Quantum Scattering over a step barrier using Discrete Variable Representations

E. R. Bittner

Started: 7-Sept.-1999

Introduction

Description

Here we consider the scattering of a quantum wavepacket over a step barrier using a time dependent description.  The original purpose of these calculations was to provide a set of test cases for my Bohmian dynamics code.  They also provide a nice series of pedagogic examples demonstrating what happens when a quantum wavefunction encounteres a sudden change in the potential energy.  In each case, the potential is a step function:  

The initial wavefunction is taken to be a gaussian centered left and is given an initial momentum kick towards the barrier.  We also calculate the expectation value of the position operator, ⟨x(t)⟩ = ⟨ψ(t)|x|ψ(t)⟩.  In the animations, this is shown as a red dot.

Implementation

For details regarding the implementation of these Mathematica routines, refer to the QuantumDVR.nb.  

Initialization

Loads the basic module

Load a memory conservation package:

<<Utilities`MemoryConserve`;

Scattering over step--below the barrier

Calculations

The basic code used in each of these calculations is as follows.

Graphical Analysis

Evolution of

Scattering over step--Above the barrier

Energy = 0.02325
Barrier Height = 0.004

Graphical Analysis/Animation

Evolution of

Scattering over step--just over the top

Calculation at E = 0.0405029, Barrier Height = 0.04

Graphical Analysis

Evolution of

Notice the fraction transmitted and reflected.  Also note that  ⟨x(t)⟩ (red dot) almost goes to a stop at when ψ(t) reaches the barrier and spits into a reflected and transmitted portion.  At this point  ⟨x(t)⟩ no longer gives a good description of the evolution of the center of the wavefunction.  However, it is the average of the two distribitions.