Quantum Physics III

What this course covers

• Perturbation Theory Motivation
• Perturbation Hamiltonian Structure
• The Lambda Parameter
• Non-Degenerate vs Degenerate Energy Levels
• Perturbation-Induced Splitting of Degeneracy
• Non-Degenerate Perturbation Theory Setup
• Power-Series Expansion in λ
• Perturbation theory series expansion
• Schrödinger equation in λ
• Coefficient matching by order in λ
• Zeroth-order equation
• First-order equation and unknowns
• Orthogonality condition on corrected states
• First-order energy correction via projection
• Recursive structure of higher-order equations
• Non-normalization of perturbed states
• First-order energy correction formula
• Energy correction without state correction
• Kth-order energy correction formula
• Vector equation component condition
• Projection onto unperturbed states
• Matrix elements of the perturbation
• Energy denominators
• Non-degeneracy requirement
• First-order state correction
• Second-order energy correction
• Perturbation theory expansion summary
• Perturbation theory formula review
• Variational principle and ground-state energy overestimate
• Second-order correction sign for ground state
• Level repulsion from second-order corrections
• Validity of the perturbation series
• 2×2 matrix example for perturbation theory
• Taylor expansion of the exact eigenvalue formula
• Radius of convergence of the perturbation series
• Anharmonic oscillator definition
• X⁴ perturbation term
• Natural length scale of the harmonic oscillator
• Dimensionally consistent perturbation Hamiltonian
• Expressing x in terms of ladder operators
• First-order energy correction computation
• Selection rules from operator parity
• Second-order energy correction computation
• Asymptotic versus convergent series
• Failure of non-degenerate perturbation theory
• Ambiguity of eigenvectors in degenerate subspaces
• Structure of the degenerate subspace V_N
• Perturbative expansion for degenerate states
• Degeneracy breaking at first order
• Orthogonality condition on state corrections
• Hierarchy of perturbation equations at each order
• Split of the first-order state correction into V_N and V̂ parts
• Three-step perturbation procedure overview
• First-order energy correction derivation
• Delta H must be diagonal in the degenerate subspace
• Choosing the correct starting basis
• Energy corrections as diagonal matrix elements
• Degeneracy breaking and higher-order corrections
• Degenerate subspace setup
• Perturbative expansion for degenerate states
• Diagonalization condition for the perturbation
• First-order energy corrections in degenerate case
• Lifting of degeneracy
• Good basis concept
• Commutator rule for identifying a good basis
• First-order state correction along V̂
• Reconstructing a state from its components
• Incompleteness of the order-λ equation
• Necessity of the order-λ² equation for the degenerate piece
• Projecting the order-λ² equation onto degenerate states
• Diagonality of δH in the degenerate subspace
• Order counting in the degenerate piece formula
• Second-order energy correction in degenerate PT
• Proof that degenerate states must mix under perturbation
• Degeneracy not lifted at first order
• Degeneracy splitting at second order
• The good basis problem
• Parametrizing zeroth-order states with coefficients
• Perturbation expansion for degenerate states
• Order-by-order Schrödinger equations
• Projection onto the complementary subspace
• First-order state correction in terms of good-basis coefficients
• Why second-order perturbation theory is needed
• Second-order energy equation setup
• Cancellation of V-hat cross terms
• ΔH acting as an eigenvalue on degenerate states
• Reduced second-order equation
• Substituting the known first-order state
• Second-order perturbation matrix M²
• Eigenvalue equation for second-order corrections
• Hydrogen atom unperturbed Hamiltonian
• Bohr radius as the fundamental length scale
• Hydrogen energy levels and principal quantum number
• Fine structure constant and energy scale
• Electron velocity and non-relativistic nature
• Hydrogen spectrum and spectroscopic notation
• Degeneracy formula and quantum number relation
• Hydrogen wave function structure
• Semi-classical interpretation of L degeneracy
• Electron spin doubling of states
• General angular momentum quantum numbers
• Electron spin quantum numbers
• Uncoupled basis states
• Total angular momentum and addition of angular momentum
• Good and bad quantum numbers in the coupled basis
• J values from coupling L with spin-1/2
• Spectroscopic notation for hydrogen states
• Coupled basis states for the hydrogen atom
• Magnetic Moment of an Electron
• The g-factor and Its Mystery
• Electron Spin Hamiltonian in a Magnetic Field
• Pauli Spinor Wave Function
• Pauli Matrix Identity: σ·A σ·B
• Minimal Electromagnetic Coupling (P → Π)
• The Pauli Hamiltonian
• Computing Π × Π and the Magnetic Field
• G=2 from the Pauli Equation
