The quantum Arnold transformation

TL;DR

This paper introduces a quantum version of the Arnold transformation that relates a broad class of quantum systems to free-particle dynamics, facilitating analytical solutions and operator algebra construction.

Contribution

It develops a quantum Arnold transformation applicable to systems with linear second-order differential equations, including friction, and constructs associated operator algebras.

Findings

Provides a method to relate complex quantum systems to free particles.

Enables quick computation of wave functions and evolution operators.

Establishes a complete Schrödinger algebra for these systems.

Abstract

By 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 Schr\"odinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wave functions and closed analytic expressions for time-evolution operators.