On singular Lagrangian underlying the Schrödinger equation

A. Gersten, Abed Shaqir

D. Arsenović, Nikola Burić, Dragomir M Davidović +1 more

Alexei A. Deriglazov

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.

Ian M Davies

The author obtains an expression for the stochastic Lagrangian as used in stochastic mechanics in a straightforward manner without obtaining the singular terms of Guerra et. at. (1983). He then presents a new derivation of the Schrodinger equation.

Waldemar Puszkarz

We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schrödinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schrödinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schrödinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.

Kálmán Varga, Joseph A. Driscoll