Hamiltonian Systems Inspired by the Schrödinger Equation

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 Schrödinger equations are obtained.