• Relativistic energy-momentum dispersion relation
• Non-relativistic expansion of the relativistic Hamiltonian
• Dirac's motivation for linearizing the Hamiltonian
• Dirac alpha and beta matrix algebra
• Four-by-four matrix representation of α and β
• The Dirac Hamiltonian
• Dirac equation coupled to electromagnetism
• Effective hydrogen atom Hamiltonian from Dirac equation
• Spin-orbit coupling term
• Darwin term
• Fine structure energy scale
• Corrections to the hydrogen atom Hamiltonian
• Three perturbation terms: relativistic, spin-orbit, Darwin
• The Darwin term and its delta-function form
• First-order perturbation energy correction calculation
• Darwin term affects only L = 0 states
• Wave function value at the origin for s-states
• Final Darwin energy correction formula
• Physical interpretation of the Darwin term
• Electron as a smeared charge distribution
• True potential energy of a smeared electron
• Taylor expansion of the potential around R
• Spherical symmetry and vanishing odd integrals
• Reduction of second-order term to the Laplacian
• Compton wavelength as the electron's effective size
• First-order relativistic energy correction
• Degeneracy problem in hydrogen atom perturbation theory
• Selecting a 'good basis' using commuting Hermitian operators
• Rotational invariance of P⁴ and commutativity with angular momentum
• Evaluating the relativistic correction via Hermiticity of P²
• Final relativistic energy correction formula
• Equivalence of coupled and uncoupled basis for spin-independent perturbations
• Spin-orbit coupling Hamiltonian
• S·L in terms of J², S², L²
• Coupled basis for angular momentum
• Validity of non-degenerate perturbation theory here
• Commutation of perturbation with good quantum numbers
• Expectation value of 1/R³
• Independence of ⟨1/R³⟩ from m_l
• First-order spin-orbit energy correction
• Combining fine structure perturbation terms
• Spin-orbit coupling vanishes for L=0
• Coupled basis for summing corrections
• Combined energy correction formula
• Fixed-J states from different L values
• L-independence of the fine structure correction
• Fine structure energy formula
• Connection to the Dirac equation
• Fine structure spectrum of hydrogen
• Zeeman Effect Overview
• Magnetic Moments of the Electron
• Zeeman Hamiltonian
• Full Hydrogen Atom Hamiltonian with Perturbations
• Internal Magnetic Field from Spin-Orbit Coupling
• Weak Zeeman Effect Regime
• Strong Zeeman Effect Regime
• Coupled basis states and spin-orbit energies
• Degeneracy structure in the coupled basis
• Zeeman perturbation matrix in the coupled basis
• L² commuting with δH_Zeeman simplifies off-diagonal blocks
• JZ commuting with δH_Zeeman diagonalizes MJ blocks
• Reduction to non-degenerate perturbation theory
• First-order Zeeman energy correction formula
• Splitting of energy levels proportional to MJ
• Rewriting LZ + 2SZ as JZ + SZ
• Vector operators under total angular momentum
• Spin as a vector operator under J
• Double commutator identity for vector operators
• Projection lemma for vector operators
• Matrix element of S_z in J-eigenstates
• Scalar operators and M_J independence
• Evaluating S·J using J = L + S
• Landé g-factor
• Weak-field Zeeman splitting formula
• Strong-field Zeeman Hamiltonian structure
• Exact solvability of Zeeman Hamiltonian
• Uncoupled states as exact eigenstates
• Exact Zeeman energy eigenvalues
• Degeneracy structure after Zeeman splitting
• Choosing perturbation theory type for fine structure
• WKB / Semiclassical Approximation Overview
• Semiclassical Limit and ℏ→0
• Local Momentum and Local de Broglie Wavelength
• Schrödinger Equation in Local-Momentum Form
• Classically Allowed vs. Forbidden Regions
• Polar Form of the Wave Function
• Probability Current from Phase Gradient
• Phase–Momentum Correspondence and Classical Action
• Position-dependent momentum and de Broglie wavelength
• WKB wave function ansatz (amplitude-phase form)
• Probability current from the WKB form
• Semiclassical validity condition (slowly varying λ)
• Pure-exponential WKB ansatz with complex S
• Nonlinear ODE for S from the Schrödinger equation
• Smallness of the iℏS″ term for slowly varying potentials
• Semiclassical power-series expansion in ℏ
• Zeroth- and first-order WKB equations
• Momentum function P(X) in semi-classical approximation
• Solving the zeroth-order semi-classical equation
• S0(X) as the integral of momentum
• Validity condition of the WKB approximation
• Local de Broglie wavelength
• Physical meaning of the semi-classical condition
• Slowly varying potential condition
• Turning points and breakdown of WKB
• Linear potential approximation near turning points
• WKB connection formulas
• WKB solutions away from turning points
• Breakdown of WKB near turning points
• WKB connection formulas
• Boundary condition at a hard wall
• Selecting the decaying exponential in the forbidden region
• Splitting the integral for boundary application
• Applying the hard-wall boundary condition to fix the phase
