The effect of linear terms in a quadratic Hamiltonian

TL;DR

This paper demonstrates that linear terms in a quadratic Hamiltonian cause only a spatial translation and phase change in the wave function, which can be described using classical trajectories, simplifying the analysis of such quantum systems.

Contribution

It provides a clear analytical description of how linear terms affect wave functions in quadratic Hamiltonians, linking quantum shifts to classical trajectories.

Findings

Linear terms cause spatial translation of the wave function.

Phase of the wave function is altered by linear terms.

Moments about the wave function's centroid evolve independently of linear terms.

Abstract

For a non-relativistic particle subject to a Hamiltonian that is quadratic in position and momentum, with coefficients that may vary with time, it is shown that the effect of the linear terms in the Hamiltonian is just a spatial translation of the wave function and a change in its phase. The shifts in position and phase can be expressed in terms of classical trajectories. This simple effect of the linear terms is related to the fact that all moments about the centroid of the wave function evolve independently of the linear terms.