unitary transformation.

Problem 3. (Infinitesimal generator extraction) A continuous family of unitary operators $\hat{U}(\theta)$ satisfies $\hat{U}(0) = \hat{I}$.

(a) If $\hat{U}(\theta) = e^{-i\theta\hat{G}/\hbar}$, show that $\hat{G} = i\hbar\frac{d\hat{U}}{d\theta}\bigg|_{\theta=0}$.

(b) For the rotation operator about $z$: $\hat{R}_z(\phi)$ acts on position eigenstates as $\hat{R}_z(\phi)|x, y, z\rangle = |x\cos\phi - y\sin\phi, x\sin\phi + y\cos\phi, z\rangle$. Compute $\frac{d}{d\phi}\hat{R}_z(\phi)\bigg|_{\phi=0}$ acting on $\psi(x,y,z)$ and identify the generator.

(c) Verify that the generator you found in (b) is $\hat{L}_z/\hbar$.

Problem 4. (Operator transformation under translation) Show that translation by $a$ transforms the position and momentum operators as:

(a) $\hat{T}^\dagger(a)\hat{x}\hat{T}(a) = \hat{x} + a\hat{I}$. (Use the Baker-Campbell-Hausdorff lemma: $e^{\hat{A}}\hat{B}e^{-\hat{A}} = \hat{B} + [\hat{A}, \hat{B}] + \frac{1}{2!}[\hat{A}, [\hat{A}, \hat{B}]] + \cdots$.)

(b) $\hat{T}^\dagger(a)\hat{p}\hat{T}(a) = \hat{p}$.

(c) Interpret physically: translation shifts position but not momentum. Why does this make sense classically?

Problem 5. (Composition of symmetry operations) Consider a system with Hamiltonian $\hat{H}$ that is invariant under two symmetry transformations $\hat{U}_1$ and $\hat{U}_2$: $[\hat{H}, \hat{U}_1] = [\hat{H}, \hat{U}_2] = 0$.

(a) Prove that the composition $\hat{U}_1\hat{U}_2$ is also a symmetry.

(b) Prove that $\hat{U}_1^{-1}$ is a symmetry.

(c) If $\hat{G}_1$ and $\hat{G}_2$ are the generators of $\hat{U}_1$ and $\hat{U}_2$, is $\hat{G}_1 + \hat{G}_2$ necessarily the generator of $\hat{U}_1\hat{U}_2$? Under what condition is this true?

(d) If $[\hat{G}_1, \hat{G}_2] \neq 0$, prove that $[\hat{G}_1, \hat{G}_2]$ also commutes with $\hat{H}$. (This is how the angular momentum algebra closes.)

Problem 6. (Galilean boost) The Galilean boost operator transforms to a frame moving with velocity $v$: $\hat{B}(v) = e^{-imv\hat{x}/\hbar}$.

(a) Show that $\hat{B}^\dagger(v)\hat{p}\hat{B}(v) = \hat{p} + mv$.

(b) Show that $\hat{B}^\dagger(v)\hat{x}\hat{B}(v) = \hat{x}$.

(c) The free particle Hamiltonian is $\hat{H} = \hat{p}^2/2m$. Show that $\hat{B}^\dagger(v)\hat{H}\hat{B}(v) \neq \hat{H}$. Conclude that the Galilean boost is not a symmetry of $\hat{H}$. What does this mean for the conservation of $m\hat{x}$?

(d) Show that $[\hat{T}(a), \hat{B}(v)] \neq 0$ and compute the commutator. What does this tell you about the structure of the Galilean group?

Problem 7. (Time evolution as a symmetry) Time evolution is generated by $\hat{H}$: $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$.

(a) What is the conserved quantity associated with time-translation symmetry?

(b) Under what conditions is energy conserved? (Think about explicit time dependence of $\hat{H}$.)

(c) The Hamiltonian generates time translation, momentum generates space translation, angular momentum generates rotation. In each case, identify the general pattern: the generator of a symmetry transformation is the conserved quantity associated with that symmetry. Why is this not circular?

Section B: Quantum Noether Theorem and Conservation Laws (Problems 8--14)

Problem 8. (Direct verification of Noether's theorem) Consider a particle in a one-dimensional potential $V(x)$.

(a) Compute $[\hat{H}, \hat{p}]$ where $\hat{H} = \hat{p}^2/2m + V(\hat{x})$. Express the result in terms of $V'(\hat{x}) = dV/dx$.