• Bohr–Sommerfeld quantisation condition
• Physical interpretation of the quantisation condition
• The Airy Equation
• Integral Representation via Fourier Transform
• Regions of Convergence in the Complex K-Plane
• Contour Deformation in the Complex Plane
• Boundary Terms and Validity of Integration by Parts
• Second Airy function Bi(u)
• Real integral representations of Ai and Bi
• Stokes phenomenon in asymptotic expansions
• Stationary phase method for contour integrals
• Stationary points of Ai(u) for positive u
• Gaussian integral evaluation near stationary point
• Asymptotic decay of Ai(u) for large positive u
• Asymptotic oscillation of Ai(u) for large negative u
• Validity regions of asymptotic expansions
• Turning points and WKB breakdown
• Linear potential approximation near a turning point
• WKB solutions in classically forbidden and allowed regions
• Rescaling WKB expressions using dimensionless variable U
• Airy equation as the exact solution for a linear potential
• Asymptotic expansions of Airy functions
• Connection formulas linking WKB solutions across a turning point
• Bi solution asymptotic expansions
• Matching Bi solution across a turning point
• WKB connection formulas
• Directionality of connection formulas
• Decaying exponential → oscillatory wave function (safe direction)
• Growing exponential → oscillatory wave function (safe direction)
• Connection formulas for a right-side turning point
• Application to tunneling through a barrier
• Time-Dependent Perturbation Theory Setup
• Loss of Energy Eigenstates Under Time-Dependent H
• Physical Picture: Transition Between Eigenstates
• Transitions to the Continuum
• The Interaction Picture
• Unitary Time Evolution and Bringing States to Rest
• Definition of the Interaction Picture State ψ̃
• Schrödinger Equation in the Interaction Picture
• Orthonormal basis of unperturbed Hamiltonian eigenstates
• Expanding psi-tilde in the interaction picture
• Recovering psi(t) from psi-tilde
• Schrodinger equation for the interaction-picture coefficients
• Matrix elements of the perturbation in the interaction picture
• Transition frequency omega_MN
• Final coupled ODE system for C_M(t)
• Two-state quantum system setup
• Delta-function perturbation Hamiltonian
• Coupled differential equations for amplitudes
• Ambiguity of delta function at a discontinuity
• Regulating the delta function
• Solving the regulated coupled equations
• Transition amplitude independence of regulator
• Transition probability between states
• Setting up time-dependent perturbation theory
• Power series expansion in λ
• The interaction-picture state ψ̃
• Interaction-picture Schrödinger equation
• Zeroth-order equation and constant state
• First-order perturbative equation
• Iterative hierarchy of perturbative equations
• Fermi's Golden Rule and transitions to a continuum
• Constant perturbations
• Harmonic perturbations
• Density of states concept
• Box normalization and periodic boundary conditions
• Counting states in momentum space
• Density of states as a function of energy
• Constant perturbation setup
• First-order transition amplitude formula
• Evaluating the time integral
• Transition probability expression
• Validity condition for perturbation theory
• Oscillatory behavior for E_F ≠ E_I
• Quadratic growth for E_F = E_I
• Transition rate and linear-in-time probability
• Role of continuum final states and energy conservation
• Summing over a continuum of final states
• Transition probability integral form
• Narrow-band approximation
• Shape of the sinc-squared integrand
• Linear growth of transition probability with time
• Change of variables to evaluate the integral
• Extending integration limits to ±∞
• Fermi's Golden Rule
• Fermi's Golden Rule Overview
• Helium Atom Hamiltonian
• Hydrogenic Energy Levels for Helium
• Discrete Energy Spectrum of Helium
• Continuum States in Helium
• Auger (Autoionization) Transition
• Energy Conservation in Auger Emission
• Coulomb Repulsion as the Transition Matrix Element
• Harmonic perturbations setup
• First-order transition amplitude formula
• Evaluating the time integral
• Role of ω_{FI} and energy matching
• Resonance and denominator divergence
• Hydrogen ionization rate as application
• Stimulated emission process
• Absorption process
• Rotating wave approximation
• Transition probability for harmonic perturbation
• Integration over final states using density of states
• Fermi's golden rule for harmonic perturbations
• Transition rate and its physical interpretation
• Hydrogen atom photoionization overview
• Ejected electron kinetic energy and the Rydberg
• Long-wavelength (dipole) approximation
• Validity range for the approximation
• Non-relativistic ejected electron
• Hamiltonian perturbation from the electric field
• Identification of H′ for harmonic perturbation theory
• Initial and final states for the transition
• Geometric setup and relevant angles
• Coordinate system with multiple angle variables
