What's Next?

What's Next

You have implemented the core toolkit of mathematical physics. Here are natural next steps:

  • Signal Processing — The DFT you implemented is the foundation; FFT algorithms, digital filters, and spectral analysis extend it to real-world signals.
  • Quantum Mechanics — Hermite and Legendre polynomials are the wavefunctions of the harmonic oscillator and hydrogen atom. The perturbation theory lesson is the first step toward time-dependent quantum mechanics.
  • General Relativity — Tensor operations and Christoffel symbols are the language of curved spacetime and Einstein's field equations.
  • Differential Equations — The RK4 and heat equation methods generalize to stiff ODEs, finite element methods, and spectral solvers.

Resources

  • Mathematical Methods for Physics and Engineering by Riley, Hobson & Bence — The most comprehensive undergraduate reference; covers every topic in this course.
  • Mathematical Physics by Hassani — Rigorous graduate-level treatment of all the methods here.
  • Numerical Recipes by Press et al. — The practical guide to implementing numerical algorithms.
  • Methods of Mathematical Physics by Courant & Hilbert — The classic two-volume reference for PDEs and spectral theory.
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