The variable phase method as a tool for scattering calculations of gases at ultracold temperatures

A collision experiment is the most common tool for probing interatomic interactions. In all types of collisions, the common ground between theoreticians and experimentalists is the cross section, which occupies a position in “microscopic” kinetics that is equivalent to that of the rate constant in “macroscopic” kinetics^[1]. The conventional method of calculating the cross section is to integrate the radial Schrödinger equation in order to obtain phase shifts, from which one can then obtain any other scattering property (e.g. the cross section, scattering amplitude, the S-matrix, etc.). An alternative and less common approach, termed the “variable phase method”^[2] or the “phase function method”^[3], obtains the phase shifts from calculating the phase function. The advantage of working with a phase function is that instead of comparing the phase of a radial wave function with that of an asymptotic form, we get the phase shift as a function of r corresponding to a potential truncated at our point of choice. Therefore, this approach provides physical intuition about the relation between the potential and the phase shift. Here we apply the phase function to study cold collisions between atoms and molecules. We present preliminary results for the ultracold collision of metastable He* atom with H₂ molecule^[4], including phase shifts, differential and integral cross sections.

[1] Ian W. M. Smith, Kinetics and dynamics of elementary gas reactions (Butterworth, London, 1980)

[2] F. Calogero, Variable phase approach to potential scattering (Academic, New York, 1967)

[3] V. V. Babikov, Method of Phase Functions in Quantum Mechanics (Nauka , Moscow, 1976)

[4] M. Pawlak, Y. Shagam, A. Klen, E. Narevicius, N. Moiseyev, J. Phys. Chem. A, 121, 2194 (2017)