On singular Lagrangian underlying the Schrödinger equation

Details On singular Lagrangian underlying the Schrödinger equation On singular Lagrangian underlying the Schrödinger equation

2009-03-08

arxiv.org/abs/0903.1428v3

Phys.Lett.A373:3920-3923,2009

Physics

Mathematical Physics

04 pages, published version

Scientific paper

10.1016/j.physleta.2009.08.050

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.

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