Phase-Integral Approximation in Momentum Space and the Bound States of an Atom

The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrödinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman-type functional integral which was recently proposed by Garrod for the energy-momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound-state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.