Exact Path Integral Treatment of the Hydrogen Atom

Applying a space-time transformation discussed in a previous paper, the Feynman path integral for the Coulomb potential is calculated exactly by reducing it to gaussian form. The resulting Green function leads to a simple derivation of the energy spectrum and the complete (normalized) wavefunctions of the hydrogen atom. In a previous paper [ 1 ] I derived a transformation formula for a class of non-linear space-time transformations in the radial path integral. The formula makes it possible to transform non-gaussian path integrals into gaussian ones. As a very simple illust ration and consistency check we computed in ref. [1] the Feynman kernel of a free particle and (in imaginary time) the distribution function of a particle undergoing brownian motion (Bessel process). In this note we shall apply the transformation formula to the Coulomb potential. To our knowledge, the following treatment represents the simplest path integral determination of the energy spectrum and the complete (nor-realized) wavefunctions of the hydrogen atom. Earlier attempts to treat the Coulomb potential by means of path integrals can be found in the papers listed in ref. [2]. The first complete path integral treatment of the hydrogen atom was carried out by Kleinert and Duru [3] (see also ref. [4]). In ref. [ 1 ] we considered three-dimensional quantum systems described by the hamiltonian H = p2/2 + V with spherically symmetric potentials V = V(r) (h = m = 1). In spherical coordinates the Feynman kernel K can be expanded into "partial waves" (see ref. [1 ], eq. (1)), where the radial kernel K t (with fixed angular momentum l) is given by the radial path integral r(T)=r b T) Kl(T:rb.ralV)= f Dr(t) exp(i f 1)/2r 2-V(r)]. (1) r(O)=r a 0 K l determines for a given angular momentum I the time evolution of the system from lime t a to the later time t b (T = t b-ta) for fixed radii r a = Ix(ta)h r b = IX(tb)l. For the class of space-time transformations t ~ r, r(t) R(r) specified by dr=~(2-v)2r-V dt, R=r l-v~2, v<2, (2) we derived in ref. [1] the transformation formula kl(E;rb,ratV)= 22~_.v (rbra) v/4 ; dr KLv(r;rd-U/2, rl-v/2lWv). o Here k I denotes the time-independent radial kernel defined by