Verstrooiing

This applet illustrates nonrelativistic quantum mechanical scattering from a square potential well of width L and finite energy depth V>0. The well is located between x=0 and x=L, and the incident wave approaches along the negative x axis.

The energy eigenstates are

where k and are the wavenumbers outside and inside the well, respectively. The e^ikx term is the incident wave, the R term is the reflected wave, and the T term is the transmitted wave.

The unit of distance is chosen such that the state with k=0 has wavenumber

inside the well.

The coefficients are given by

||² is the reflection coefficient and ||² is the transmission coefficient. (They satisfy ||²+||²=1.) The reflection coefficient is given by

This has maxima at those values of k (the scattering resonances) for which

The wavefunction whose probability distribution is animated by the applet has the form

This is a finite sum of energy eigenstates weighted by a Gaussian centered at k₀ with width proportional to Dk. The sum runs in equal steps from k₀-Dk/2 to k₀+Dk/2. A phase factor is included to center the incident wave packet around x=x₀ at t=0. The unit of time is chosen such that the frequency is k²/2.

Notes:

	The location of the well is indicated by the red bar on the graph of the probability density.
	The momentum spectrum is superimposed in blue on the graph of the reflection coefficient.
	The incident wavepacket looks irregular because of interference with the reflected waves.

Instructions for use

	Set the width L of the well (fixed at 30 in the demo) using the scrollbar provided.
	Set the initial position x0 of the incident wavepacket (fixed at -40 in the demo) using the scrollbar provided.
	Set the number of k values (fixed at 6 in the demo) using the popup menu.
	Set the central momentum k0 and width Dk using the scrollbars provided.
	Start, stop, resume, and reset using the buttons provided.
	After a run is stopped, any change in parameters resets the time to t=0.