Quantum Mechanics II

Symmetry, transitions, scattering, and the road to relativistic quantum theory

About this course

Symmetry and rotations, time-dependent perturbation theory and light–atom interaction, scattering theory, open systems and the density matrix, then path integrals, second quantization and the Klein–Gordon and Dirac equations.

What you will be able to do

  • Use rotation group structure and the Wigner–Eckart theorem to evaluate matrix elements
  • Derive transition rates with Fermi's golden rule and state the selection rules for atomic transitions
  • Compute scattering cross sections with the Born approximation and partial waves
  • Describe mixed states with the density matrix and explain decoherence in open systems
  • Quantize a field by second quantization and solve the Dirac equation for a free particle

What this course covers

  • Symmetry as probability-preserving map
  • Two kinds of symmetry: unitary and anti-unitary
  • Time reversal as an anti-linear, anti-unitary map
  • Kramers degeneracy from time-reversal
  • Continuous symmetries and conservation laws
  • Conservation law vs. accidental constancy
  • Selection rules from symmetry
  • Differential cross-section defined from apparatus
  • Attenuation cross-section via exponential beam decay
  • Hard-sphere classical cross-section
  • Form of the outgoing scattered wave
  • Differential cross-section equals |f(θ)|²
  • To-scale visualisation of cross-section areas
  • Cross-section beyond the geometric shadow: wave regime
  • The interaction picture: a change of bookkeeping
  • Integrating the equation of motion exactly
  • First-order amplitude as a Fourier component
  • Three switching regimes: slow, pulsed, and sudden
  • First-order breakdown and self-reporting error
  • Resonant driving: no weak-enough coupling
  • The line-drawing choice and series convergence
  • Five foundational loans
  • Sum over paths forced by drilling screens
  • Path weight as unit-modulus phase
  • Free-particle propagator and its three pins
  • Path-integral construction and its honest limitations
  • Classical world as a width, not a limit
  • Stationary action vs. least action
  • Schrödinger equation recovered from path integral
  • Superposition vs. mixture: the central question
  • State vector runs out: entanglement theorem
  • The density matrix: definition and three properties
  • The diagonal vs. off-diagonal: what the corner encodes
  • Purity and the Bloch ball geometry
  • Partial trace: taking half of an entangled pair
  • Purity is conserved under unitary evolution: the corner cannot shrink
  • The combinatorial cost of labeled identical particles
  • Swap symmetry and the two kinds of matter
  • Occupation-number representation of many-body states
  • Creation and annihilation operators and their square-root norms
  • The Pauli exclusion principle as a line of algebra
  • Direct and exchange terms from the two-body operator
  • Fock space: the occupation-number Hilbert space across all particle numbers
  • Rotations of space: generators and brackets
  • The spin group SU(2): closed form and bracket table
  • The two-to-one map from SU(2) to SO(3)
  • The kernel of the map: two-to-one correspondence
  • Full-turn sign and the two-valued invariant across all J
  • Why spin-½ is allowed: the single-valuedness argument revisited
  • Three factors of two: correcting the 'half-speed' misconception
  • The decoherence problem: why closed-system proof fails
  • Entanglement with the environment and taking the part
  • Decoherence rate and the two timescales
  • Pointer basis: which states survive decoherence
  • Lindblad form: the most general valid master equation
  • Decoherence versus the measurement problem
  • Closing summary and the Markov debt
  • Cross-section as modulus squared of scattering amplitude
  • Lippmann–Schwinger source reformulation
  • Born approximation: substituting the incident beam
  • Scattering amplitude as Fourier transform of the potential
  • Square-well amplitude and shape-reading from diffraction zeros
  • Coulomb scattering and the screened-potential limit
  • Quantum–classical agreement: the Rutherford formula
  • Validity criterion and limits of the Born approximation
  • Why a single decay needs a rate, not just a probability
  • Shape of the single-final-state probability: peak and width
