Scattering over a step potential

Using Mathematica to actually do Quantum Mechanics

Copyright @1999 E. R. Bittner & Univ. of Houston

In this set of examples, I show how one can use Mathematica to set up and solve the wave equations for scattering over a step discontinuity.  The basic trick is in matching the right and left hand components of the wave function at the step discontinuity.   

This notebook will be available on the web page and you will able to play with this yourself.

Scattering over a step barrier

Here we take the "classic" scattering problem.  In this case, the initial wave approaches the step in the potential from the left and is either transmitted or reflected.

Calculations

The next cell is "closed". To open up the "closed" cells, click on Cell->Cell Properties-> Cell Open under the "Cell" menu.

Note the transmission coefficient is undefined for E < 1.  In otherwords, in order to be transmitted, I must have enough energy to surmount the barrier.  For E < 1, I have perfect reflection.

Plots of the Wavefunction for various energies

In these plots the real part of the wave is shown in red and the imaginary part in blue.  Note that after the wave passes the barrier, its wavelength becomes much longer.   The function waveplot[energy] plots the wavefunction for a given energy.

Scattering over a sudden change in V

Here the potential energy decreases suddenly at x = 0.

Calculations

Basically, the same as above, just taking V = -1.

Transmission and Reflection Coeficients.

Notice that at as E -> 0, the transmission coef goes to zero and the particle is perfectly reflected from the step!!!

Plots of the wavefunction for various energies.

Again, red is the real part and blue the imaginary part