Numerical Solution of the 1-D Schrodinger Equation

Research question

How can numerical methods approximate bound-state solutions of the one-dimensional Schrodinger equation, and how do the results compare with known quantum oscillator behaviour?

Background

This project sits between Year 2 quantum physics, mathematical methods, and planned Year 3 quantum mechanics and numerical methods. It is intended to become a reproducible computational quantum mechanics notebook or report.

Mathematical model

The central equation is the time-independent one-dimensional Schrodinger equation. For oscillator models, the potential can be chosen so that numerical eigenvalues and eigenfunctions have known analytic structure for comparison.

Methods

The planned methods include finite-difference discretisation, matrix eigenvalue problems, boundary-condition checks, grid-refinement studies, and comparison with analytic or reference solutions where available.

Implementation

The implementation will likely use Python with NumPy, SciPy, Matplotlib, and possibly PyTorch for later experimentation. Code will be linked only after it is reproducible and documented.

Results

Expected outputs include eigenvalue tables, eigenfunction plots, convergence checks, and comparisons between numerical and analytic quantum oscillator results.

Validation

Validation should include dimensional checks, convergence under smaller grid spacing, boundary sensitivity, and comparison with known oscillator eigenvalues and eigenfunctions.

Limitations

Early versions may overfit to textbook cases. A public version should explain which potentials are solved, which assumptions are numerical rather than physical, and where the method breaks down.

What I learned

The project is a practical route into computational quantum mechanics and numerical linear algebra.

Future work

Future extensions could include time-dependent evolution, non-harmonic potentials, uncertainty visualisation, and a short technical note on the discretisation method.