Exactly solvable Hermite, Laguerre, and Jacobi type quantum parametric oscillators

Abstract

We introduce exactly solvable quantum parametric oscillators, which are generalizations of the quantum problems related with the classical orthogonal polynomials of Hermite, Laguerre, and Jacobi type, introduced in the work of Büyükaşık et al. [J. Math. Phys. 50, 072102 (2009)]. Quantization of these models with specific damping, frequency, and external forces is obtained using the Wei-Norman Lie algebraic approach. This determines the evolution operator exactly in terms of two linearly independent homogeneous solutions and a particular solution of the corresponding classical equation of motion. Then, time-evolution of wave functions and coherent states are found explicitly. Probability densities, expectation values, and uncertainty relations are evaluated and their properties are investigated under the influence of the external terms.

Semiclassical interpretation of Wei–Norman factorization for SU(1, 1) and its related integral transforms

“…This provides a geometric and semiclassical interpretation of the WN factorization for the SU(1, 1) case, since all the information provided by the evolution operator is obtained from the solutions of the EL equation. Similar results can be found in the literature 9,10 (see also Refs. 11 and 12), but in those papers, just one ordering has been considered.…”

Squeezing and resonance in a generalized Caldirola-Kanai type quantum parametric oscillator

“…in our previous work, 28 using the Wei-Norman Lie algebraic approach the evolution operator was found asÛ…”

“…(1) the mixed term B(t)(qp +pq)/2, and the linear terms D 0 (t)q, E 0 (t)p, with time-dependent squeezing parameter B(t) and displacement parameters D 0 (t), E 0 (t). Then, the evolution operator for the generalized problem is obtained using the Wei-Norman algebraic approach, 28 which allows us to find explicitly the time-evolution of given initial states. The main goal of this work is to study the squeezing and displacement properties of the CK-oscillator under the influence of the external terms and the corresponding time-dependent parameters.…”

“…36,[43][44][45] Moreover, for a general quadratic oscillator, it is known that parameter B(t) effects both the squeezing and displacement properties, while the driving linear forces and their parameters D 0 (t) and E 0 (t) contribute only to the displacement of the wave packets. 23,28,43,44 According to this, choosing periodic driving forces, resonance in coherent states of a quantum damped oscillator was addressed long time ago in Ref. 18.…”

Exact time-evolution of a generalized two-dimensional quantum parametric oscillator in the presence of time-variable magnetic and electric fields

“…[7][8][9][10][11][12][13][14][15] Recently, in Ref. 16, by a straightforward application of the Wei-Norman technique and by properly choosing the ordering of the exponential operators, we found the evolution operator for a quantum parametric oscillator described by a Hamiltonian with a SU(1, 1) ⊕ h(4) group structure. The significance of our results is that for a time-dependent one-dimensional Schrödinger equation (SE) with the most general quadratic in the position and momentum Hamiltonian, we were able to determine the evolution operator explicitly in terms of two linearly independent homogeneous solutions and a particular solution to the corresponding classical equation of motion.…”