(b) For $V(x) = \frac{1}{2}m\omega^2 x^2$, evaluate $[\hat{H}, \hat{p}]$. Is momentum conserved?

(c) For $V(x) = V_0$ (constant), evaluate $[\hat{H}, \hat{p}]$. Is momentum conserved?

(d) Using the Ehrenfest theorem $\frac{d}{dt}\langle\hat{p}\rangle = -\langle V'(\hat{x})\rangle$, interpret the result of (b) physically. How does it relate to Newton's second law?

Problem 9. (Conservation in a central potential) For a particle in a central potential $V(r)$ in three dimensions:

(a) Show that $[\hat{H}, \hat{L}_z] = 0$ by computing the commutator directly. (Use $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$ and the canonical commutation relations.)

(b) Show that $[\hat{H}, \hat{L}^2] = 0$. (This is harder. You may use the result $[\hat{L}_i, \hat{p}^2] = 0$ and $[\hat{L}_i, r^2] = 0$ without proof, or prove them as a bonus.)

(c) What are the conserved quantities? How do they manifest in the quantum numbers of the hydrogen atom?

Problem 10. (Two-body system) Two particles interact via a potential that depends only on their separation: $V = V(|\mathbf{r}_1 - \mathbf{r}_2|)$.

(a) Show that the total momentum $\hat{\mathbf{P}} = \hat{\mathbf{p}}_1 + \hat{\mathbf{p}}_2$ commutes with $\hat{H}$.

(b) Is the individual momentum $\hat{\mathbf{p}}_1$ conserved? Prove or disprove.

(c) Show that the total angular momentum $\hat{\mathbf{L}} = \hat{\mathbf{L}}_1 + \hat{\mathbf{L}}_2$ is conserved.

(d) What symmetry corresponds to conservation of total momentum? What symmetry corresponds to conservation of total angular momentum?

Problem 11. (Degeneracy from symmetry) Let $\hat{U}$ be a symmetry: $[\hat{H}, \hat{U}] = 0$. Let $|E\rangle$ be a non-degenerate eigenstate of $\hat{H}$ with energy $E$.

(a) Show that $\hat{U}|E\rangle$ is also an eigenstate with energy $E$.

(b) Since the eigenstate is non-degenerate, conclude that $\hat{U}|E\rangle = c|E\rangle$ for some $c$. Show that $|c| = 1$.

(c) What does this mean? A non-degenerate energy eigenstate must be an eigenstate of every symmetry operator. Verify this for the QHO: the non-degenerate ground state $|0\rangle$ is an eigenstate of $\hat{\Pi}$ with eigenvalue $+1$.

(d) Now suppose $|E\rangle$ is degenerate (multiplicity $d$). Show that $\hat{U}$ maps the $d$-dimensional eigenspace into itself. Conclude that the matrix representation of $\hat{U}$ in this eigenspace is a $d \times d$ unitary matrix.

Problem 12. (Simultaneous eigenstates) Consider $\hat{H} = \hat{p}^2/2m + \frac{1}{2}m\omega^2 x^2$ (QHO).

(a) Verify that $[\hat{H}, \hat{\Pi}] = 0$ by showing that $V(-x) = V(x)$ and that $\hat{\Pi}\hat{p}^2\hat{\Pi} = \hat{p}^2$.

(b) The QHO eigenstates are $|n\rangle$ with $n = 0, 1, 2, \ldots$. Show that $\hat{\Pi}|n\rangle = (-1)^n|n\rangle$. (Hint: use the Hermite polynomial relation $H_n(-\xi) = (-1)^n H_n(\xi)$, or use the ladder operator definition $|n\rangle = (\hat{a}^\dagger)^n|0\rangle/\sqrt{n!}$ with $\hat{\Pi}\hat{a}^\dagger\hat{\Pi} = -\hat{a}^\dagger$.)

(c) What is $\langle m|\hat{x}|n\rangle$ when $m + n$ is even? Prove your answer using the parity selection rule.

Problem 13. (Energy-time Noether connection) The time-evolution operator is $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$.

(a) What symmetry does $\hat{H}$ generate?

(b) For a time-independent Hamiltonian, show that $\frac{d}{dt}\langle\hat{H}\rangle = 0$ — energy is conserved.

