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SIGMA 3 (2007), 016, 18 pages quant-ph/0603077
https://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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