The velocity operator in quantum mechanics in noncommutative space

We tested the consequences of noncommutative (NC from now on) coordinates x(k), k = 1, 2, 3 in the framework of quantum mechanics. We restricted ourselves to 3D rotationally invariant NC configuration spaces with dynamics specified by the Hamiltonian (H) over cap = (H) over cap (0) + (U) over cap, where (H) over cap (0) is an analogue of kinetic energy and (U) over cap = (U) over cap((r) over cap) denotes an arbitrary rotationally invariant potential. We introduced the velocity operator by (V) over cap (k) = -i[(X) over cap (k), (H) over cap] ((X) over cap (k) being the position operator), which is a NC generalization of the usual gradient operator (multiplied by -i). We found that the NC velocity operators possess various general, independent of potential, properties: (1) uncertainty relations [(V) over cap (i), (X) over cap (j)] indicate an existence of a natural kinetic energy cut-off, (2) commutation relations [(V) over cap (i), (V) over cap (j)] = 0, which is non-trivial in the NC case, (3) relation between (V) over cap (2) and (H) over cap (0) that indicates the existence of maximal velocity and confirms the kinetic energy cut-off, (4) all these results sum up in canonical (general, not depending on a particular form of the central potential) commutation relations of Euclidean group E(4) = SO(4)(sic)T(4), (5) Heisenberg equation for the velocity operator, relating acceleration (V) over cap over dot(k) = -i[(V) over cap (k), (H) over cap] to derivatives of the potential. (C) 2013 AIP Publishing LLC.