time-dependent-schrodinger

Modelling of Time-dependent Schrödinger equation using finite difference method

This project presents the numerical analysis of time-dependent Schrödinger equation (TDSE) in 1D and 2D. Potential term is considered time-independent, to model some well-known examples of harmonic oscillator / finite-well / tunnelling / free-particle. Moreover, single-slit and double-slit experiments were numerically simulated in 2D.

The numerical analysis of 1D TDSE is currently limited with explicit methods: Explicit Euler and Crank-Nicolson methods. On the other hand, 2D TDSE is experimented with both explicit and implicit methods: Explicit Euler and Alternating Direction Implicit (ADI) methods. While the animations given below demonstrate some exemplary cases, the derivation of numerical models are reported, (see report).

The animation below shows the tunneling phenomena in 1D, where the particle is confined within 1D infinite potential well while a finite potential barrier (width = 0.5, at $x = 1$, V(x) = 0.5) is introduced.

The animation below shows the single and double slit experiments, where the particle is confined within 2D infinite potential well while finite potential barriers creating the slit geometries (width = 0.5, slit size = 1, at $x = 1$, V(x,y) = 100) are introduced.