• Matrix element integral setup
• Expressing cosγ via dot product of unit vectors
• Vanishing of the cos(φ−φ′) term under φ′ integration
• Factoring out constants and performing the φ′ integral
• Final form and dimensional analysis of the matrix element
• Hydrogen photoionization matrix element
• Validity conditions for the plane-wave approximation
• Electron momentum from photon energy
• Density of states for free plane waves
• Fermi's golden rule application
• Differential ionization rate and angular dependence
• Solid-angle integration of cos²θ
• Total photoionization rate and high-momentum simplification
• Atomic units for the ionization rate
• Light-atom interaction overview
• Two-level atom model
• Photon absorption
• Stimulated emission
• Why stimulated emission requires a field
• Laser principle and population inversion
• Optical pumping to achieve population inversion
• Thermal equilibrium populations of atomic states
• Boltzmann factor for level populations
• Planck black-body radiation spectrum
• Absorption rate and Einstein B coefficient
• Stimulated emission rate and Einstein B coefficient
• Failure of two-process equilibrium
• Spontaneous emission and Einstein A coefficient
• Rate equation for photon interactions
• Solving for spectral energy density U(ω)
• Using Boltzmann population ratio N_A/N_B
• Comparison with Planck thermal radiation formula
• Equality of stimulated emission and absorption coefficients
• Relationship between A and B Einstein coefficients
• Calculating spontaneous emission rate via B coefficient
• Vacuum fluctuations as the origin of spontaneous emission
• Temperature dependence of emission and absorption rates
• Electric dipole approximation
• Electric field as spatially uniform over the atom
• Scalar potential for a uniform electric field
• Perturbing Hamiltonian for atom-light coupling
• Electric dipole operator
• H′ in the dipole interaction
• Transition probability under harmonic perturbation
• Dipole matrix elements and transition selection
• Einstein's stimulated and spontaneous emission
• Dipole moment operator
• Transition probability formula for a monochromatic field
• Incoherent superposition of blackbody radiation modes
• Summing over radiation modes near resonance
• Spectral energy density U(ω)
• Relating field amplitude to energy density
• Transition probability expressed via energy density
• Incoherent mode summation as an integral
• Transition probability integral formula
• Peak dominance of the sinc-squared factor
• Orientational averaging of the dipole coupling
• Linear growth of transition probability with time
• Definition of the transition rate
• Complex dipole matrix element vector
• Isotropy identity ⟨N_i N_j⟩ = δ_ij/3
• Final stimulated transition rate formula
• Einstein B coefficient from quantum theory
• Einstein A coefficient for spontaneous emission
• Exponential decay and atomic lifetime
• Multiple decay channels and total decay rate
• Dipole matrix elements for hydrogen transitions
• Selection rule for magnetic quantum number (ΔM)
• Selection rule for orbital quantum number (ΔL)
• Decay pathways and metastability in hydrogen
• Potentials vs. fields in quantum mechanics
• Deriving potentials from Maxwell's equations
• Gauge transformations
• Physical equivalence of gauge-related potentials
• Same E and B does not imply gauge equivalence
• Wilson loops and topology
• Magnetic flux quantization on a torus
• Schrödinger equation with electromagnetic fields
• Kinematic vs. canonical momentum
• Velocity operator in quantum mechanics
• Gauge invariance principle
• Gauge transformation of the wave function
• Space- and time-dependent phase transformation
• Gauge-invariant operators and expectation values
• Hamiltonian for a charged particle in an electromagnetic field
• Gauge transformations of electromagnetic potentials
• Gauge invariance of the Schrödinger equation
• Phase transformation of the wave function under gauge change
• Operator ordering in the expanded Hamiltonian
• Explicit form of the electromagnetic Hamiltonian
• Gauge covariant derivative
• Schrödinger equation as covariant derivatives
• Canonical momentum versus kinetic momentum
• Gauge invariant observables
• Momentum operator's lack of gauge invariance
• Gauge-invariant kinetic momentum P − QA/c
• Torus geometry and identifications
• Vector potential on a torus
• Gauge transformations patching potentials on a torus
• Single-valuedness of the gauge transformation U = e^{iQλ/ℏc}
• Magnetic flux quantization on a torus
• The flux quantum
• Landau levels and classical cyclotron motion
• Vector potential and gauge choice
• Hamiltonian for a charged particle in a magnetic field
• Conservation of PX and separability
• Factored wavefunction and effective 1D problem
• Mapping onto a harmonic oscillator
• Magnetic length scale
• Quantized Landau energy levels
• Infinite degeneracy of each Landau level
• Gauge choice and solution equivalence
