New Exact Solution of the Bound States for the Potential Family V(r)=A/r<sup>2</sup>-B/r+Cr<sup>k</sup> (k=0,-1,-2) in both Noncommutative Three Dimensional Spaces and Phases: Non Relativistic Quantum Mechanics

Abstract

In present work we obtain the modified bound-states solutions for central family V(r)=A/r2-B/r+Crk (k=0,-1,-2) in both noncommutative three dimensional spaces and phases. It has been observed that the energy spectra in ordinary quantum mechanics was changed, and replaced degenerate new states, depending on two infinitesimals parameters Θ and θ corresponding the noncommutativity of space and phase, in addition to the discrete atomic quantum numbers: j, l, sz=+-1/2 and corresponding to the two spins states of electron by (up and down) and non polarized electron. The deformed anisotropic Hamiltonian formed by three operators: the first describes usual the usual family potential, the second describe spin-orbit interaction while the last one describes the modified Zeeman effect (containing ordinary Zeeman effect).