Nonrelativistic quantum mechanics in a noncommutative space

Luigi Accardi

DongShik Kang

Pulak Ranjan Giri, P. Roy

Halqem Nizamidin, Abduwali Anwar, Sayipjamal Dulat A +1 more

We study the noncommutative nonrelativistic quantum dynamics of a neutral particle, which possesses an electric qaudrupole moment, in the presence of an external magnetic field. First, by introducing a shift for the magnetic field, we give the Schrödinger equations in the presence of an external magnetic field both on a noncommutative space and a noncommutative phase space, respectively. Then by solving the Schrödinger equations both on a noncommutative space and a noncommutative phase space, we obtain quantum phases of the electric quadrupole moment, respectively. We demonstrate that these phases are geometric and dispersive.

Tiago C. Adorno, Mario Cesar Baldiotti, Dmitri M Gitman

We construct a nonrelativistic wave equation for spinning particles in the noncommutative space (in a sense, a $\ensuremath{\theta}$ modification of the Pauli equation). To this end, we consider the nonrelativistic limit of the $\ensuremath{\theta}$-modified Dirac equation. To complete the consideration, we present a pseudoclassical model (\`a la Berezin-Marinov) for the corresponding nonrelativistic particle in the noncommutative space. To justify the latter model, we demonstrate that its quantization leads to the $\ensuremath{\theta}$-modified Pauli equation. We extract $\ensuremath{\theta}$-modified interaction between a nonrelativistic spin and a magnetic field from such a Pauli equation and construct a $\ensuremath{\theta}$ modification of the Heisenberg model for two coupled spins placed in an external magnetic field. In the framework of such a model, we calculate the probability transition between two orthogonal Einstein-Podolsky-Rosen states for a pair of spins in an oscillatory magnetic field and show that some of such transitions, which are forbidden in the commutative space, are possible due to the space noncommutativity. This allows us to estimate an upper bound on the noncommutativity parameter.

ANTON Z. CAPRI