• Finite-size sample and boundary conditions
• Quantization of KX in a finite sample
• Guiding-center coordinate y₀
• Counting degenerate states in a Landau level
• Degeneracy as area over magnetic length squared
• Degeneracy as flux divided by flux quantum
• Numerical scale of Landau level degeneracy
• Wavefunction structure of Landau level states
• Relevance to the quantum Hall effect
• Adiabatic approximation overview
• Classical pendulum with varying frequency
• Energy change in a time-dependent Hamiltonian
• Hamilton's equations of classical mechanics
• Cancellation of equations-of-motion terms in dH/dt
• Adiabatic condition: time scale separation
• Adiabatic condition as a dimensionless inequality
• Adiabatic invariant definition
• Adiabatic invariant I = E/ω
• Quantum number analogy for adiabatic invariance
• Computing dI/dt
• dI/dt as kinetic minus potential energy
• Oscillation of V − K in a harmonic oscillator
• Change in I over one period
• Double suppression of adiabatic invariant change
• Phase space and elliptical orbits
• Semi-axes of the phase-space ellipse
• Area of the phase-space ellipse as adiabatic invariant
• Phase-space area as a contour integral ∮P dX
• Adiabatic invariant becoming a quantum number
• Bohr-Sommerfeld (WKB) quantization condition
• Suppression of transitions under slow perturbations
• Adiabatic evolution overview
• Instantaneous eigenstates
• Instantaneous eigenstates do not solve the Schrödinger equation
• Ansatz for the adiabatic solution
• Differential equation for C(t)
• Purely imaginary nature of ⟨ψ|ψ̇⟩
• Berry's phase vs. dynamical phase
• Missing slow-variation condition — the gap in the derivation
• Instantaneous energy eigenstates
• Dynamical phase θ(t)
• Geometric (Berry) phase γ(t)
• Adiabatic theorem statement
• Energy-gap condition for adiabaticity
• Approximate nature of the adiabatic result
• Observability of relative phase
• Superposition of instantaneous energy eigenstates
• Applying the Schrödinger equation to the expansion
• Projecting onto a basis state to isolate coefficients
• Equation of motion for each coefficient C_k
• Adiabatic approximation: neglecting coupling terms
• Solution in the adiabatic approximation: dynamical and geometric phases
• Coupling between modes and breakdown of the adiabatic picture
• Coupling term between instantaneous eigenstates
• Differentiating the instantaneous eigenvalue equation
• Projecting onto a different eigenstate (k ≠ n)
• Coupling term expressed via Ḣ matrix element
• Rate of Hamiltonian change and adiabatic validity
• Linear ramp Hamiltonian model
• Adiabatic errors scale as 1/T
• Formal statement of the adiabatic theorem
• Born-Oppenheimer molecular picture
• Energy level diagram as a function of separation R
• Instantaneous energy eigenstates from static solutions
• Time-dependent energy level diagram
• Non-adiabatic (Landau-Zener) transition probability
• Toy two-level Hamiltonian with crossing levels
• Exact solutions as dressed instantaneous eigenstates
• Energy level crossing vs. avoided crossing
• Two-level Hamiltonian with off-diagonal coupling
• Energy eigenvalues of the coupled two-level system
• Avoided level crossing
• Limit H12 → 0 and non-adiabatic passage
• Characteristic timescale τD of Hamiltonian change
• Rabi oscillation period T12
• Adiabatic condition for two-level passage
• Landau–Zener transition probability
• Adiabatic theorem review
• Dynamic phase (theta_n)
• Geometric (Berry's) phase (gamma_n)
• Time-independence of the geometric phase
• Configuration (parameter) space
• Solving the Hamiltonian across parameter space
• Path in configuration space as system evolution
• Instantaneous eigenstates from parameter-space solutions
• Chain rule for time-dependent coordinates
• Gradient notation for state differentiation
• Geometric phase as a path integral in configuration space
• Berry connection
• Gauge freedom in instantaneous energy eigenstates
• Gauge transformation of the Berry connection
• Gauge invariance of the Berry phase for closed paths
• Geometric phase with real eigenstates
• Berry connection imaginary proof
• Berry phase vanishes in 1D configuration space
• Berry connection as a vector potential
• Stokes' theorem and Berry curvature
• Spin-½ in a rotating magnetic field
• Why molecules lack spherical symmetry
• Electron-to-nuclear mass ratio
• Electronic energy scale estimate
• Physical picture of molecular bonding
• Hydrogen molecule ion H₂⁺
• Nuclear vibrational energy scale
• Separation of electronic and nuclear timescales
• Rotational energy scale of molecules
• Molecular energy scale hierarchy
• Adiabatic approximation for electrons
• Born-Oppenheimer approximation overview
• Many-body molecular Hamiltonian
• Fixed nuclear skeleton approximation
• Electronic eigenstates parametrized by nuclear positions
• Full wavefunction expansion in electronic states
• Single-term Born-Oppenheimer ansatz
• Variational method in quantum mechanics
• Born-Oppenheimer product wave function ansatz
