Twist Deformation of Rotationally Invariant Quantum Mechanics
arXiv:1002.1019 · doi:10.1063/1.3506406
Abstract
Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, anomalous (the anomaly vanishes when the deformation parameter goes to zero). The deformed operators, Taylor-expanded in the deformation parameter, can be selected to minimize the anomaly. We present the deformations (and their anomalies) of undeformed rotationally-invariant operators corresponding to the harmonic oscillator (quadratic potential), the anharmonic oscillator (quartic potential) and the Coulomb potential.
20 pages
References in corpus (11)
- Spectrum of the non-commutative spherical well
- Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products
- Quantum Fields on the Groenewold-Moyal Plane
- Noncommutativity due to spin
- Wigner Oscillators, Twisted Hopf Algebras and Second Quantization
- Thermodynamics of a non-commutative fermion gas
- Lectures on Hopf Algebras, Quantum Groups and Twists
- Noncommutative quantum mechanics as a gauge theory
- Noncommutative version of an arbitrary nondegenerated mechanics
- Bound state energies and phase shifts of a non-commutative well
- Noncommutative quantum mechanics: uniqueness of the functional description