Mathematical physics
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Mathematical Physics — scope drawn from open mathematical physics notes and typical US graduate mathematical physics syllabi.
Classical mechanics and variational principles
- Lagrangian and Hamiltonian formulations — Lagrangian and Hamiltonian formulations
- Legendre transform — Legendre transform
- Canonical equations — Canonical equations
- Poisson brackets and symmetries — Poisson brackets and symmetries
- Noether's theorem — Noether's theorem
- Conserved quantities — Conserved quantities
- Integrable systems (introduction) — Integrable systems (introduction)
Quantum mechanics (mathematical)
- Hilbert spaces — Hilbert spaces
- Self-adjoint operators — Self-adjoint operators
- Spectral theorem and functional calculus — Spectral theorem and functional calculus
- Schrödinger and Heisenberg pictures — Schrödinger and Heisenberg pictures
- Path integrals (introduction) — Path integrals (introduction)
- Perturbation theory for quantum systems — Perturbation theory for quantum systems
PDEs and special functions in physics
- Heat, wave, and Schrödinger equations — Heat, wave, and Schrödinger equations
- Green's functions — Green's functions
- Fundamental solutions — Fundamental solutions
- Bessel and Legendre equations — Bessel and Legendre equations
- Hermite functions in separation of variables — Hermite functions in separation of variables
- Distribution theory for physics (Dirac delta, introduction) — Distribution theory for physics (Dirac delta, introduction)
- Scattering theory (introduction) — Scattering theory (introduction)
Computational and applied topics
- Numerical methods for Schrödinger-type equations — Numerical methods for Schrödinger-type equations
- WKB and semiclassical approximations — WKB and semiclassical approximations
- Group theory in crystallography — Group theory in crystallography
- Band structure (introduction) — Band structure (introduction)
- Statistical mechanics — Statistical mechanics
- Ergodicity (overview) — Ergodicity (overview)
- Field theories — Field theories
- Gauge symmetry (mathematical introduction) — Gauge symmetry (mathematical introduction)
Cross-disciplinary methods
- Inverse problems in imaging — Inverse problems in imaging
- Geophysics — Geophysics
- Stability — Stability
- Bifurcation in nonlinear PDE models — Bifurcation in nonlinear PDE models
- Renormalization — Renormalization
- Scaling in critical phenomena (introduction) — Scaling in critical phenomena (introduction)
- Symplectic integrators for Hamiltonian systems — Symplectic integrators for Hamiltonian systems
- Research-level problem sets aligned to physics department syllabi — Research-level problem sets aligned to physics department syllabi
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Classical mechanics and variational principles
Lagrangian and Hamiltonian formulations
Coming soon - Quantum mechanics (mathematical)
Hilbert spaces
Coming soon - PDEs and special functions in physics
Heat, wave, and Schrödinger equations
Coming soon - Computational and applied topics
Numerical methods for Schrödinger-type equations
Coming soon - Cross-disciplinary methods
Inverse problems in imaging
Coming soon
Notes
Graduate Mathematical Physics — scope drawn from open mathematical physics notes and typical US graduate mathematical physics syllabi. Topic outline: `content/topics/graduate/math_physics.json`.