SCATTERING

Film Loop: Scattering in One Dimension - 3)Edge Effects

Length(min.):4:00, 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 potential well; the sharpness of the edges of the well are varied. The behavior of the wave packet is shown for three well shapes: (a) sharp edges and infinitely steep walls; potential has zero surface thickness; (b) rounded edges and slightly sloped walls; 90% to 10% falloff distance is about 1/8 the well width, or a thin potential surface; (c) gently varying well shape; 90% to 10% falloff distance is about 1/4 the well width, or a thick potential surface.

DISCUSSION: The horizontal coordinate used in the display is the X-axis: the potential well is symmetrical about X = 0. For the well, the vertical coordinate is the potential energy; a Wood-Saxon potential has been described in American Journal of Physics, 35, 177 (August 1967). For the wave packet, the vertical coordinate is the position probability density, Phi(x,t)-squared. The average energy of the wave packets is one-half the maximum depth of the well, and is the same for all three examples. As the boundary of the potential becomes more diffuse, the structure of the wave packet during the scattering event becomes less complicated. The figure, taken from the last sequence in the film, compares the structure of the wave packet at similar times during an interaction with a square well (top) and the softer Wood-Saxon well (bottom). Other pertinent information concerning the formulation of this type of problem, integration techniques, and computer-input parameters has been published in American Journal of Physics, 35, 177 (March 1967).