Cartesian Operator Factorization Method for Hydrogen

We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique this approach is fact that Hamiltonian represented as sum over factorizations in terms of coupled operators depend on momenta each Cartesian direction. determine eigenstates energies, wavefunctions both coordinate momentum space, we also illustrate how technique can be employed develop confluent hypergeometric equation approach. The methodology developed here could potentially other Hamiltonians Schrödinger factorizations.