schrödinger

master thesis · university of zurich · supervised by dr. p. saha

The time-dependent nonlinear Schrödinger equation appears across physics — in fibre optics, Bose-Einstein condensates, and as a macroscopic description of the theoretical dark matter particle, the axion. It governs how a quantum mechanical wave function evolves under an external potential and its own self-interaction.

Computing the wave function directly on a 2D grid is expensive. The split-step Fourier method splits the Hamiltonian into a position-space part (the potential) and a momentum-space part (the kinetic energy), solving each alternately via a leapfrog scheme. Each timestep requires a forward and inverse FFT.

The thesis applies proper orthogonal decomposition (POD) — extracting an optimal basis from simulation data through singular value decomposition. A small number of spatial modes captures most of the wave function's energy. The Galerkin projection then reduces the full PDE to a system of ordinary differential equations in the mode coefficients.

The reduced model reproduces the full simulation with errors on the order of 10⁻⁸ using as few as 15 modes, for both the linear and nonlinear cases. For systems with multiple initial conditions, 50–200 modes suffice to describe an entire family of wave functions.

The work was motivated by the possibility that dark matter axions may obey a macroscopic Schrödinger equation at galactic scales, where direct simulation is computationally prohibitive.

„Denn jetzt sehen wir durch einen Spiegel, undeutlich, dann aber von Angesicht zu Angesicht. Jetzt erkenne ich stückweise, dann aber werde ich erkennen, gleichwie ich erkannt bin." — 1. Korinther 13, 12