  • Area under the shape is linear in time (Fermi's Golden Rule mechanism)
  • Fermi's Golden Rule: assembling the rate formula
  • Validity window and finite-band correction
  • Why amplitudes — not probabilities — are squared and summed
  • Neutron lifetime: counting final states and comparing with measurement
  • Modal orthogonality and field decomposition
  • Quantizing a field mode: photon as a rung
  • Field operator and locality
  • Zero-point energy and the non-empty vacuum
  • Casimir force: vacuum energy between plates
  • Spontaneous emission as vacuum stimulation
  • Vacuum energy cosmological discrepancy and negative-frequency roots
  • Four inherited results and lecture roadmap
  • Plane wave as a sum over angular momenta
  • Two closed doors: what a spherical potential cannot do
  • Phase shifts as the complete description of spherical scattering
  • Total cross-section, unitarity ceiling, and optical theorem
  • Centrifugal barrier, truncation, and scattering length
  • Hard-sphere phase shift and the ruler picture
  • Transparency of a resonant spherical well (Ramsauer-type)
  • Vector (tensor) operator definition via commutation
  • Selection rule from a single rotation axis
  • Wigner–Eckart theorem derivation
  • Convention-independence of measurable quantities
  • Closed-form Clebsch–Gordan coefficients from the ladder
  • Projection theorem and atomic g-factors
  • One number per rank: limits of the theorem
  • Dipole approximation and the atom-to-wavelength ratio
  • Einstein coefficients and forced spontaneous emission
  • Parity selection rule from inversion symmetry
  • Angular momentum selection rule from the rank-1 tensor property of position
  • Combined electric-dipole selection rules and survivor counting
  • Forbidden transitions and the next term in the expansion
  • "Dark is a rate, not a rule" — the 21-cm line as capstone
  • Geometric phase introduced via Foucault pendulum
  • Dynamical vs. geometric phase: two kinds of phase
  • Adiabatic condition as a dimensionless ratio
  • Leak is second-order in speed
  • Splitting the phase: dynamical and geometric terms
  • Geometric phase equals half the enclosed solid angle
  • Interference fringe as observable measurement of geometric phase
  • Asymmetry of the Schrödinger equation under Lorentz boosts
  • Deriving the Klein-Gordon equation
  • Two-branch energy spectrum and the negative-energy root
  • Conserved density and its negative values on the lower branch
  • Instability from a spectrum with no ground state
  • Compton wavelength as the single-particle breakdown scale
  • Yukawa potential and the Klein-Gordon equation's correct predictions
  • The unitarity ceiling and what saturates it
  • Time delay at resonance
  • Breit-Wigner shape from the straight-line cotangent
  • Resonance as an almost-bound state; width vanishing at threshold
  • ℏ as the width–lifetime exchange rate; reading it from a graph
  • Fano profiles: a real state need not produce a peak
  • Threshold cusps: a bump with no state underneath
  • The Relativistic Mismatch: Why the Schrödinger Equation Breaks
  • The First-Order Demand and the Anti-Commutation Conditions
  • Minimum Matrix Size: Why 4×4 Is Forced
  • Spin-½ as an Unavoidable Consequence
  • Negative-Energy States and the Prediction of Antimatter
  • The Electron's Magnetic Moment: G = 2
  • Residual Disagreement and the Limits of the Equation

Syllabus

  1. Symmetry and Rotations

    1. L1 · Symmetry, and Why It Is the Deepest Idea Here
    2. L2 · Rotations, and the Group Spin Belongs To
    3. L3 · The Wigner–Eckart Theorem

  2. Time-Dependent Phenomena

    1. L4 · Time-Dependent Perturbation Theory
    2. L5 · Fermi's Golden Rule, and Why Rates Are Squared
    3. L6 · Light and Atoms — Absorption, Emission, Selection Rules
    4. L7 · The Adiabatic Theorem and Berry's Phase

  3. Scattering

    1. L8 · Scattering — Cross Sections from First Principles
    2. L9 · The Born Approximation
    3. L10 · Partial Waves and Phase Shifts
    4. L11 · Resonances, and What a Bump Really Is

  4. Mixed States and Open Systems

    1. L12 · The Density Matrix
    2. L13 · Open Systems and Decoherence

  5. Toward Quantum Fields

    1. L14 · The Path Integral
    2. L15 · Second Quantization
    3. L16 · Quantum Fields, Gently
    4. L17 · Klein–Gordon and the Trouble with Relativity
    5. L18 · The Dirac Equation, and Where Antimatter Came From