Abstract
We generalize Schrödinger’s factorization method for Hydrogen from the conventional separation into angular and radial coordinates to a Cartesian-based factorization. Unique to this approach is the fact that the Hamiltonian is represented as a sum over factorizations in terms of coupled operators that depend on the coordinates and momenta in each Cartesian direction. We determine the eigenstates and energies, the wavefunctions in both coordinate and momentum space, and we also illustrate how this technique can be employed to develop the conventional confluent hypergeometric equation approach. The methodology developed here could potentially be employed for other Hamiltonians that can be represented as the sum over coupled Schrödinger factorizations.
Highlights
The Hydrogen atom was originally solved by Pauli employing operator methods by discovering the Lie algebra of the SO(4) symmetry in the problem [1]
Similar to how the wave-equation approach can be solved directly in Cartesian coordinates [7], we develop here the Cartesian factorization method for Hydrogen
While it shares some of the characteristics of the spherical coordinate-based factorization method, it is distinctly different
Summary
The Hydrogen atom was originally solved by Pauli employing operator methods by discovering the Lie algebra of the SO(4) symmetry in the problem [1]. In 1940, Schrödinger developed the factorization method for quantum mechanics [4] and employed it to solve Hydrogen as well [5]. This factorization method was extensively reviewed by Infeld and Hull [6]. Similar to how the wave-equation approach can be solved directly in Cartesian coordinates [7], we develop here the Cartesian factorization method for Hydrogen. While it shares some of the characteristics of the spherical coordinate-based factorization method, it is distinctly different. It is surprising that one can discover a new methodology for solving
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