Solutions to the time-dependent Schrodinger equation in the continuous spectrum case

TL;DR

This paper extends the Lewis-Riesenfeld method to solve the time-dependent Schrödinger equation for systems with continuous spectra, providing explicit formulas and an example calculation for a particle in a linear potential.

Contribution

It introduces a generalized Lewis-Riesenfeld approach applicable to continuous spectrum cases, expanding the method's scope beyond discrete eigenvalues.

Findings

Derived an explicit formula for the generalized Lewis-Riesenfeld phase.

Applied the method to a particle in a time-dependent linear potential.

Demonstrated the approach's effectiveness through an explicit example.

Abstract

We generalize the Lewis-Riesenfeld technique of solving the time-dependent Schrodinger equation to cases where the invariant has continuous eigenvalues. An explicit formula for a generalized Lewis-Riesenfeld phase is derived in terms of the eigenstates of the invariant. As an illustration the generalized phase is calculated for a particle in a time-dependent linear potential.