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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 eikx 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 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.


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