(c) If $\hat{H}$ depends explicitly on time, $\hat{H} = \hat{H}(t)$, does $\frac{d}{dt}\langle\hat{H}\rangle = 0$ still hold? What additional term appears?

(d) Give a physical example where energy is not conserved (i.e., where $\hat{H}$ depends on $t$). Identify which symmetry is broken.

Problem 14. (Building intuition: symmetry identification) For each of the following Hamiltonians, identify all symmetries and the corresponding conserved quantities:

(a) Free particle in 3D: $\hat{H} = \hat{\mathbf{p}}^2/2m$.

(b) Particle in a uniform electric field: $\hat{H} = \hat{p}^2/2m + eEx$.

(c) Particle in a central potential: $\hat{H} = \hat{p}^2/2m + V(r)$.

(d) Particle in a cylindrically symmetric potential: $\hat{H} = \hat{p}^2/2m + V(\rho, z)$ where $\rho = \sqrt{x^2 + y^2}$.

(e) Particle on a ring: $\hat{H} = \hat{L}_z^2/2I$.

(f) 1D particle in a double well: $\hat{H} = \hat{p}^2/2m + V_0(x^4 - ax^2)$ with $a > 0$.

Section C: Parity (Problems 15--21)

Problem 15. (Parity of standard operators) Determine the parity (even or odd) of each operator:

(a) $\hat{x}^3$

(b) $\hat{p}^2$

(c) $\hat{x}\hat{p} + \hat{p}\hat{x}$

(d) $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$

(e) $\hat{\mathbf{r}} \cdot \hat{\mathbf{p}}$

(f) $\hat{L}^2$

Problem 16. (Parity selection rules) A system with Hamiltonian $\hat{H}$ is parity-invariant: $[\hat{H}, \hat{\Pi}] = 0$. The energy eigenstates $|n\rangle$ have definite parity $\pi_n = \pm 1$.

(a) Show that $\langle n|\hat{x}|n\rangle = 0$ for any energy eigenstate with definite parity. (Use the parity selection rule.)

(b) More generally, show that $\langle n|\hat{A}|m\rangle = 0$ whenever $\hat{A}$ is parity-odd and $|n\rangle$, $|m\rangle$ have the same parity.

(c) For the QHO, which matrix elements $\langle m|\hat{x}^2|n\rangle$ are non-zero? (Hint: $\hat{x}^2$ is parity-even.)

(d) The electric dipole moment operator is $\hat{d} = q\hat{x}$. Using parity, explain why atoms in energy eigenstates do not have permanent electric dipole moments (in the absence of degeneracy).

Problem 17. (Parity and the infinite square well) The infinite square well has $V(x) = 0$ for $0 < x < a$ and $V = \infty$ otherwise.

(a) As written, the potential is not symmetric about $x = 0$. Redefine coordinates so that the well is symmetric: $V(x) = 0$ for $-a/2 < x < a/2$. Write the energy eigenstates in the new coordinates.

(b) Verify that the eigenstates separate into even and odd classes. What are their parities?

(c) Compute $\langle n|\hat{x}|m\rangle$ for the first four states ($n, m = 1, 2, 3, 4$). Verify that the parity selection rule is satisfied.

Problem 18. (Parity in 3D) The parity of the hydrogen atom eigenstates $|n, l, m\rangle$ is $(-1)^l$.

(a) What is the parity of the $1s$, $2s$, $2p$, $3d$, and $4f$ states?

(b) Using the parity selection rule, show that electric dipole transitions require $\Delta l = \pm 1$. (Hint: the electric dipole operator $\hat{d} = -e\hat{\mathbf{r}}$ is parity-odd.)

(c) Can the transition $3d \to 1s$ occur via electric dipole radiation? What about $3d \to 2p$?

(d) Magnetic dipole transitions involve the operator $\hat{\mathbf{L}}$, which is parity-even. What is the parity selection rule for magnetic dipole transitions?

Problem 19. (Parity and degeneracy) Prove that if $[\hat{H}, \hat{\Pi}] = 0$ and the energy spectrum is non-degenerate, then every energy eigenstate has definite parity.

(a) Start with $\hat{H}|E\rangle = E|E\rangle$ and apply $\hat{\Pi}$ to both sides.

(b) Use non-degeneracy to conclude $\hat{\Pi}|E\rangle = c|E\rangle$.

