Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space

Gáliková, Veronika; Kováčik, Samuel; Prešnajder, P.

doi:10.1063/1.4835615

Cited by 35 publications

(45 citation statements)

References 11 publications

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“…The vectorŴ i has not been introduced yet, but it turns out to be closely related to the kinetic part of the Laplace-Runge-Lenz vector, see Ref. 13. Time derivative of velocity expectation is fully determined by expectation of this commutator (Ehrenfest theorem).…”

Section: The Acceleration Operatormentioning

confidence: 99%

The velocity operator in quantum mechanics in noncommutative space

Kováčik

Prešnajder

2013

Journal of Mathematical Physics

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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.Comment: 19 pages, no figure

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Section: The Acceleration Operatormentioning

confidence: 99%

The velocity operator in quantum mechanics in noncommutative space

Kováčik

Prešnajder

2013

Journal of Mathematical Physics

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“…In next section, it will allow us to define the kinetic part of a scalar field theory action as in the commutative case, in terms of the ordinary Laplacian. The closure condition (5.7), written as 5) implies that the integrand is a total derivative, i.e., there exist such a bidifferential operator…”

Section: Jhep08(2015)024mentioning

confidence: 99%

“…Besides their appearence in the quantum gravity context, noncommutative structures on R 3 are very interesting because of their application in the quantization of standard dynamical systems as for example the hydrogen atom [4][5][6]. As for their implications in quantum field theory, such as their renormalization properties and the UV/IR behaviour, which are quite different from noncommutative field theories on Moyal space-time, they are analyzed in [7][8][9]within the noncommutative space R 3 λ first introduced in [10].…”

Section: Introductionmentioning

confidence: 99%

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Kupriyanov

Vitale

2015

J. High Energ. Phys.

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Abstract:We consider linear star products on R d of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, Tr (f g) = Tr (f · g). We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibniz rule holds true up to a total derivative. As a particular example we study the space R 3 θ with su(2) type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibniz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.

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“…Depending on the construction that is used, it can be totally [24] or partially [9] broken, or even totally retained [25].…”

Section: The Relation L · M = 0 and The Energy Shellmentioning

confidence: 99%

Physics of the so q (4) hydrogen atom

Castro

Kullock

2015

Theor Math Phys

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In this work we investigate the q-deformation of the so(4) dynamical symmetry of the hydrogen atom using the theory of the quantum group su q (2). We derive the energy spectrum in a physically consistent manner and find a degeneracy breaking as well as a smaller Hilbert space. We point out that using the deformed Casimir as was done before leads to inconsistencies in the physical interpretation of the theory.

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Laplace-Runge-Lenz vector in quantum mechanics in noncommutative space

Cited by 35 publications

References 11 publications

The velocity operator in quantum mechanics in noncommutative space

The velocity operator in quantum mechanics in noncommutative space

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Physics of the so q (4) hydrogen atom

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