SCATTERING

Film Loop: Scattering in One Dimension 4)Momentum Space

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

This computer-animated sequence shows the time development of a Gaussian wave packet in two representations: configuration space and momentum space. In each representation the same wave packet moves into and out of the region of a finite square-potential well. In each case, the energy of the packet is equal to one-half the well depth. The event in configuration space is shown first (same as first sequence in 80-4013 and 80-4021); then the same event in momentum space; finally, a simultaneous comparison of both representations.

DISCUSSION: The displays with the dark background represent one-dimensional configuration space; the origin is in the center of the horizontal axis. The vertical axis is the position probability density Phi(x,t)-squared. The displays with the light background represent one-dimensional momentum space; zero momentum is in the center of the horizontal axis. The vertical axis is in the momentum-probability density M(k,t)-squared A note on this momentum-space sequence has been published in American Journal of Physics, 36, May 1968. The figure shows the wave packet near the middle of the scattering event in both X-space (top) and Phi-space (bottom). The probability density in X-space moves into the region of the potential, develops rapid oscillations, and begins to reflect part of the packet. At the same time, the probability density in Phi-space develops high-momentum components and a packet begins to grow at negative momenta. Because a free-particle momentum-probability density is independent of time, the wave packet in momentum space does not change until the particle, in configuration space enters the region of the potential; after the particle, in configuration space, has left the region of the potential, the altered shape of the packet, in momentum space, again remains constant. (The behavior of the free-particle position-probability density is shown in 80-4054.) 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).