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Journal of Physics B Atomic and Molecular Physics

Correspondence identities. V. The zero-energy Coulomb problem

A Norcliffe

Published Feb 1, 1971• 6 citations
Publisher

Abstract

For Pt. IV see abstr. A41769 of 1969. The classical and quantal problem of a charged particle moving in an attractive Coulomb potential with total energy zero is considered. A complete correspondence identity is obtained by expressing the kernel of the quantum-mechanical spectral operator IE= delta (E-H) exactly in terms of the classical action function in momentum representation. Use is made of the dynamical symmetry properties exhibited by both the classical and quantal problems, and a path-integral method of evaluating the spectral operator kernel is also presented.

DOI
10.1088/0022-3700/4/2/001

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For Pt. IV see abstr. A41769 of 1969. The classical and quantal problem of a charged particle moving in an attractive Coulomb potential with total energy zero is considered. A complete correspondence identity is obtained by expressing the kernel of the quantum-mechanical spectral operator IE= delta (E-H) exactly in terms of the classical action function in momentum representation. Use is made of the dynamical symmetry properties exhibited by both the classical and quantal problems, and a path-integral method of evaluating the spectral operator kernel is also presented.
The complete quantum-dynamical solutions for a free particle moving in one dimension and for a free particle moving on a ring are obtained in terms of classical path sums. The propagators and Green operator kernels, and, in particular, the kernel of the spectral operator I E = δ( E - H ) are expressed explicitly in this way. Relations between quantum and classical mechanics which hold for low quantum numbers and relatively small values of the classical action are discussed. They are the correspondence identities.
An exact path-integral solution of the Dirac equation with a Coulomb potential is found. The energy-dependent Green's function of the second-order Dirac-Coulomb equation, expressed as a polar-coordinate path integral, is reduced to the exactly solvable path integral for an isotropic harmonic oscillator by a coordinate transformation combined with a local time rescaling. The Green's function of the linear Dirac equation is evaluated with the help of the Biedenharn transformation. The energy spectrum for the bound states is also obtained.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.
Abstract The quantum mechanical kernel K(q″,t″; q′,t′) in Feynman's form of a sum over paths is evaluated by means of the canonical transformation to the system Q, P for which the Hamiltonian vanishes identically. The kernel is found to be completely determined by the classical paths, and is given by The sum is over all classical paths leading from q′ at t′ to q″ at t′. S(q″, P, t′; q′, P, t′) is the classical action along the classical path characterized by the constant momentum P.