The noncommutative harmonic oscillator in more than one dimension

Hatzinikitas, Agapitos; Smyrnakis, Ioannis

doi:10.1063/1.1416196

Cited by 85 publications

(75 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, in particular, the position of the particle on the corresponding sphere cannot be localized. In this respect the WQO belongs to the class of models of non-commutative quantum oscillators [34]- [38] and, more generally, to theories with non-commutative geometry [39,40]. It is shown, however, that the non-commutativity between our position and momentum operators is different from the non-commutativity appearing in the most commonly adopted form of generalized Heisenberg commutation relations (see eq.…”

Section: P2mentioning

confidence: 99%

“…The relations (3.3) and (3.4) imply that the WQO belongs to the class of models of non-commutative quantum oscillators [34]- [38] and, more generally, to theories with noncommutative geometry [39,40]. The literature on this subject is vast.…”

Section: Known Properties Of 3d Wqosmentioning

confidence: 99%

“…often adopted in the literature on non-commutative quantum mechanics (see for instance [34]- [38]). In the right hand side of (3.…”

Section: Known Properties Of 3d Wqosmentioning

confidence: 99%

See 2 more Smart Citations

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

King

Palev

Stoilova

et al. 2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(1|3), are further investigated. Within each state space W (p), p = 1, 2, . . ., the energy E q , q = 0, 1, 2, 3, takes no more than 4 different values. If the oscillator is in a stationary state ψ q ∈ W (p) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called "nests". These lie on a sphere centered on the origin of fixed, finite radius ̺ q . The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases (p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases (p = 1, 2), but it is never less than two. In certain states in W (2) (resp. in W (1)) the oscillator is "polarized" so that all the nests lie on a plane (resp. on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations. †

show abstract

Section: P2mentioning

confidence: 99%

Section: Known Properties Of 3d Wqosmentioning

confidence: 99%

See 1 more Smart Citation

The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

King

Palev

Stoilova

et al. 2003

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

show abstract

“…Along the same line of development, quantum mechanics on the noncommutative space has been studied extensively [11,12,13,14,15,16,17,18,19,20]. The complete eigenstates of the noncommutative oscillator [14,15,18,19] have been found analytically in two and higher dimensions.…”

Section: Introductionmentioning

confidence: 98%

“…The complete eigenstates of the noncommutative oscillator [14,15,18,19] have been found analytically in two and higher dimensions. In particular, the spectrum of the noncommutative oscillator is shown to be identical to that of an anisotropic oscillator on the corresponding commutative space.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric quantum mechanics on non-commutative space

Ghosh

2005

Eur. Phys. J. C

View full text Add to dashboard Cite

We construct supersymmetric quantum mechanics in terms of two real supercharges on noncommutative space in arbitrary dimensions. We obtain the exact eigenspectra of the two and three dimensional noncommutative superoscillators. We further show that a reduction in the phase-space occurs for a critical surface in the space of parameters. At this critical surface, the energy-spectrum of the bosonic sector is infinitely degenerate, while the degeneracy in the spectrum of the fermionic sector gets enhanced by a factor of two for each pair of reduced canonical coordinates. For the two dimensional noncommutative 'inverted superoscillator', we find exact eigenspectra with a welldefined groundstate for certain regions in the parameter space, which have no smooth limit to the ordinary commutative space.

show abstract