On a q-analogue of the spin-orbit coupling

Based on the tensor method, a $q$-analogue of the spin-orbit coupling is introduced in a $q$-deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations $j = \ell \pm 1/2$ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of an Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.