Abstract:
The Fock-Tani (unitary) transformation of the second- quantized Hamiltonian gives a representation which treats reactants and products symmetrically, and composites exactly. Each term in the Fock-Tani potential corresponds to a specific physical process and contains terms orthogonalizing continuum states to the bound states. The difficulty in carrying out this transformation can be lessened by working in a center of mass system, giving (n-1) reduced mass particles. After a general analysis of such systems, the Fock- Tani transformations in the 3→2-body case are carried out for the reactions a⁺+(b⁺c⁻)→(a⁺c⁻)+b⁺ and a⁻+(b⁺c⁻)→(a⁻b⁺)+c⁻. It is found that for (2) the transformation in the symmetrical reduced mass system can easily be carried out, but the Jacobi reduced mass system requires the more complicated d-matrix approach. This transformation has not yet been attempted in the full 3-body system but is likely to be as difficult as that for (1). First order differential and total cross sections are computed for resonant charge transfer in (1) for a proton- hydrogen initial state. The Fock-Tani T-matrix for the initial-state Jacobi system is found to be identical to that for the full 3-body system. That for the symmetrical reduced mass system gives an error of order l/mprot in the incident wave vector. A comparison of the Jacobi version and a previous special case Fock-Tani transformation, where the proton mass is taken as infinite, is also made. Cross sections for (ls→ls) positronium formation in positron-hydrogen collisions, calculated using the same program as for the proton-hydrogen case, are found to disagree with the previous Fock-Tani result, probably due to lack of convergence of the previous result. Cross sections for reactions (1) involving muons in hydrogenic isotopes (of interest in quantum electrodynamics and catalyzed fusion) are also calculated. Finally, extension of the results to higher order is considered. Polarized Schrodinger wave functions for a system containing a hydrogenic atom and a fully kinetic external charge are found to first order. These would be used in the Fock-Tani matrix elements to account for some initial- and final-state effects. Calculations of distorted second-quantized states and second and third order T-matrix elements are also outlined.