Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 016, 18 pages      quant-ph/0603077      http://dx.doi.org/10.3842/SIGMA.2007.016

Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics

Christiane Quesne a and Volodymyr M. Tkachuk b
a) Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
b) Ivan Franko Lviv National University, Chair of Theoretical Physics, 12 Drahomanov Str., Lviv UA-79005, Ukraine

Received November 22, 2006; Published online January 31, 2007

Abstract
Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators x, p. The resulting Hamiltonians contain a contribution proportional to p4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials.

Key words: deformed algebras; uncertainty relations; supersymmetric quantum mechanics; shape invariance.

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