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arXiv:1405.5099 · Foundations of Physics

Lagrangian form of Schrödinger equation

D. Arsenović, Nikola Burić, Dragomir M Davidović, Slobodan Prvanović

Published Jun 4, 2014• 4 citations
arXivHTMLPDF
DOI
10.1007/s10701-014-9810-4

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We analyze the properties that manifest hamiltonian nature of the Schrödinger 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 of the Schrödinger equation with time independent potential can be presented in the form Ψ = (-h2 2m + V )φ + ih∂ t φ , where the real field φ (t, x i ) is some solution of nonsingular Lagrangian theory being specified below.Preservation of probability turns out to be the energy conservation law for the field φ .After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.
In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation.By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T * M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T * M, H) is obtained.This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T * M, H).Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.