The velocity operator in quantum mechanics in noncommutative space

Abstract

We tested in the framework of quantum mechanics the consequences of a noncommutative (NC from now on) coordinates. We restricted ourselves to 3D rotationally invariant NC configuration spaces with dynamics specified by the Hamiltonian H = H(kin) + U, H(kin) is an analogue of kinetic energy and U = U(r) denotes an arbitrary rotationally invariant potential. We introduced the velocity operator by Heisenberg relation using the commutator of the coordinate and the Hamiltonian operators. We found that the NC velocity operator possesses various general, independent of potential, properties: 1) uncertainty relations indicate an existence of a natural kinetic energy cut-off, 2) vanishing commutator relations for velocity components, which is non-trivial in the NC case, 3) modified relation between the velocity operator and H(kin) 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 the Euclidean group E(4), 5) NC Heisenberg equation for the velocity operator, relating acceleration to derivatives of the potential.

19 pages, no figures

References in corpus (4)

• Spectrum of the non-commutative spherical well
• Gauge Covariance of the Aharonov-Bohm Phase in Noncommutative Quantum Mechanics
• The velocity operator in quantum mechanics in noncommutative space
• Bound state energies and phase shifts of a non-commutative well

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• Eigenvalue problem for radial potentials in space with SU(2) fuzziness
• Non-commutativity in Unified Theories and Gravity
• Magnetic monopoles and symmetries in noncommutative space
• Quantum Mechanics in Noncommutative space