Gonzalo E. Ordonez
My supervisor T. Petrosky, together with T. Miyasaka and I have worked on a system of three quantum-mechanical particles, interacting through a short-range potential. For simplicity we have considered particles in a one-dimensional space. The particles are represented by large wavepackets in space representation.
The pictures below (created by T. Miyasaka and I) show the distribution of scattering probability, as a function of the momenta of the particles, expressed in Jacobi coordinates (the horizontal axis is proportional to the relative momentum of two of the particles and the vertical axis is proportional to the momentum of the third particle). The collision sequence considered is a 1-2 collision followed by a 2-3 collision.
The first picture is taken at an early time during the collision of the particles. Since the wavepackets representing the particles are large, they are able to reach an asymptotic state while they are still colliding. This is shown in the second picture. We have performed the theoretical analysis of this asymptotic state using the Liouville-space formulation, which has been worked out in recent years by Prof. Prigogine's group. The third picture is taken at a time much larger than the duration of the collisions, and corresponds to the S-matrix regime. Note that the two horizontal lines that appeared during the collision have vanished. The horizontal lines correspond to transient processes that occur during the collision, and disappear after the collision between the three particles is finished.
We predicted the transient effects using the Liouville space formulation, and verified their existence through numerical solutions of Schrodinger's equation shown in the pictures below.
The Liouville-space formalism makes explicit the appearance of irreversible phenomena in both classical and quantum systems. These phenomena are caused by the continued interaction between many particles. The scattering of three particles during the collision already contains the "seed" of such irreversible phenomena.
The usual scattering theory (based on the S-matrix asymptotic description) does not consider the time scales during the interaction. In order to have a more accurate description of non-equilibrium irreversible processes an extension of scattering theory is thus necessary. The Liouville-space formulation provides a framework for this extension.
Figure 1: During the collision - early stage
Figure 2: During the collision - asymptotic state
Figure 3: After the collision - S matrix result
These results are published in Physical Review A vol. 53, 4075 (1996). The title is "Extension of scattering theory for finite times: three-body scattering", by T. Petrosky, G. Ordonez and T. Miyasaka.