Solution of Coulomb Path Integral in Momentum Space

The path integral for a point particle in a Coulomb potential is solved in momentum space. The solution permits us to answer for the first time an old question of quantum mechanics in curved spaces raised in 1957 by DeWitt: The Hamiltonian of a particle in a curved space must not contain an additional term proportional to the curvature scalar R, since this would change the level spacings in the hydrogen atom. ∗[email protected] , http://www.physik.fu-berlin.de/ ̃kleinert 1. One should think that by now everything interesting is known about the path integral of the Coulomb problem describing the physics of the hydrogen atom. There exists a comprehensive textbook [1] in which this subject is treated at great length. However, the existing solution applies only to the fixed-energy amplitude in position space. The momentum space problem has so far remained untackled, and the purpose of this note is to fill this gap. Apart from our desire to complete the path integral description of the simplest physical object of atomic physics, the present note is motivated by another long-standing open problem in the quantum mechanics of curved spaces, first raised by Bryce DeWitt in 1957 [2]: Is the Hamilton operator for a particle in curved space obtained by merely replacing the euclidean Laplace operator in the kinetic energy by the Laplace-Beltrami operator ∆, or must we add a term proportional to h̄R, as suggested by various path integral formulations of the problem in the literature [3]? Only experiment can decide what is right, but up to now no physical system has been contrived where the presence of an extra R-term could be detectable. All experimentally accessible systems in curved space have either a very small R caused by gravitation, whose detection is presently impossible, or a constant R which does not change level spacings, an example for the latter being the spinning symmetric and asymmetric top [1]. Surprisingly, the solution developed in this note supplies an answer to this problem by forbidding an extra R-term, which although being constant would change the level spacings in the hydrogen atom. 2. Starting point for our treatment is the path integral formulation for the matrix elements in momentum space fo the resolvent operator R̂ ≡ i/(E − Ĥ) with the Coulomb Hamilton