This applet illustrates nonrelativistic quantum mechanical scattering from a one-dimensional 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

|R|^{2} is the **reflection coefficient** and |T|^{2} is the **transmission coefficient**. (They satisfy |R|^{2}+|T|^{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 k_{0} with width proportional to Dk. The sum runs in equal steps from k_{0}-Dk/2 to k_{0}+Dk/2. A phase factor is included to center the incident wave packet around x=x_{0} at t=0. The unit of time is chosen such that the frequency is k^{2}/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 x
_{0}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 k
_{0}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.