# PHYS 130B Homework Solutions

← <a href="../index.html">Back to the PHYS 130B lecture notes</a>

A self-contained collection of homework problems with worked solutions, one
page per lesson. This index lists the problems for each lesson so you can
jump directly to the one you need.

## Contents

### Chapter 1 — Qubit

**[1.1.1 What is a Qubit](1-1-1_solutions.md)**

1. Born rule and probabilities
2. State reconstruction from probabilities
3. Global phase invariance
4. Bloch sphere parametrization
5. Quantum information
6. Distinguishing non-orthogonal states
7. Photon polarization vs. transmon qubit

**[1.1.2 State and Representation](1-1-2_solutions.md)**

1. Qubit state decomposition
2. State reconstruction from Pauli expectations
3. State representation in different bases
4. Bloch vector calculation
5. Antipodal states on Bloch sphere
6. Bloch vector rotation
7. Spin expectation along an arbitrary axis
8. Parametrization of n-level systems

**[1.1.3 Hermitian Operators](1-1-3_solutions.md)**

1. Spectral decomposition of a two-level Hamiltonian
2. Hermitian and anti-Hermitian decomposition
3. Bloch vector from amplitudes
4. Eigenstate of spin along an arbitrary axis
5. Pauli decomposition of a 2x2 Hermitian operator
6. Non-Hermitian eigenstates
7. Pauli square along an arbitrary axis
8. Pauli multiplication from commutator and anti-commutator

**[1.2.1 Measurement Postulate](1-2-1_solutions.md)**

1. Measurement probabilities and collapse
2. Sequential measurements and filters
3. Measurement of a non-Pauli observable
4. State inference from measurement frequencies
5. Phase erasure by complementary measurement
6. Quantum interference
7. Filter vs measurement misconception

**[1.2.2 Uncertainty and Incompatibility](1-2-2_solutions.md)**

1. Most general operator commuting with Z
2. Robertson relation on a specific state
3. Spin commutator along arbitrary axes
4. Total Pauli uncertainty for a pure qubit
5. Sharp observable does not violate Heisenberg
6. Saturation and maximum of the Robertson bound
7. Maximising the Z-uncertainty
8. Minimum of the X-Z uncertainty product

**[1.2.3 Measurement Operators](1-2-3_solutions.md)**

1. Spin-axis projector
2. Born rule via projector formula
3. Spectral decomposition
4. **Two-qubit measurement**
5. Sequential projectors and non-commutativity
6. Repeatability from idempotence
7. Operator functions via spectral decomposition

**[1.3.1 Unitary Evolution](1-3-1_solutions.md)**

1. Conservation of observables commuting with H
2. Exponential of a Pauli matrix
3. Extracting the generator
4. Two-level Hamiltonian
5. Normalization and unitarity
6. Unitary as Bloch-sphere rotation
7. Superposition time evolution
8. Composition and the time-evolution group
9. **Observable evolution in time**

**[1.3.2 Schrödinger Picture](1-3-2_solutions.md)**

1. General Bloch precession
2. Larmor evolution of a Y-basis eigenstate
3. Survival probability
4. Two-pulse phase-accumulation sequence
5. Pi-pulse implements the NOT gate
6. Sequential pulses about different axes
7. Larmor precession vs Rabi oscillation
8. **Three-state evolution**

**[1.3.3 Heisenberg Picture](1-3-3_solutions.md)**

1. Picture equivalence on a concrete example
2. Heisenberg evolution under a tilted Hamiltonian
3. Pauli precession as a cross product
4. **Harmonic oscillator dynamics**
5. Conservation at the expectation level
6. SU(2) generator and operator rotation
7. Cyclic evolution and the half-angle
8. Algebra of conserved quantities

### Chapter 2 — Identical Particles

**[2.1.1 Tensor Product](2-1-1_solutions.md)**

1. Identifying entangled states
2. Operator products via the mixed-product rule
3. **The SWAP operator**
4. Expectation in product states
5. Single-body and two-body Z measurements
6. Heisenberg interaction matrix
7. Pauli string decomposition
8. Hilbert-space parametrization

