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arXiv:nucl-th/9709034 · Journal of Physics G Nuclear and Particle Physics

An application of the three-dimensionalq-deformed harmonic oscillator to the shell model

P. P. Raychev, R. P. Roussev, Nicola Lo Iudice, P. A. Terziev

Published Oct 1, 1998• 20 citations
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Abstract

A procedure for the construction of a q -deformed version of the Hamiltonian of the three-dimensional harmonic oscillator (HO), based on the application of q -deformed algebras, is presented. The spectrum of this Hamiltonian is not degenerated in the quantum number of the q -deformed angular momentum. The results together with their applicability to the shell model are compared with the predictions of the modified HO.

DOI
10.1088/0954-3899/24/10/009

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Journal of Physics Conference Series•Jan 1, 2021•1 citation
A procedure for the construction of a q -deformed version of the Hamiltonian of the three-dimensional harmonic oscillator (HO), based on the application of q -deformed algebras, is presented. The spectrum of this Hamiltonian is not degenerated in the quantum number of the q -deformed angular momentum. The results together with their applicability to the shell model are compared with the predictions of the modified HO.
The boson expansion method is applied to find the spectrum of a q-deformed harmonic oscillator. We use two different boson expansions, each of them including a deformation parameter, defined in terms of exponential and logarithmic functionals. The resulting Hamiltonians are bilinear forms of the transformed operators. Physical effects resulting from the deformation of the generators of the algebra are studied by comparing known finite-range potentials and the effective potentials obtained for each of the considered Hamiltonians.
The possibility of q-deformation of known exactly solvable quantum mechanical models is considered via the dynamical symmetry of the radial problem. The algebraic structure of the D-dimensional simple harmonic oscillator is analysed from this point of view. In the two-dimensional case, q-deformed bosons are introduced for the radial degrees of freedom, which together with the standard angular operators yield the so(2)(+)so q (2,1) dynamical algebra. The corresponding Hilbert space is identified and possible q-deformed Hamiltonians described.
The Liouville equation for the q-deformed 1-D classical harmonic oscillator is derived for two definitions of q-deformation.This derivation is achieved by using two different representations for the q-deformed Hamiltonian of this oscillator corresponding to undeformed and deformed phase spaces.The resulting Liouville equation is solved by using the method of characteristics in order to obtain the classical probability distribution function for this system.The 2-D and 3-D behaviors of this function are then investigated using a computer visualization method.The results are compared with those for the classical anharmonic oscillator.This comparison reveals that there are some similarities between these two models, where the results for the q-deformed oscillator exhibit similar whorl shapes that evolve with time as for the anharmonic oscillator.It is concluded that studying the Liouville dynamics gives more details about the physical nature of q-deformation than using the equation of motion method.It is also concluded that this result could have reflections on the interpretation of the quantized version of this q-deformed oscillator.