Correspondence identities.

Abstract

Bohr's correspondence principle relates quantum phenomena to classical mechanics in the limit ~h / S→0, as the dynamical action variables S become large in comparison with Planck's constant. Relations between quantum and classical mechanics which hold even for low quantum numbers and relatively small values of the classical action are called correspondence identities. For the Coulomb potential the following three correspondence identities are known: 1) The Rutherford Scattering identity The quantum-mechanical and classical angular differential cross sections for the scattering of a charged particle by a fixed charge are the same. (2) The Bohr-Sommerfeld identity The old quantum theory, which postulated that only those orbits occur for which the action around the classical path of a periodic system is a multiple of 2π~h gives the correct energy levels of the hydrogen atom and hydrogenic ions. (3) The Fock identity The classical and quantal microcanonical distributions in momentum of the electron in the hydrogen atom are equal for all values of the classical energy equal to the levels En. These correspondence identities concern the system of electron and proton and in this thesis it is shown how each of the identities follows from a complete correspondence identity whereby the non-relativistic quantum dynamics of the system is obtained from the solution of the corresponding classical problem. A complete correspondence identity is provided by expressing the kernel of the spectral operator IE = δ(E — H) in momentum representation for all real non-zero energies E, as a sum over paths of terms containing the classical action. For the bound states the paths are the classical paths. For positive energies they are the generalised classical paths which arise from the analytic continuation in energy of the bound state paths. The generalised classical paths are built up from the paths of scattering of both electrons and positrons and are needed to obtain the quantal barrier penetration in momentum space. Because of the similarities between the techniques used in this thesis to provide a complete correspondence identity and those of the phase-integral approximation the results are compared wherever possible with those of Gutzwiller (1967). Finally, a general derivation of a scattering cross section from the spectral operator is presented which does not require an explicit treatment of the long-range distortion in the case of the Coulomb potential.