Noncommutative Phase Space Schrödinger Equation with Minimal Length

Abstract

We consider the Schrödinger equation under an external magnetic field in two-dimensional noncommutative phase space with an explicit minimal length relation. The eigenfunctions are reported in terms of the Jacobi polynomials, and the explicit form of energy eigenvalues is reported.

Non-commutativity in polar coordinates

“…An interesting avenue for future work would be to investigate black hole solutions in this curious limit of high non-commutativity to complement previous work in this area. There, we are immediately confronted with the minimal length scale set by quantum gravity or models that involve minimal length uncertainty relations [25,26] which imply it might be more appropriate to consider r 2 ∼ θ . However, the functional form of the polar commutator remains unchanged (see footnote 5) at lowest order when r is of the order √ θ , so that (23) remains valid in this regime.…”

Exact Solutions of the (2+1)-Dimensional Dirac Oscillator under a Magnetic Field in the Presence of a Minimal Length in the Non-commutative Phase Space

“…In this context, many papers were published where a different quantum system in space with the Heisenberg algebra was studied. They are the Abelian Higgs model [19], the thermostatics with minimal length [20], the one-dimensional hydrogen atom [21], the casimir effect in minimal length theories [22], the effect of minimal lengths on electron magnetism [23], the DO in one and three dimensions [24][25][26][27][28], the noncommutative (NC) (2+1)-dimensional DO and quantum phase transition [10], the solutions of a two-dimensional Dirac equation in the presence of an external magnetic field [29], the NC phase space Schrödinger equation [30], and the Schrödinger equation with harmonic potential in the presence of a magnetic field [31].…”

The exact solutions of a (2 + 1)-dimensional Dirac oscillator under a magnetic field in the presence of a minimal length

“…In this context, many papers were published where a different quantum system in space with Heisenberg algebra was studied. They are: the Abelian Higgs model [15], the thermostatics with minimal length [16], the one-dimensional Hydrogen atom [17], the casimir effect in minimal length theories [18], the effect of minimal lengths on electron magnetism [19], the Dirac oscillator in one and three dimensions [20][21][22][23][24], the solutions of a two-dimensional Dirac equation in presence of an external magnetic field [25], the noncommutative phase space Schrödinger equation [26], Schrödinger equation with Harmonic potential in the presence of a Magnetic Field [27].…”