Invariants and States Generating Symmetry of Nonstationary Systems

Abstract

The aim of this work is to consider, following Ref. [1–3], the relation of spectrum generating algebras [4] and dynamical groups [5] (or noninvariance groups [6]) with integrals of motion and symmetries of the Schrodinger equation. We consider these concepts for examples of stationary and nonstationary quantum systems. The nonstationary systems have no energy levels. For this reason, there are no spectrum generating algebras for these systems. On the other hand, using the integrals of motion, it is possible to construct a generalization of the spectrum generating algebra for non-stationary systems as well. This construction can be called state generating symmetry of nonstationary quantum system. This construction was used in [2], [7] for quantum parametric oscillators. For stationary quantum systems nonstationary integrals of motion as elements of a spectrum generating algebra have been introduced by Dothan [8]. The spectrum generating algebra O(4,2) for the hydrogen atom has been studied in Ref. [1], [9]–[12]. Symmetry of potential in quantum mechanics is considered to be responsible for level degeneration. So, the degeneration of hydrogen atom discrete levels was explained by Fock [13] as a consequence of O(4) hidden symmetry.