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

Description/Abstract

The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl(13), are further investigated. Within each state space W(p), p = 1, 2, ..., the energy Eq, q = 0, 1, 2, 3, takes no more than four different values. If the oscillator is in a stationary state ψq 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 centred 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 2. In certain states in W(2) (respectively in W(1)) the oscillator is 'polarized' so that all the nests lie on a plane (respectively 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.