The hydrogen atom: Quantum mechanics on the quotient of a conformally flat manifold

Abstract

The regularization of the Kepler problem proposed by Kustaanheimo and Stiefel provides an example of quantum mechanics on the quotient of a conformally flat manifold.

Path integral for a relativistic spinless Coulomb system

“…1) While the path integral of the nonrelativistic Coulomb system has been solved some 15 years ago [1] and further discussed by many authors [2] - [14], so that it has become textbook material [15], the relativistic problem has remained open. The purpose of this note is to fill this gap.…”

Recent progress in path integrals

“…He had been active in this field long before I stumbled into it. I still remember my first encounter with his work in a paper he had written with Ringwood in 1980 [1], where he criticized my earliest contribution to the subject published with Duru in 1979. We had found the solution of the path integral of the hydrogen atom [2], and Jozef called it ''gentleman-like".…”

Group theory and orbital fluctuations of the hydrogen atom

“…When I met Professor Barut in 1965 as a graduate student in Colorado he suggested to me to try and calculate all dynamical properties of the H atom in terms of group operations within some extension of the symmetry group SO (4). In this way we should be able to avoid the use of the full algebra of canonical position and momentum operators 2,/~.…”

“…In this way we should be able to avoid the use of the full algebra of canonical position and momentum operators 2,/~. The first idea was to relate the matrix elements of/~ and 2 to those of generators of a suitable Lie algebra extending that of SO (4). However, when evaluating the matrix elements of the position operator between the H atom wave functions, there was an immediate obstacle: The operator 2 connected all principal quantum numbers n with each other.…”

“…We therefore removed this factor by a scale transformation. After this, the matrix elements of 2, ~0 did indeed connect only neighboring n. They turned out to be generators of the Lie algebra of the group SO (4,2). There are 15 generators denoted by LA~ with LA~ = --LEA and A, B = 1 ..... 6.…”