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arXiv:1302.1316 · Physica Scripta

The quantum Arnold transformation and the Ermakov–Pinney equation

Julio Guerrero, Francisco F. López-Ruiz

Published Feb 11, 2013• 22 citations
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Abstract

The previously introduced quantum Arnold transformation, a unitary operator mapping the solutions of the Schrödinger equation for time dependent quadratic Hamiltonians into the solutions for the free particle, is revised and some interesting extensions are introduced, providing in particular a generalization of the Ermakov–Pinney equation.

DOI
10.1088/0031-8949/87/03/038105

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The previously introduced quantum Arnold transformation, a unitary operator mapping the solutions of the Schrödinger equation for time dependent quadratic Hamiltonians into the solutions for the free particle, is revised and some interesting extensions are introduced, providing in particular a generalization of the Ermakov–Pinney equation.
We revise the Lewis–Riesenfeld invariant method for solving the quantum time-dependent harmonic oscillator in light of the quantum Arnold transformation previously introduced and its recent generalization to the quantum Arnold–Ermakov–Pinney transformation. We prove that both methods are equivalent and show the advantages of the quantum Arnold–Ermakov–Pinney transformation over the Lewis–Riesenfeld invariant method. We show that, in the quantum time-dependent and damped harmonic oscillator, the invariant proposed by Dodonov and Man’ko is more suitable, and provide some examples to illustrate it, focusing on the damped case.
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schrödinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schrödinger equation. We further show that the first and the second Pöschl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.
Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
We discuss the extension of the Lewis-Riesenfeld technique of solving the time-dependent Schr\"odinger equation to cases where the invariant has continuous eigenvalues. The extension is carried out for a general Ermakov system. We also discuss extensions to NNN-dimensional systems and the calculation of the propagator for a general Ermakov system.
Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.
Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg–Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wavefunctions and closed analytic expressions for time-evolution operators.