• Separation of fast and slow degrees of freedom
• Effective Hamiltonian for nuclear degrees of freedom
• Electronic energy as an effective potential
• Berry connection in molecular systems
• Berry connection formula for nuclear dynamics
• Born-Oppenheimer approximation (lowest order)
• H₂⁺ hydrogen molecular ion system
• Bound state energy and dissociation
• Electron Hamiltonian for two-center problem
• Variational method for the electron wave function
• LCAO (linear combination of atomic orbitals)
• Symmetry of the molecular wave function
• Validity regime of the LCAO ansatz
• Electronic energy as a function of proton separation
• Nuclear repulsion and total effective potential
• Equilibrium bond length and binding energy
• Purpose of scattering experiments
• Experimental setup: beam, target, detectors
• Particle reactions and identity change
• Definition of scattering (no identity change)
• Elastic vs. inelastic scattering
• Franck–Hertz experiment as inelastic scattering example
• Simplifying assumptions: no spin, non-relativistic
• Center-of-mass frame reduction for two-body potentials
• Energy eigenstates vs. wave packets in scattering
• Scattering problem Hamiltonian and energy eigenstates
• Finite-range potentials
• Energy parametrized by wave vector K
• Incident plane wave as the incoming state
• Outgoing spherical scattered wave
• Scattering amplitude f(θ,φ)
• Asymptotic full scattering wave function
• Scattering amplitude F(θ,φ)
• Differential cross section definition
• Formal definition: dσ as a ratio
• Incident probability flux
• Scattered particle count per unit time
• Cross section in terms of F(θ,φ)
• Partial waves in 3D scattering
• Central potential and azimuthal symmetry
• Free particle in spherical coordinates
• Radial Schrödinger equation with centrifugal barrier
• No energy quantization for the free particle
• Spherical Bessel functions j_l and n_l
• Plane wave expansion in spherical harmonics
• Ingoing and outgoing spherical waves
• Scattering wavefunction structure
• Scattering amplitude and differential cross section
• Central potential and azimuthal symmetry
• Radial solutions with spherical Bessel functions
• Plane wave expansion in spherical waves
• Large-argument asymptotics of spherical Bessel functions
• Plane wave expansion in spherical harmonics
• Large-argument expansion of Bessel functions
• Incoming and outgoing spherical waves
• Independence of partial wave solutions
• Definition of partial waves
• 1D scattering with a hard wall and finite-range potential
• Reference solution with zero potential (1D)
• Phase shift in 1D scattering
• Scattered wave as difference from reference wave
• Form of the scattered wave in terms of phase shift
• Asymptotic wavefunction expansion (r >> a)
• Identifying incoming and outgoing waves
• Matching incoming wave boundary condition
• Independent scattering of each partial wave (L)
• Equal amplitudes of incoming and outgoing partial waves
• Phase shift as the scattering parameter
• Total cross section from partial waves
• Orthogonality of spherical harmonics in integration
• Partial wave cross section formula
• Non-interference of partial waves in total cross section
• Physical significance of phase shifts
• Physical intuition behind the optical theorem
• Forward scattering amplitude calculation
• The optical theorem
• Phase shift as coefficient ratio
• Asymptotic radial wavefunction form
• Three equivalent definitions of phase shift
• Hard-sphere scattering setup
• Boundary condition at the hard-sphere surface
• Computing phase shifts from Bessel functions
• Total cross section formula for the hard sphere
• Low-energy (long-wavelength) cross section
• Partial wave expansion for central potentials
• Phase shifts as scattering parameters
• Scattering amplitude in terms of phase shifts
• Cross section from phase shifts
• Radial Schrödinger equation inside the potential
• Boundary condition at the potential edge
• Logarithmic derivative method for phase shifts
• Explicit formula for tan(δL)
• When phase shifts dominate the cross section
• Condition KA << 1: short range and low energy
• Impact parameter and angular momentum
• Semiclassical identification of impact parameter with partial wave L
• Negligible scattering for L > KA
• Centrifugal barrier and the turning point R*
• Classical estimate of partial wave cross section
• Partial wave unitarity bound
• Classical cylinder picture of partial wave contributions
• Integral scattering equation formulation
• Rescaling the potential (U of R)
• Green's function definition
• Integral equation solution via superposition
• Born approximation motivation
• Explicit form of the outgoing Green's function
• Boundary condition choice (ψ₀ and G⁺)
• Far-field approximation and scattering amplitude formula
• Lippmann-Schwinger integral equation
• Green's function G+
• Far-field (large-r) approximation setup
• Approximate expansion of |r−r′|
• Amplitude vs. phase accuracy in approximations
