Factorizations of two vector operators for the Coulomb problem

We consider the factorization of two vector constants of the motion ${\mathrm{C}}^{\pm }$, which were recently introduced for the Coulomb problem [O. L. de Lange and R. E. Raab, Phys. Rev. A 34, 1650 (1986)]. The operators ${\mathrm{C}}^{\pm }$ are quantum-mechanical analogs of the classical conserved vectors (1∓iL^×${)\mathrm{A}}_c$ where L is the orbital angular momentum vector and ${\mathrm{A}}_c$ is the Laplace-Runge-Lenz vector. It is shown that ${\mathrm{C}}^{\pm }$ can be factored in two different ways to yield operators which, apart from their dependence on the constants of the motion ${\mathrm{L}}^2$ and H, are linear in either p or r. In this way we obtain 16 abstract operators. The properties of these operators are investigated and the following observations are made: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of ${\mathrm{L}}^2$ and $L_z$. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) Two operators factorize an operator related to the Hamiltonian. In linearized, differential form they are the two ladder operators for the radial part of the coordinate-space wave functions which were discovered by Schrödinger. (iii) Two operators yield a factorization which is related to Hylleraas’s equation. In linearized, differential form they are ladder operators for the radial part of the momentum-space wave function.