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

Abstract

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 p 4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials.

Minimal areas fromq-deformed oscillator algebras

“…These relations are very reminiscent of the q-deformed oscillator algebra studied in this context for instance in [9,10,11,12,13,14,15,16]. This example and the one in the previous subsection indicate that dynamical spacetime relations will naturally lead to deformed Fock spaces.…”

“…This means in such spaces one has almost inevitably definite fundamental distances below which no substructure can be resolved [9,10,11,12,13,14,15,16,17]. Recently some of us proposed [18] a consistent dynamical noncommutative space-time structure in a two dimensional space which leads to a fundamental length in one direction, implying that objects in these spaces are of string type.…”

“…Minimal lengths have been known and studied for some time [9,10,11,12,13,14,15,16] in simultaneous x, p-measurements as a consequence of a deformation of the x, pcommutator. In [18] it was demonstrated explicitly that they also result in simultaneous x, y-measurements as a consequence of the dynamical noncommutativity of space-time.…”

Quadratic algebras and position-dependent mass Schrödinger equations

“…They are indeed very useful in the study of many physical systems, such as electronic properties of semiconductors and quantum dots, nuclei, quantum liquids, 3 He clusters, metal clusters, etc. Furthermore, the PDM presence in quantum mechanical problems may reflect some other unconventional effects, such as a deformation of the canonical commutation relations or a curvature of the underlying space.…”

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

“…Several authors have investigated the exact solution of Schrödinger equation with position dependent mass using SUSY techniques [8,9,10]. so(2,1), su(1,1) Lie algebras and quadratic algebra approach for PDM Schrödinger equation in two dimensions were used as generating algebra as a potential algebra to obtain exact solutions of the effective mass wave equation [11,12,13]. Exact solutions of Schrödinger equation in D dimensions, quantum well problem includes PDM approach [14,15] and the point canonical transformations (PCT) are other studies and approaches providing exact solution of energy eigenvalues and corresponding eigenfunctions [16,17,18,19].…”