On singular Lagrangian underlying the Schr\"odinger equation

TL;DR

This paper explores the Hamiltonian structure of the Schrödinger equation, revealing it as arising from a singular Lagrangian with constraints, and introduces a real field representation linking probability conservation to energy conservation.

Contribution

It demonstrates that the Schrödinger equation can be derived from a singular Lagrangian framework and introduces a real field formulation analogous to electrodynamics.

Findings

Schrödinger equation originates from a singular Lagrangian with constraints.

Solution representation involves a real field satisfying a nonsingular Lagrangian.

Probability conservation corresponds to energy conservation of the real field.

Abstract

We analyze the properties that manifest Hamiltonian nature of the Schr\"odinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution to the Schr\"odinger equation with time independent potential can be presented in the form $\Psi=(-\frac{\hbar^2}{2m}\triangle+V)\phi+i\hbar\partial_t\phi$, where the real field $\phi(t, x^i)$ is some solution to nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field $\phi$. After introducing the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics: the real field $\phi$ turns out to be a kind of potential for a wave function.