Coulomb problem in an angular-momentum basis: An algebraic formulation.

Abstract

We show that a representation-independent, spectrum-generating algebra for the Coulomb problem in an angular momentum basis can be obtained by quantizing two complex, time-dependent, classical vectors, ${\mathrm{D}}_{c}$${=\mathrm{F}}_{c}$+i${\mathrm{G}}_{c}$ and ${\mathrm{D}}_{c}^{\mathrm{*}}$. The approach is based on an analogy with a treatment of the isotropic harmonic oscillator [A. J. Bracken and H. I. Leemon, J. Math. Phys. 21, 2170 (1980)], and on work in which classical constants of the motion were quantized to yield shift operators for angular momentum in the Coulomb problem [O. L. de Lange and R. E. Raab, Phys. Rev. A 34, 1650 (1986)]. By construction ${\mathrm{F}}_{c}$ and ${\mathrm{G}}_{c}$ are orthogonal to the orbital angular momentum L, their moduli have equal, constant magnitude, and they rotate about L. In this construction we use ${\mathrm{A}}_{c}$ (the Laplace-Runge-Lenz vector) and ${\mathrm{A}}_{c}$\ifmmode\times\else\texttimes\fi{}L^ as basis vectors. ${\mathrm{F}}_{c}$ and ${\mathrm{G}}_{c}$ contain an undetermined phase factor exp(i\ensuremath{\delta}). ${\mathrm{D}}_{c}$ and ${\mathrm{D}}_{c}^{\mathrm{*}}$ are quantized by requiring that the resulting operators should be shift operators for energy and angular momentum in the bound-state kets \ensuremath{\Vert}nlm〉. This determines the operators ${\ensuremath{\Delta}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the classical phase factors exp(\ifmmode\pm\else\textpm\fi{}i\ensuremath{\delta}). In the coordinate and momentum representations of wave mechanics respectively, ${\ensuremath{\Delta}}^{\ifmmode\pm\else\textpm\fi{}}$ are the dilatation operators for coordinate-space and momentum-space wave functions. The shift operators can be factorized to yield 20 abstract operators. Apart from their dependence on ${\ensuremath{\Delta}}^{\ifmmode\pm\else\textpm\fi{}}$ and constants of the motion, ten of these are linear in p, eight are linear in r, and two are quadratic in r. Apart from ${\ensuremath{\Delta}}^{\ifmmode\pm\else\textpm\fi{}}$, these operators can be linearized by replacing constants of the motion with their eigenvalues: In the coordinate and momentum representations of wave mechanics they are first-order differential operators. The shift operators are part of a Hermitian basis for a spectrum-generating algebra which is shown to be SO(2,1)\ensuremath{\bigoplus}SO(3,2).