Example of quantum systems reduction

TL;DR

This paper presents a formalism for reducing quantum systems to classical subspaces using group generators, demonstrated on the hydrogen atom model, showing the system's exact quasiclassical and integrable nature.

Contribution

It introduces a reduction method mapping quantum problems onto classical phase spaces, exemplified by the hydrogen atom, revealing its quasiclassical and integrable properties.

Findings

Motion in the reduced space is purely classical.

Kepler orbits are inherently closed, independent of initial conditions.

Quantum corrections are confined to the bifurcation line at infinite angular momentum.

Abstract

To solve the quantum-mechanical problem the procedure of mapping onto linear space $W$ of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem is solved noting conservation of the Runge-Lentz vector $n$ and reducing the 4-dimensional incident phase space $T$ to the 3-dimensional linear subspace $W=T^* V\times R^1$, where $T^* V$ is the (angular momentum ($l$) - angle ($\vp$)) phase space and $R^1 =n$. It is shown explicitly that (i) the motion in $R^1$ is pure classical as the consequence of the reduction, (ii) motion in the $\vp$ direction is classical since the Kepler orbits are closed independently from initial conditions and (iii) motion in the $l$ direction is classical since all corresponding quantum corrections are defined on the bifurcation line ($l=\infty$) of the problem. In our terms the H-atom problem is exactly quasiclassical and is completely integrable by this reasons.