Exact wave functions for generalized harmonic oscillators

Abstract

Abstract. We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.

On a hidden symmetry of quantum harmonic oscillators

“…The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresponding Ermakov-type system [17], [39], [41], [44]). These "hidden parameters" usually disappear after evaluation of matrix elements and cannot be observed from the spectrum.…”

“…[39]. As a result, the functions a n (p, t) are of the same form (1.2)-(1.9), if ψ n → a n and x → p, with the initial data…”

“…More applications include, but are not limited to charged particle traps and motion in uniform magnetic fields, molecular spectroscopy and polyatomic molecules in varying external fields, crystals through which an electron is passing and exciting the oscillator modes, and other mode interactions with external fields. Quadratic Hamiltonians have particular applications in quantum electrodynamics because the electromagnetic field can be represented as a set of generalized driven harmonic oscillators [13], [20].The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresponding Ermakov-type system [17], [39], [41], [44]). These "hidden parameters" usually disappear after evaluation of matrix elements and cannot be observed from the spectrum.…”

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

“…For the latter case, in [36], [37] and [38], multiparameter solutions in the spirit of Marhic in [30] have been presented. The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem Equation (1) with h(t) = 0 and in the process of finding a general solution of a Riccati system [38] and [39].…”

“…The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem Equation (1) with h(t) = 0 and in the process of finding a general solution of a Riccati system [38] and [39]. In addition, Ermakov systems with solutions containing parameters [36] have been used successfully to construct solutions for the generalized harmonic oscillator with a hidden symmetry [37], and they have also been used to present Galilei transformation, pseudoconformal transformation and others in a unified manner, see [37]. More recently, they have been used in [40] to show spiral and breathing solutions and solutions with bending for the paraxial wave equation.…”