The Quantum Arnold Transformation and the Ermakov-Pinney equation
Julio Guerrero, Francisco F. L\'opez-Ruiz
TL;DR
This paper revises the Quantum Arnold Transformation, a unitary operator linking solutions of Schrödinger equations for quadratic Hamiltonians to free particle solutions, and introduces extensions including a generalized Ermakov-Pinney equation.
Contribution
It provides a revised formulation of the Quantum Arnold Transformation and extends it to include a generalized Ermakov-Pinney equation, broadening its applicability.
Findings
Revised the Quantum Arnold Transformation for quadratic Hamiltonians.
Introduced extensions leading to a generalized Ermakov-Pinney equation.
Enhanced understanding of solution mappings in quantum mechanics.
Abstract
The previously introduced Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schr\"odinger 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.
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