The main point of this paper is to examine a “hidden” dynamical symmetry connected with the conservation of Laplace-Runge-Lenz vector (LRL) in the hydrogen atom problem solved by means of non-commutative quantum mechanics (NCQM). The basic features of NCQM will be introduced to the reader, the key one being the fact that the notion of a point, or a zero distance in the considered configuration space, is abandoned and replaced with a “fuzzy” structure in such a way that the rotational invariance is preserved. The main facts about the conservation of LRL vector in both classical and quantum theory will be reviewed. Finally, we will search for an analogy in the NCQM, provide our results and their comparison with the QM predictions. The key notions we are going to deal with are non-commutative space, Coulomb-Kepler problem, and symmetry.
The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate
∣n,l,m⟩ with other degenerate states
∣n,l±1,m′⟩. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.