• Incident and scattered wave vectors k_i and k_n
• Simplified far-field wave function expression
• Born approximation
• First Born Approximation Definition
• Validity of the Born Approximation
• Born Approximation Wave Function Formula
• Scattering Amplitude in the Born Approximation
• Momentum Transfer Vector K
• Scattering Amplitude as Fourier Transform of Potential
• Angular Dependence Hidden in K
• Magnitude of the Momentum Transfer Vector
• Born Approximation for Non-Spherically-Symmetric Potentials
• Diagrammatic interpretation of the Born series
• Green's function as a propagator
• Analogy between Born diagrams and Feynman diagrams
• First Born approximation formula
• Spherical symmetry simplification of the Born formula
• Angular integration in spherical coordinates
• Reduced radial Born amplitude formula
• Yukawa potential and its physical motivation
• Born scattering amplitude for the Yukawa potential
• Definition of identical particles
• Indistinguishability of identical particles
• Identical particles beyond elementary particles
• Isospin and approximate identity
• Classical distinguishability via trajectories and tagging
• Symmetry of the Hamiltonian under particle exchange
• Tensor product states for distinguishable particles
• Ambiguity of tensor product for identical particles
• Failure of declaring all exchange-related states equivalent
• Exchange degeneracy problem
• Two-particle tensor product Hilbert space
• Permutation operator P₂₁
• P₂₁ is its own inverse (involution)
• P₂₁ is Hermitian and unitary
• Symmetrizer S and antisymmetrizer A operators
• Orthogonal projectors: definition and conditions
• S and A are orthogonal projectors
• Operators acting on tensor product subspaces
• Conjugation action of operators on operators
• Permutation operator conjugating B₁ to B₂
• Distributing conjugation over operator products
• Relabelling composite operators under permutation
• Symmetric operators and commutativity with P₂₁
• Symmetric group S_N and N! permutations
• Permutation operator notation
• General permutation as function α on {1,…,N}
• Inverse of a permutation operator
• Transpositions as elementary permutations
• All permutation operators are unitary
• Even and odd permutations
• Equal numbers of even and odd permutations in S_N
• Symmetrizers and anti-symmetrizers
• Limits of simultaneously diagonalizing permutation operators
• Special simultaneous eigenstates of all permutation operators
• N-particle Hilbert space as a tensor product
• Symmetric states and their defining property
• Even and odd permutations and the sign factor
• Antisymmetric states and their defining property
• Symmetric and antisymmetric subspaces
• Symmetric and anti-symmetric quantum states
• Even and odd permutations via transpositions
• Unitarity of permutation operators and hermiticity of transpositions
• Equal count of even and odd permutations
• Symmetric and anti-symmetric states as subspaces
• The symmetrizer and anti-symmetrizer operators
• Projector condition: P² = P
• Hermiticity of S and A via permutation group structure
• Permutation operators acting on S and A
• Rearrangement of permutation operator lists
• Sign factor multiplication rule
• S and A as orthogonal projectors
• Proof that S²=S
• Proof that A²=A
• Proof that AS=0 (orthogonality)
• S and A project onto symmetric and anti-symmetric subspaces
• Two-particle case: S+A equals the identity
• Three-particle case and mixed-symmetry states
• The Symmetrization Postulate
• Bosons and Fermions Defined Statistically
• Non-Physical Arbitrary States
• Statistics in Lower Dimensions
• The Spin-Statistics Theorem
• Statistics of Composite Particles
• Exchange degeneracy problem resolution
• Vector space VU of permuted states
• Uniqueness of symmetric and anti-symmetric states in VU
• Applying symmetrizer and anti-symmetrizer projectors
• Slater determinant for fermionic states
• Determinant formula and permutation signs
• Occupation numbers for many-particle states

Syllabus

1. Lectures

   1. L1.1 General problem. Non-degenerate perturbation theory
   2. L1.2 Setting up the perturbative equations
   3. L1.3 Calculating the energy corrections
   4. L1.4 First order correction to the state. Second order correction to energy
   5. L2.1 Remarks and validity of the perturbation series
   6. L2.2 Anharmonic Oscillator via a quartic perturbation
   7. L2.3 Degenerate Perturbation theory: Example and setup
   8. L2.4 Degenerate Perturbation Theory: Leading energy corrections
   9. L3.1 Remarks on a 'good basis'
   10. L3.2 Degeneracy resolved to first order; state and energy corrections
   11. L3.3 Degeneracy resolved to second order
   12. L3.4 Degeneracy resolved to second order (continued)
   13. L4.1 Scales and zeroth-order spectrum
   14. L4.2 The uncoupled and coupled basis states for the spectrum
   15. L4.3 The Pauli equation for the electron in an electromagnetic field
   16. L4.4 Dirac equation for the electron and hydrogen Hamiltonian
