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Journal of Physics A Mathematical and General

Energy spectrum, potential and inertia functions of a generalizedf-oscillator

Salomon Sylvain Mizrahi, J P Camargo Lima, V. V. Dodonov

Published Mar 2, 2004• 44 citations
Publisher

Abstract

We consider a generalized four-parameter q -algebra AA † − q γ A † A = q α N +β , [ N , A ] = − A , [ N , A †] = A †, associating with operators A and A † the nonlinear f -oscillator operators, defined in terms of the usual harmonic oscillator operators as A ≡ af ( N ) and A † ≡ f *( N ) a † (where a and a † are operators of the Weyl–Heisenberg algebra and N = a † a ). The function f ( N ) is determined from the commutation relations. We write the Hamiltonian for the free f -oscillator and obtain its energy spectrum. Besides, expressing the Hamiltonian in terms of coordinate and momentum, we determine the potential and inertia functions (coordinate-dependent mass) and analyse their behaviour by varying the parameters.

DOI
10.1088/0305-4470/37/11/012

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We consider a generalized four-parameter q -algebra AA † − q γ A † A = q α N +β , [ N , A ] = − A , [ N , A †] = A †, associating with operators A and A † the nonlinear f -oscillator operators, defined in terms of the usual harmonic oscillator operators as A ≡ af ( N ) and A † ≡ f *( N ) a † (where a and a † are operators of the Weyl–Heisenberg algebra and N = a † a ). The function f ( N ) is determined from the commutation relations. We write the Hamiltonian for the free f -oscillator and obtain its energy spectrum. Besides, expressing the Hamiltonian in terms of coordinate and momentum, we determine the potential and inertia functions (coordinate-dependent mass) and analyse their behaviour by varying the parameters.
Adopting the framework of the generalized q -deformed Heisenberg–Weyl algebra , we present a mathematical procedure which leads us to obtain analytical expressions for a general class of q -deformed coherent states associated with the different patterns of the energy spectrum exhibited by the nonlinear f -oscillator. In particular, we establish the properties of a small group of q -deformed coherent states for α > γ > 0 with emphasis on the resolution of unity. As an application of these properties, we investigate the Robertson–Schrödinger uncertainty relation and the squeezing effect for the deformed coordinate and momentum operators, which are defined in terms of the abstract elements of this algebra. Furthermore, we also obtain the Wigner function and the correct quantum-mechanical marginal distributions in phase space.
Abstract The main objective of this research paper is to develop an approach within phase-space quantum mechanics for the pseudo-harmonic oscillator potential, enabling the derivation of analytical expressions for the Wigner distribution function and the formulation of a general expression for the moments associated with higher-order position and momentum operators. This approach also allows for the identification of a key dimensionless parameter α in the phase-space transformations. This parameter contributes to improving the general analytical expression for the energy levels. These energy levels are evaluated numerically and show strong agreement with existing numerical results reported in the literature for various diatomic molecular systems. Furthermore, it has been verified that our results are accurate and satisfy both the Heisenberg-Born-Jordan-Dirac commutation relation and the Heisenberg uncertainty principle.
Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
The recently introduced by us, two- and three-parameter (p, q)- and (p, q, [Formula: see text])-deformed extensions of the Heisenberg algebra were explored under the condition of their direct link with the respective (nonstandard) deformed quantum oscillator algebras. In this paper, we explore certain Hermitian Hamiltonians build in terms of non-Hermitian position and momentum operators obeying definite [Formula: see text](N)-pseudo-hermiticity properties. A generalized nonlinear (with the coefficients depending on the particle number operator N) one-mode Bogoliubov transformation is developed as main tool for the corresponding study. Its application enables to obtain the spectrum of “almost free” (but essentially nonlinear) Hamiltonian.
This work addresses the study of the oscillator algebra, defined by four parameters ppp, qqq, ααα, and ννν. The time-independent Schrödinger equation for the induced deformed harmonic oscillator is solved; explicit analytic expressions of the energy spectrum are given. Deformed states are built and discussed with respect to the criteria of coherent state construction. Various commutators involving annihilation and creation operators and their combinatorics are computed and analyzed. Finally, the correlation functions of matrix elements of main normal and antinormal forms, pertinent for quantum optics analysis, are computed.
We derive quantum solutions of a generalized inverted or repulsive harmonic oscillator with arbitrary time-dependent mass and frequency using the quantum invariant method and linear invariants, and write its wave functions in terms of solutions of a second-order ordinary differential equation that describes the amplitude of the damped classical inverted oscillator. Afterwards, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum, the associated uncertainty relation, and the quantum correlations between coordinate and momentum. As a particular case, we apply our general development to the generalized inverted Caldirola–Kanai oscillator.