Noether's Theorem and time-dependent quantum invariants

TL;DR

This paper explores how Noether's theorem can be used to derive time-dependent quantum invariants, such as integrals of motion, for systems like the harmonic oscillator with varying parameters, linking symmetries to conserved quantities.

Contribution

It provides explicit constructions of linear time-dependent invariants for non-stationary quantum systems with variable mass and coupling, extending the application of Noether's theorem.

Findings

Time-dependent integrals of motion are derived from symmetry considerations.

Explicit formulas for invariants in systems with varying parameters are provided.

The approach links classical equations of motion to quantum invariants.

Abstract

The time dependent-integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the non-stationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are obtained also from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated, and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.