   17. L5.1 Evaluating the Darwin correction
   18. L5.2 Interpretation of the Darwin correction from nonlocality
   19. L5.3 The relativistic correction
   20. L5.4 Spin-orbit correction
   21. L5.5 Assembling the fine-structure corrections
   22. L6.1 Zeeman effect and fine structure
   23. L6.2 Weak-field Zeeman effect; general structure
   24. L6.3 Weak-field Zeeman effect; the projection lemma
   25. L6.4 Strong-field Zeeman
   26. L6.5 Semiclassical approximation and local de Broglie wavelength
   27. L7.1 The WKB approximation scheme
   28. L7.2 Approximate WKB solutions
   29. L7.3 Validity of the WKB approximation
   30. L7.4 Connection formula stated and example
   31. L8.1 Airy functions as integrals in the complex plane
   32. L8.2 Asymptotic expansions of Airy functions
   33. L8.3 Deriving the connection formulae
   34. L8.4 Deriving the connection formulae (continued) logical arrows
   35. L9.1 The interaction picture and time evolution
   36. L9.2 The interaction picture equation in an orthonormal basis
   37. L9.3 Example: Instantaneous transitions in a two-level system
   38. L9.4 Setting up perturbation theory
   39. L10.1 Box regularization: density of states for the continuum
   40. L10.2 Transitions with a constant perturbation
   41. L10.3 Integrating over the continuum to find Fermi's Golden Rule
   42. L10.4 Autoionization transitions
   43. L11.1 Harmonic transitions between discrete states
   44. L11.2 Transition rates for stimulated emission and absorption processes
   45. L11.3 Ionization of hydrogen: conditions of validity, initial and final states
   46. L11.4 Ionization of hydrogen: matrix element for transition
   47. L12.1 Ionization rate for hydrogen: final result
   48. L12.2 Light and atoms with two levels, qualitative analysis
   49. L12.3 Einstein's argument: the need for spontaneous emission
   50. L12.4 Einstein's argument: B and A coefficients
   51. L12.5 Atom-light interactions: dipole operator
   52. L13.1 Transition rates induced by thermal radiation
   53. L13.2 Transition rates induced by thermal radiation (continued)
   54. L13.3 Einstein's B and A coefficients determined. Lifetimes and selection rules
   55. L13.4 Charged particles in EM fields: potentials and gauge invariance
   56. L13.5 Charged particles in EM fields: Schrodinger equation
   57. L14.1 Gauge invariance of the Schrödinger Equation
   58. L14.2 Quantization of the magnetic field on a torus
   59. L14.3 Particle in a constant magnetic field: Landau levels
   60. L14.4 Landau levels (continued). Finite sample
   61. L15.1 Classical analog: oscillator with slowly varying frequency
   62. L15.2 Classical adiabatic invariant
   63. L15.3 Phase space and intuition for quantum adiabatic invariants
   64. L15.4 Instantaneous energy eigenstates and Schrodinger equation
   65. L16.1 Quantum adiabatic theorem stated
   66. L16.2 Analysis with an orthonormal basis of instantaneous energy eigenstates
   67. L16.3 Error in the adiabatic approximation
   68. L16.4 Landau-Zener transitions
   69. L16.5 Landau-Zener transitions (continued)
   70. L17.1 Configuration space for Hamiltonians
   71. L17.2 Berry's phase and Berry's connection
   72. L17.3 Properties of Berry's phase
   73. L17.4 Molecules and energy scales
   74. L18.1 Born-Oppenheimer approximation: Hamiltonian and electronic states
   75. L18.2 Effective nuclear Hamiltonian. Electronic Berry connection
   76. L18.3 Example: The hydrogen molecule ion
   77. L19.1 Elastic scattering defined and assumptions
   78. L19.2 Energy eigenstates: incident and outgoing waves. Scattering amplitude
   79. L19.3 Differential and total cross section
   80. L19.4 Differential as a sum of partial waves
   81. L20.1 Review of scattering concepts developed so far
   82. L20.2 The one-dimensional analogy for phase shifts
   83. L20.3 Scattering amplitude in terms of phase shifts
   84. L20.4 Cross section in terms of partial cross sections. Optical theorem
   85. L20.5 Identification of phase shifts. Example: hard sphere
   86. L21.1 General computation of the phase shifts
   87. L21.2 Phase shifts and impact parameter
   88. L21.3 Integral equation for scattering and Green's function
   89. L22.1 Setting up the Born Series
   90. L22.2 First Born Approximation. Calculation of the scattering amplitude
   91. L22.3 Diagrammatic representation of the Born series. Scattering amplitude for spherically symm...
   92. L22.4 Identical particles and exchange degeneracy
   93. L23.1 Permutation operators and projectors for two particles
   94. L23.2 Permutation operators acting on operators
   95. L23.3 Permutation operators on N particles and transpositions
   96. L23.4 Symmetric and Antisymmetric states of N particles
   97. L24.1 Symmetrizer and antisymmetrizer for N particles
   98. L24.2 Symmetrizer and antisymmetrizer for N particles (continued)
   99. L24.3 The symmetrization postulate
   100. L24.4 The symmetrization postulate (continued)