SCATTERING

Film Loop: Scattering in One Dimension - Part1)Barriers

Length(min.):3:00, Color: No, Sound: No

This computer-animated sequence shows the time development of a Gaussian wave packet as it moves into and out of the region of a finite square-potential barrier. The reflection from the barrier and the penetration into or through the barrier are shown for incident particle energies equal in magnitude to (a) one-half the barrier height, (b) the barrier height, and (c) twice the barrier height.

DISCUSSION:The horizontal coordinate used in the display is the X-axis; the potential barrier is symmetrical about X = 0. For the barrier, the vertical coordinate is potential energy. For the wave packet, the vertical coordinate is the position probability density, Phi(x,t)-squared. In each example the initial value of the probability density is the same even though the particle energy increases by a factor of two in each successive example. In the last case, where the particle energy is twice the barrier potential, two weak reflections are seen - from the near and from the far barrier walls. In the second case, where the average particle energy equals the barrier potential, a portion of the wave packet is trapped inside the barrier for a relatively long period of time; note that the peak of the trapped part of the packet appears to bounce back and forth between the barrier walls as the probability leaks out from both walls.

The rapid oscillations which appear when the packet is close to the potential (see figure) are accurate solutions to the time-dependent Schrodinger equation and are not the result of computer error or programming approximations. Detailed information concerning the formulation of the problem, integration techniques, initial conditions, and computer input parameters has been published in American Journal of Physics, 35, 177 (March 1967).