Exact Solution of a Time-Dependent Quantal Harmonic Oscillator with a Singular Perturbation

Abstract

The quantal problem of a particle interacting in one dimension with an external time-dependent quadratic potential and a constant inverse square potential is exactly solved. The solutions are found both in the Schrödinger representation, by using a generating function or a time-dependent raising operator, and in the Heisenberg picture. They depend only on the solution of the classical harmonic oscillator. The generalizations to the n-dimensional problem and to the problem of N particles in one dimension, interacting pairwise via a quadratic time-dependent potential and a constant inverse square potential, are finally sketched.

Cat-states in the framework of Wigner–Heisenberg algebra

“…On the other hand, this potential was generalized by Camiz and Dodonov et al to the non-stationary (varying frequency) PHO potential [62,63,64]. In addition, such a physical problem was also studied in arbitrary dimension D [60][61][62][63][64]. Also, Dong et .al have studied its dynamical group in two dimensions [65].…”

“…A few years later, Calogero studied the one-dimensional three-and N-body problems interacting pairwise via harmonic and inverse square (centrifugal) potential [60,61]. On the other hand, this potential was generalized by Camiz and Dodonov et al to the non-stationary (varying frequency) PHO potential [62,63,64]. In addition, such a physical problem was also studied in arbitrary dimension D [60][61][62][63][64].…”

Exact solutions for the general nonstationary oscillator with a singular perturbation

“…For brevity system (2) should be referred to as the general SO. Exact invariants and wave functions for various particular cases of (2) have been considered in the literature: b(t) = 0, constant m, ω and g -in [12,14]; b(t) = 0, constant m, g and varying ω(t) -in [11,13]; b(t) = 0, varying m(t), ω(t), g(t) with the constrain (3) with c = 1 -in [2]; varying b(t) and ω(t) and constant m and g -in [9]. In [7] three Heisenberg operators and their correlation functions for (2) with (3) were considered (in different parametrization).…”

“…Previously this singular oscillator (SO) has been treated in a number of paper [6,7,8,9,10,11,12,13,14], exact invariants and wave functions being obtained for the case of stationary SO (constant m, ω and g) in [12,14] and of SO with varying frequency ω(t) (but constant m, g) in [11,13]. The more general cases treated in [7,9] corresponds (as in [2]) to m(t)g(t) = const.…”

“…The more general cases treated in [7,9] corresponds (as in [2]) to m(t)g(t) = const. The potential (1) has vide applications in molecular and solid state physics: the radial motion of such systems as the hydrogen atom, the n-dimensional oscillator, the charged particle in an uniform magnetic plus electric field with scalar potential proportional to 1/(x 2 + y 2 ) and the N identical particles interacting pairwise with potential energy V ij = (mω 2 /2)(x i −x j ) 2 +g/(x i −x j ) 2 can be reduced [12,13,15] to the case of SO. The potential (1) was recently applied to describe a two-ion trap [1].…”