(c) Use $\hat{\Pi}^2 = \hat{I}$ to show $c = \pm 1$.

(d) Give an example where degenerate eigenstates need not have definite parity. (Consider the free particle and states $e^{ikx}$.)

Problem 20. (Perturbation and broken parity) The QHO has parity symmetry. Now add a perturbation $\hat{H}' = \lambda\hat{x}^3$.

(a) Does the perturbed Hamiltonian commute with $\hat{\Pi}$? Why or why not?

(b) The first-order energy correction is $E_n^{(1)} = \langle n|\hat{H}'|n\rangle$. Use parity to show that $E_n^{(1)} = 0$ for all $n$.

(c) The second-order correction involves $|\langle m|\hat{H}'|n\rangle|^2$. Which matrix elements $\langle m|\hat{x}^3|n\rangle$ are non-zero? (Determine the parity constraint.)

(d) This is a preview of perturbation theory (Chapter 17): parity can tell you which corrections vanish without computing any integrals. Identify one more perturbation $\hat{H}'$ for which all first-order corrections vanish by parity.

Problem 21. (Parity in the Dirac equation — preview) In relativistic quantum mechanics (Chapter 29), the Dirac equation has a parity operator $\hat{\Pi}_D = \gamma^0\hat{\Pi}$, where $\gamma^0$ is a $4 \times 4$ matrix. The intrinsic parity of a spin-1/2 particle is $+1$, and the intrinsic parity of its antiparticle is $-1$.

(a) Why must antiparticles have opposite intrinsic parity to particles? (Hint: think about electron-positron annihilation into photons and parity conservation.)

(b) A $\pi^0$ meson decays into two photons: $\pi^0 \to \gamma\gamma$. Photons have intrinsic parity $-1$ (since the electromagnetic field is a polar vector). What is the parity of the $\pi^0$? (Consider the final state parity.)

(c) Why is this question labeled "preview"? What do you need from Chapter 29 to make this fully rigorous?

Section D: Time Reversal and Advanced Topics (Problems 22--28)

Problem 22. (Antilinearity basics) The time reversal operator $\hat{\Theta}$ is antilinear: $\hat{\Theta}(c|\psi\rangle) = c^*\hat{\Theta}|\psi\rangle$.

(a) Show that $\hat{\Theta}(|\psi\rangle + |\phi\rangle) = \hat{\Theta}|\psi\rangle + \hat{\Theta}|\phi\rangle$ (additivity is preserved).

(b) Show that for an antiunitary operator, $\langle\hat{\Theta}\psi|\hat{\Theta}\phi\rangle = \langle\psi|\phi\rangle^* = \langle\phi|\psi\rangle$.

(c) Explain why you cannot define eigenvalues of an antilinear operator in the usual sense. (Hint: if $\hat{\Theta}|\psi\rangle = \lambda|\psi\rangle$, consider $\hat{\Theta}(c|\psi\rangle)$.)

Problem 23. (Time reversal of a spinless particle) For a spinless particle, the time reversal operator is simply complex conjugation in the position representation: $\hat{\Theta}\psi(x) = \psi^*(x)$.

(a) Verify that $\hat{\Theta}\hat{x}\hat{\Theta}^{-1} = \hat{x}$.

(b) Verify that $\hat{\Theta}\hat{p}\hat{\Theta}^{-1} = -\hat{p}$. (Use $\hat{p} = -i\hbar\partial/\partial x$ and the antilinearity of $\hat{\Theta}$.)

(c) Show that $\hat{\Theta}^2 = \hat{I}$ for a spinless particle.

(d) If $\hat{H}$ is real (no magnetic field), show that $[\hat{H}, \hat{\Theta}] = 0$ in the sense that $\hat{\Theta}\hat{H}\hat{\Theta}^{-1} = \hat{H}$. Conclude that for every eigenstate $\psi(x)$, $\psi^*(x)$ is also an eigenstate with the same energy.

Problem 24. (Kramers' theorem) For a spin-1/2 particle, $\hat{\Theta} = -i\hat{\sigma}_y\hat{K}$, where $\hat{K}$ is complex conjugation.

(a) Compute $\hat{\Theta}^2$ and show it equals $-\hat{I}$. (Use $\hat{\sigma}_y^2 = \hat{I}$ and the antilinearity of $\hat{K}$.)

