PU-301 · Quantum Mechanics II
Building on the one-dimensional wave mechanics of QM I, this unit develops the full abstract Hilbert-space formalism and deploys it against the problems where quantum mechanics becomes predictive: angular momentum and the hydrogen atom, time-independent and time-dependent perturbation theory, identical particles, and the emission and absorption of radiation. The intellectual arc moves from the axioms and their representation theory to approximation methods and scattering, equipping the student to compute real spectra, transition rates, and cross-sections rather than merely idealized bound states.
Lectures
| L01 | From Wave Mechanics to Hilbert Space — |
| L02 | Postulates: States, Observables, and Measurement |
| L03 | Dirac Notation and Representations |
| L04 | Commutators and the Uncertainty Principle |
| L05 | Schrodinger, Heisenberg, and Interaction Pictures — |
| L06 | The Classical Limit and Ehrenfest's Theorem |
| L07 | Angular Momentum from Symmetry |
| L08 | Orbital Angular Momentum and Spherical Harmonics |
| L09 | The Central Potential and the Radial Equation |
| L10 | The Hydrogen Atom |
| L11 | Spin-1/2 and the Pauli Algebra |
| L12 | Adding Angular Momenta: Clebsch-Gordan |
| L13 | Non-Degenerate Perturbation Theory |
| L14 | Degenerate Perturbation Theory |
| L15 | Fine Structure and the Zeeman Effect |
| L16 | The Variational Method |
| L17 | Helium and Multi-Electron Atoms |
| L18 | The WKB Approximation |
| L19 | Time-Dependent Perturbation Theory |
| L20 | Fermi's Golden Rule |
| L21 | Interaction of Atoms with Radiation |
| L22 | Selection Rules and Spectral Lines |
| L23 | Identical Particles and Exchange |
| L24 | Scattering: Cross-Sections and Amplitudes |
| L25 | The Born Approximation |
| L26 | Partial Waves, Phase Shifts, and the Optical Theorem |
Derivations homed in this unit
Spectral Theorem for Hermitian Observables
Derives that Hermitian operators have real eigenvalues and a complete orthonormal eigenbasis, grounding the measurement postulate.
Generalized Uncertainty Relation
Derives the Robertson-Schrodinger bound on the product of variances of two observables from the Cauchy-Schwarz inequality.
Angular Momentum Spectrum from Commutators
Derives the allowed eigenvalues of J^2 and J_z purely from the SO(3) commutation relations using raising and lowering operators.
Spherical Harmonics as Angular Momentum Eigenstates
Derives the spherical harmonics as the position-space eigenfunctions of orbital L^2 and L_z with integer quantum numbers.
Separation of the Central-Potential Schrodinger Equation
Reduces the three-dimensional Schrodinger equation for a central potential to a one-dimensional radial equation with an effective potential.
Hydrogen Atom Bound-State Spectrum
Solves the Coulomb radial equation to obtain the Rydberg energy levels and the associated Laguerre radial wavefunctions.
Spin-1/2 and the Pauli Matrices
Constructs the two-dimensional representation of angular momentum and derives the Pauli matrix algebra and rotation operator for spinors.
Addition of Angular Momenta
Derives the decomposition of a tensor product of angular momenta into irreducible multiplets and the Clebsch-Gordan coefficients.
Non-Degenerate Perturbation Theory
Derives the first- and second-order corrections to energies and states for a Hamiltonian with a small perturbation.
Degenerate Perturbation Theory
Derives the good basis and level splittings by diagonalizing the perturbation within a degenerate eigenspace.
Fine Structure of Hydrogen
Derives the relativistic-kinetic, spin-orbit, and Darwin corrections that split the hydrogen levels by total angular momentum, and the Zeeman splitting in a field.
Rayleigh-Ritz Variational Principle
Proves that the expectation of the Hamiltonian in any trial state is an upper bound on the ground-state energy.
WKB Approximation and Bohr-Sommerfeld Quantization
Derives the semiclassical wavefunction and the quantization condition with connection formulae across turning points.
Time-Dependent Perturbation Theory
Derives the first-order transition amplitude between states under a time-dependent perturbation in the interaction picture.
Fermi's Golden Rule
Derives the transition rate to a continuum of final states from the long-time limit of the first-order transition probability.
Electric Dipole Transitions and Selection Rules
Derives the dipole transition rate and the angular-momentum and parity selection rules for atomic radiation from minimal coupling.
Identical Particles and Exchange Symmetry
Derives the symmetrization postulate, Slater determinants for fermions, and the exchange contribution to energy.
Born Approximation and Partial Waves
Derives the scattering amplitude via the Lippmann-Schwinger equation, the Born differential cross-section, and the partial-wave phase shifts with the optical theorem.