**[2.1.2 Symmetrization](2-1-2_solutions.md)**

1. Three-particle statistics
2. Exchange projectors
3. Generalized Pauli exclusion
4. Bosonic normalization factor
5. Insertion and deletion as inverses
6. Slater determinant overlap
7. Fermion counting
8. Boson counting

**[2.1.3 Second Quantization](2-1-3_solutions.md)**

1. Fock states from the vacuum
2. Number-ladder commutators
3. Coherent states
4. Stimulated emission and the boson enhancement factor
5. Slater determinant from creation operators
6. Particle-hole conjugation
7. Two-mode boson tunnelling
8. Fock space dimensions
9. **Equal partition theorem**
10. **Schwinger boson**

**[2.2.1 Angular Momentum Algebra](2-2-1_solutions.md)**

1. Spin-1 verification of the angular-momentum algebra
2. j=3/2 multiplet by repeated lowering
3. Ladder formula via Schwinger bosons
4. Ladder action: termination and verification
5. Transverse variance on an angular-momentum eigenstate
6. Robertson uncertainty for angular momentum
7. Vector model and the tilt angle
8. **Quantum bootstrap**

**[2.2.2 Spin Representations](2-2-2_solutions.md)**

1. Stern-Gerlach for spin-1
2. Spin-1 rotation operator via polynomial truncation
3. Spinor rotation by ninety degrees about x
4. Spinor 2-pi rotation and neutron interferometry
5. Spin-j rotation: integer vs half-integer
6. Higher spin representations
7. **Spin-1 time evolution**

**[2.2.3 Addition of Angular Momenta](2-2-3_solutions.md)**

1. Angular momentum addition
2. Total angular momentum
3. Coupling scheme
4. **Spin-orbit coupling**
5. **Spin-1 and spin-1/2 coupling**
6. Lande g-factor
7. Two-electron exchange interaction

### Chapter 3 — Path Integral

**[3.1.1 Geometric Optics](3-1-1_solutions.md)**

1. Reflection from Fermat
2. Total internal reflection
3. Stratified index gradient
4. Stationary versus minimum
5. Corpuscle versus wave speed
6. Curved mirror focusing
7. Optics-mechanics dictionary

**[3.1.2 Physical Optics](3-1-2_solutions.md)**

1. Refraction at an interface
2. Reflection from wavefronts
3. Layered medium ray tracing
4. Interference of two paths
5. Slab in an interferometer

**[3.1.3 Wave-Particle Duality](3-1-3_solutions.md)**

1. Free particle action
2. De Broglie wavelength
3. Optics-mechanics translation
4. Dimensions of action
5. Path cancellation
6. Relativistic action limit

**[3.2.1 Path Integral Formulation](3-2-1_solutions.md)**

1. **Wavepacket spreading**
2. Composition test
3. Free-particle slice action
4. Slice action with a potential
5. Phase difference between nearby slices
6. Functional equation from composition

**[3.2.2 Schrödinger Equation](3-2-2_solutions.md)**

1. Zeroth Gaussian moment
2. Second Gaussian moment
3. Verifying the slice normalization
4. Slice propagator approaches a delta function
5. **Probability conservation**
6. Higher moments do not matter
7. Phase-space spreading: loss of minimum uncertainty
8. Potential placement freedom

**[3.2.3 Free Particle Propagator](3-2-3_solutions.md)**

1. Dispersion relation
2. Momentum-space form of the free propagator
3. Direct Gaussian integration
4. Slice agreement (1D)
5. Action conjugate relations
6. Phase velocity vs group velocity
7. Magnitude and normalization
8. Macroscopic phase factors

**[3.3.1 Stationary Phase Approximation](3-3-1_solutions.md)**

1. Gaussian check
2. Cubic phase
3. Stationary-region width
4. Second variation
5. Validity breakdown
6. Mexican-hat saddles
7. Harmonic oscillator from SPA
8. Correspondence principle

**[3.3.2 WKB Approximation](3-3-2_solutions.md)**

1. Hamilton-Jacobi from WKB
2. Allowed and forbidden regions
3. Validity criterion
4. Square-barrier tunneling (heuristic)
5. Penetration depth
6. Airy connection formula
7. **Double-well tunnel splitting**

