A study of the variational aspects for the Fock expansion of the solution of hydrogenic atoms in constant magnetic fields

The study of hydrogen in a constant magnetic field has been one of the most persistent problems in non-relativistic quantum mechanics. Although it is conceptually one of the simplest problems that one can think of, the non-separability of the Schrodinger Equation containing both a Coulombic term and a constant magnetic term in theHamiltonian has made the problem especially difficult. In this dissertation, we apply a solution in the form of the Fock expansion to this problem. It is shown that the logarithmic terms which are associated with the Fock expansion vanish. We then derive and solve a three term recurrence relation in order to find a set of solutions to this Schrodinger equation. Linear combinations of these Fock solutions which satisfy the physical boundary conditions are found, and at the same time upper bounds for the binding energies are found using the Raleigh-Ritz variational principal. It is shown that these same Fock solutions produce lower bounds for the binding energies when theSchwinger variational principal is employed. Therefore, the energies for bound states of hydrogen in a constant magnetic field can be bracketed from both above and below. We furthermore examine another recent method, that of Kravchenko,Liberman and Johansson[15] for solving the hydrogen atom in a constant magnetic field problem. We show that their method is equivalent to examining an eigenchannel in the R-matrix method of Bohm and Fano, and compare their results to the ones we obtain through variational methods on the Fock solution. We find that their method is both accurate and efficient for calculating the binding energies.