Quantum Evolution in One Dimensional Disordered Systems: Localisation and Oscillations

This work is a simple example of the quantum dynamics of a particle in a disordered system in one dimension. In particular, we illustrate numerically the phenomenon of Anderson localisation of a wave packet using a basic model constructed with small random rectangular potential barriers. Also, we study the dynamics of a quantum particle in a disordered potential formed by an harmonic oscillator perturbed by random rectangular barriers. To show the effects of disorder on the dynamics of the system, we compare the time evolution of the wave function of the unperturbed harmonic oscillator with the wave function of the disordered system. We do this by taking the scalar product between the unperturbed and perturbed wave functions at each timestep for different values of the perturbation parameters affecting the disordered wave packet.