(b) Using $\hat{\Theta}^2 = -\hat{I}$, prove that $\langle\psi|\hat{\Theta}\psi\rangle = 0$ for any state $|\psi\rangle$.

(c) If $[\hat{H}, \hat{\Theta}] = 0$, show that $|E\rangle$ and $\hat{\Theta}|E\rangle$ are degenerate and orthogonal. This is Kramers' theorem.

(d) A magnetic field breaks time-reversal symmetry. Explain why, and describe how this lifts Kramers' degeneracy (Zeeman effect).

Problem 25. (Symmetry-based problem solving) Use the symmetry toolbox to answer the following without solving any differential equations:

(a) A particle is in the ground state of a parity-symmetric potential. What is $\langle\hat{x}\rangle$?

(b) A particle is in a central potential. What is $\langle\hat{L}_x\rangle$ for the state $|n, l, m\rangle$?

(c) The QHO has eigenstates $|n\rangle$. Using the parity of $\hat{x}^3$, determine which matrix elements $\langle m|\hat{x}^3|n\rangle$ are non-zero.

(d) A free particle has definite momentum $|p\rangle$. What is $\langle\hat{x}\rangle$? (Be careful — this state is not normalizable.)

Problem 26. (Bloch's theorem verification) Consider a particle in a periodic potential $V(x) = V(x + a)$ in one dimension.

(a) Show that $[\hat{H}, \hat{T}(a)] = 0$.

(b) The eigenvalue of $\hat{T}(a)$ is $e^{ika}$ for some $k$. Why must the eigenvalue have modulus 1?

(c) Write the general form of the Bloch wave $\psi_k(x) = e^{ikx}u_k(x)$ and verify that $\hat{T}(a)\psi_k(x) = e^{ika}\psi_k(x)$ if and only if $u_k(x + a) = u_k(x)$.

(d) Why is $k$ only defined up to $2\pi/a$? (Hint: show that $k$ and $k + 2\pi/a$ give the same eigenvalue of $\hat{T}(a)$.)

Problem 27. (The symmetry group of the QHO) The QHO Hamiltonian $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2)$ has more symmetry than parity alone.

(a) Define $\hat{N} = \hat{a}^\dagger\hat{a}$. Show that $[\hat{H}, \hat{N}] = 0$. What is the conserved quantity? Interpret it physically.

(b) The operator $e^{i\theta\hat{N}}$ is unitary. Show that it implements a phase rotation: $e^{i\theta\hat{N}}\hat{a}e^{-i\theta\hat{N}} = e^{-i\theta}\hat{a}$. (Use the BCH formula with $[\hat{N}, \hat{a}] = -\hat{a}$.)

(c) This is a $U(1)$ symmetry. At what value of $\theta$ does $e^{i\theta\hat{N}}$ reduce to the parity operator $\hat{\Pi}$?

(d) In quantum field theory (Chapter 34), this $U(1)$ symmetry becomes the symmetry associated with particle number conservation. Comment on the significance: the QHO already contains the seed of field theory's most fundamental symmetry.

Problem 28. (Capstone: symmetry analysis of a physical system) A charged particle (charge $q$, mass $m$) moves in a combined Coulomb and uniform magnetic field:

$$\hat{H} = \frac{1}{2m}\left(\hat{\mathbf{p}} - \frac{q}{c}\mathbf{A}\right)^2 - \frac{Ze^2}{r}$$

where $\mathbf{A} = \frac{1}{2}\mathbf{B} \times \mathbf{r}$ and $\mathbf{B} = B\hat{z}$.

(a) Which continuous symmetries of the hydrogen atom ($\mathbf{B} = 0$) survive when $\mathbf{B} \neq 0$? Which are broken?

(b) Which component(s) of angular momentum are conserved when $\mathbf{B} \neq 0$?

(c) Is parity preserved? Justify your answer.

(d) Is time-reversal symmetry preserved? (Hint: how does $\mathbf{B}$ transform under time reversal?)

(e) What can you predict about the degeneracy pattern of the energy levels in the magnetic field, compared to $\mathbf{B} = 0$? Relate this to the Zeeman effect.

(f) If $B$ is very small, perturbation theory (Chapter 17) applies. Without computing anything, use symmetry to predict which first-order matrix elements $\langle n', l', m'|\hat{H}'|n, l, m\rangle$ are non-zero, where $\hat{H}' = -\frac{qB}{2mc}\hat{L}_z$.