Quantum Mechanical Observables under a Symplectic Transformation of Coordinates

TL;DR

This paper explores how symplectic transformations affect quantum observables in finite-dimensional systems, defining eigenstates within rigged Hilbert spaces and providing explicit transformation formulas with illustrative examples.

Contribution

It introduces a formalism for eigenstates under symplectic transformations in quantum systems using rigged Hilbert spaces, with explicit transformation formulas.

Findings

Explicit form of eigenstate transformations derived

Formalism applicable to finite-dimensional quantum systems

Examples illustrating the transformation process included

Abstract

We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quantized version of a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper.