New vector operators for the nonrelativistic Coulomb problem

The purpose of this paper is to obtain a Lie algebra with the property that suitable linear combinations of its basis operators are ladder operators for the quantum number l in the eigenkets ‖nlm〉 of the nonrelativistic Coulomb problem. By analogy with the way in which the Pauli-Lenz operator A is constructed from the classical Laplace-Runge-Lenz vector ${\mathrm{A}}_c$, we deduce a new Hermitian vector operator B as a quantum-mechanical analog of a conserved classical vector ${\mathrm{B}}_c$ which is orthogonal to ${\mathrm{A}}_c$ and the orbital angular momentum L, and has the property $B_c$=$A_c$. Apart from a standard modification, B→${\mathrm{B}}_c$ as ħ→0. We show that the combination A+iB provides abstract ladder operators for the quantum numbers l, and l and m, and we calculate the coefficients for these transformations. The operators L and B are a basis for a Lie algebra which is the same as that of L and A, namely, O(4). The Hermitian operators H, $B_z$, and $L_z$ are a set of commuting operators and we determine the corresponding eigenvalues and eigenkets. Finally we show that the ten operators A, B (both suitably modified), L, and (${ħ}^{\mathrm{-}2}$ ${\mathrm{L}}^2$ ${+(1/4))}^{1/2}$ can be combined to form a Hermitian basis for a Lie algebra, the de Sitter algebra O(3,2).