Wave function propagation

Simulation of the propagation of a quantum particle's wave function through potentials such as a step potential, potential barrier, potential well, and harmonic oscillator. The initial wave function is a Gaussian wave packet whose time evolution is calculated numerically using the propagator $\hat{U}(t; t_0) = e^{-i\hat{H}(t-t_0)/\hslash}$.

The evolution is calculated numerically by applying the time-evolution operator to the initial Gaussian wave packet. One of the most counterintuitive results observable at a potential barrier is quantum tunneling: even when the energy is lower than the barrier height, a nonzero fraction of the wave packet passes through to the other side—a phenomenon with no analogue in classical mechanics.

Marco P. M. de Souza