**[3.3.3 Bohr-Sommerfeld Quantization](3-3-3_solutions.md)**

1. Harmonic oscillator
2. Particle in a box
3. Linear potential
4. Anharmonic oscillator
5. Two-dimensional anisotropic oscillator
6. Hydrogen atom (circular orbits)
7. Correspondence principle

### Chapter 4 — Phase and Gauge

**[4.1.1 Gauge Principle](4-1-1_solutions.md)**

1. Diagnosing redundancy
2. Plane-wave gauge transform
3. Covariance of derivatives
4. Gauge on punctured plane
5. What gauge transformations cannot do
6. **Second-order bilinear covariance**
7. Gauge connection on links

**[4.1.2 Electromagnetic Coupling](4-1-2_solutions.md)**

1. Velocity for the Pauli Hamiltonian
2. Velocity uncertainty in B
3. **Cyclotron motion from Heisenberg**
4. Symmetric gauge verification
5. Field strength tensor components

**[4.1.3 Gauge Invariance](4-1-3_solutions.md)**

1. Gauge-transforming an observable
2. Probability current
3. Canonical momentum shifts
4. Energy under gauge transformation
5. Coulomb gauge residual freedom
6. Gauge-invariant classification

**[4.2.1 Berry Phase](4-2-1_solutions.md)**

1. Static orthogonality
2. Reality of the Berry connection
3. Gauge transformation of the connection
4. Spin-1 Berry phase
5. Two gauges on Bloch sphere
6. Berry phase in Bloch band

**[4.2.2 Aharonov-Bohm Effect](4-2-2_solutions.md)**

1. Invisible solenoid
2. Which path matters
3. Gauge dependence of single paths
4. Surface independence
5. AB phase as Berry phase
6. Electric Aharonov-Bohm
7. Toroidal magnet

**[4.2.3 Flux Ring](4-2-3_solutions.md)**

1. Persistent current at finite temperature
2. Many fermions on the ring
3. **Half-flux doublet under perturbation**
4. Cooper pair versus electron
5. Flux-dependent tunneling
6. SQUID magnetometry
7. Ground state phase transition
8. Dimensional analysis

### Chapter 5 — Perturbation Theory

**[5.1.1 Toy Model](5-1-1_solutions.md)**

1. Accuracy window
2. Inverse design
3. Basis rotation: energies vs states
4. Controlled asymmetry
5. Avoided crossing width
6. Misconception test
7. Near-degeneracy diagnosis

**[5.1.2 Non-degenerate Perturbation Theory](5-1-2_solutions.md)**

1. Gauge choice and normalization convention
2. **Three-level perturbation**
3. Coupling over gap
4. Second-order energy correction and sign
5. Diagonal and off-diagonal perturbation
6. **Harmonic oscillator with linear perturbation**
7. Selection rules and parity
8. Near-degeneracy and breakdown

**[5.1.3 Degenerate Perturbation Theory](5-1-3_solutions.md)**

1. Why the old formula fails
2. Block first, levels later
3. Effective Hamiltonian and dark state
4. Hydrogen Stark splitting
5. Residual degeneracy
6. When to switch methods
7. **Flux ring with cosine perturbation**

**[5.2.1 Interaction Picture](5-2-1_solutions.md)**

1. Expectations across pictures
2. Transition frequencies
3. Choosing a picture
4. Operator-picture transformation
5. Two-level system under monochromatic drive
6. Misconception check

**[5.2.2 Dyson Series](5-2-2_solutions.md)**

1. Volterra integral form
2. Iteration to second order
3. Time-ordering identity
4. Bare Green's function
5. Schrödinger-picture Dyson series
6. Recursive Dyson equation
7. Two-level Feynman diagrams
8. **Three-level virtual transition**

**[5.2.3 Applications](5-2-3_solutions.md)**

1. Phase cancellation
2. Sinc-squared properties
3. Sinc-to-delta
4. Density of states
5. Adiabatic ramp Lorentzian
6. Adiabatic to static perturbation
7. **Three-level Raman (long-time limit)**
8. Minimal Kubo exercise