Hamiltonian Systems Inspired by the Schrödinger Equation

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 Schrödinger equations are obtained.

Keywords

• Schrödinger equation
• Hamiltonian systems on manifolds of scalar products
• n-level quantum systems
• scalar product as a dynamical variable
• essential non-perturbative nonlinearity
• conservation laws
• GL(n
• C)-invariance