Mathematical physics
Graduate · Math
Syllabus focus
Standard syllabus · STEM / applied
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Topics typically covered
Standard syllabus
Classical mechanics and variational principles
- Lagrangian and Hamiltonian formulations
- Legendre transform and canonical equations
- Poisson brackets and symmetries
- Noether's theorem and conserved quantities
- Integrable systems (introduction)
Quantum mechanics (mathematical)
- Hilbert spaces and self-adjoint operators
- Spectral theorem and functional calculus
- Schrödinger and Heisenberg pictures
- Path integrals (introduction)
- Perturbation theory for quantum systems
PDEs and special functions in physics
- Heat, wave, and Schrödinger equations
- Green's functions and fundamental solutions
- Bessel, Legendre, and Hermite functions in separation of variables
- Distribution theory for physics (Dirac delta, introduction)
- Scattering theory (introduction)
STEM / applied
Computational and applied topics
- Numerical methods for Schrödinger-type equations
- WKB and semiclassical approximations
- Group theory in crystallography and band structure (introduction)
- Statistical mechanics: partition functions and ergodicity (overview)
- Field theories and gauge symmetry (mathematical introduction)
Cross-disciplinary methods
- Inverse problems in imaging and geophysics
- Stability and bifurcation in nonlinear PDE models
- Renormalization and scaling in critical phenomena (introduction)
- Symplectic integrators for Hamiltonian systems
- Research-level problem sets aligned to physics department syllabi
Notes
Topics reflect common graduate mathematical physics syllabi at US universities. Exact balance of classical vs quantum content varies by department and instructor background.