Hamiltonian Systems Inspired by the Schr\"odinger Equation

TL;DR

This paper develops a Hamiltonian framework for n-level quantum systems where the scalar product is a dynamical variable, deriving generalized Schr"odinger equations and conservation laws.

Contribution

It introduces a novel Hamiltonian approach with a dynamical scalar product, extending traditional quantum mechanics formulations.

Findings

Derived equations of motion for the wave function and scalar product G.

Obtained generalized Schr"odinger equations including first- and second-order modifications.

Identified conservation laws within the dynamical G framework.

Abstract

Described is n-level quantum system realized in the n-dimensional ''Hilbert'' space H with the scalar product G taken as a dynamical variable. The most general Lagrangian for the wave function and G is considered. Equations of motion and conservation laws are obtained. Special cases for the free evolution of the wave function with fixed G and the pure dynamics of G are calculated. The usual, first- and second-order modified Schr\"odinger equations are obtained.