PU-303 · Condensed Matter Physics
The unit builds the quantum theory of solids from a single organizing principle—discrete translational symmetry—showing how periodicity forces electronic and vibrational states into Bloch waves and energy bands, and how band filling alone distinguishes metals, insulators, and semiconductors. It then confronts the failures of the independent-electron picture, developing the statistical and collective phenomena (Fermi statistics, lattice dynamics, superconductivity, magnetism) that make real condensed matter richer than the sum of its electrons.
Lectures
| L01 | What Condensed Matter Physics Asks — |
| L02 | Bravais Lattices, Bases, and Crystal Symmetry — |
| L03 | The Reciprocal Lattice and the Brillouin Zone |
| L04 | X-ray Diffraction: Laue, Bragg, and the Structure Factor |
| L05 | Reading Diffraction Patterns and Systematic Absences |
| L06 | Electrons in a Periodic Potential: Bloch's Theorem |
| L07 | The Nearly-Free-Electron Model and the Origin of Gaps |
| L08 | The Tight-Binding Limit |
| L09 | Metals, Insulators, and Semiconductors from Band Filling |
| L10 | Semiclassical Dynamics and Effective Mass |
| L11 | The Density of States and Van Hove Singularities |
| L12 | The Free Electron Gas and Sommerfeld Theory |
| L13 | Lattice Dynamics: The Diatomic Chain |
| L14 | Phonons: Quantizing the Lattice |
| L15 | Heat Capacity of Solids: Einstein and Debye |
| L16 | Screening and the Failure of the Independent Electron |
| L17 | Semiconductor Statistics and Doping |
| L18 | Transport Theory and the Boltzmann Equation |
| L19 | Electrons in Magnetic Fields: Landau Levels |
| L20 | The Integer Quantum Hall Effect and Topology |
| L21 | Magnetism I: Exchange and Mean-Field Order |
| L22 | Magnetism II: Spin Waves and Magnons |
| L23 | Superconductivity I: Phenomenology and the Cooper Problem |
| L24 | Superconductivity II: The BCS Ground State and Gap |
| L25 | Broken Symmetry, Order Parameters, and Collective Modes |
| L26 | Synthesis: From One Symmetry to the Phases of Matter — |
Derivations homed in this unit
Reciprocal Lattice from the Bravais Lattice
Construct the reciprocal lattice as the set of wavevectors giving unit plane-wave phase on every Bravais lattice point, and derive its primitive vectors.
Laue and Bragg Diffraction Conditions
Show that constructive X-ray scattering from a crystal requires the momentum transfer to equal a reciprocal lattice vector, prove equivalence to Bragg's law, and obtain the structure factor governing systematic absences.
Bloch's Theorem from Translational Symmetry
Prove that eigenstates of a periodic Hamiltonian can be chosen as a plane wave times a lattice-periodic function, labelled by a crystal momentum in the first Brillouin zone.
Band Gaps in the Nearly-Free-Electron Model
Use degenerate perturbation theory at Brillouin-zone boundaries to show a weak periodic potential opens gaps of size twice the relevant Fourier component.
Tight-Binding Band Dispersion
Derive the band energy as a lattice Fourier sum of hopping integrals by expanding Bloch states in localized atomic orbitals.
Effective Mass and Semiclassical Electron Dynamics
Derive the semiclassical equations of motion for a Bloch electron and identify the inverse effective-mass tensor with the band curvature.
Density of States and Van Hove Singularities
Express the density of states as a Brillouin-zone surface integral over a band and show that band extrema and saddle points produce dimensionality-dependent singularities.
Sommerfeld Free-Electron Heat Capacity
Apply the Sommerfeld expansion to the Fermi-Dirac distribution to derive the linear-in-temperature electronic heat capacity of a degenerate electron gas.
Phonon Dispersion of the Diatomic Chain
Diagonalize the harmonic equations of motion for a diatomic linear chain to obtain acoustic and optical branches separated by a zone-boundary gap.
Quantization of Lattice Vibrations
Canonically quantize the normal modes of the crystal to obtain phonons as bosonic quasiparticles with creation and annihilation operators.
Debye Model and the T-cubed Heat Capacity
Integrate the phonon density of states with a linear dispersion and Debye cutoff to derive the low-temperature T-cubed lattice heat capacity and high-temperature Dulong-Petit limit.
Intrinsic Carrier Concentration in Semiconductors
Integrate the Fermi-Dirac occupation over parabolic conduction and valence bands to derive the law of mass action and the intrinsic carrier density's exponential temperature dependence.
Electrical Conductivity from the Boltzmann Equation
Solve the linearized Boltzmann equation in the relaxation-time approximation to derive the conductivity tensor and recover the Drude result from Fermi-surface averages.
Landau Levels and the Integer Quantum Hall Effect
Quantize a two-dimensional electron gas in a magnetic field into Landau levels and derive the quantized Hall conductance plateaus.
Cooper Pair Instability
Show that an arbitrarily weak attractive interaction binds a pair of electrons above a filled Fermi sea, signalling the instability of the normal metal.
BCS Gap Equation and the Superconducting Gap
Mean-field decouple the pairing Hamiltonian via a Bogoliubov transformation to derive the self-consistent gap equation and its critical temperature.
Weiss Mean-Field Theory of Ferromagnetism
Derive the self-consistent magnetization and Curie-Weiss susceptibility of the Heisenberg model in mean-field approximation, locating the ferromagnetic transition.
Spin-Wave Dispersion and Magnons
Apply the Holstein-Primakoff transformation to the Heisenberg ferromagnet to derive the quadratic magnon dispersion and the Bloch T-to-the-three-halves law.