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Proceedings of the National Academy of Sciences

Lagrangian Functions and Schrödinger's Rule

H. Bateman

Published May 1, 1927• 4 citations
Publisher

Abstract

In a recent paper Schrödinger(1) has extended a rule, used by writers in the theory of gravitation, for deriving a stress energy tensor from a Lagrangian function and has illustrated its application in the case of the tensor which he has associated with the system of equations proposed by Gordon.(2)
DOI
10.1073/pnas.13.5.326

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The expressions for the density of electricity and current given recently by W. Gordon are modified so as to give the wave-equations used by de Broglie and Schroedinger. The new equations are derived from a variational principle which also gives some other equations of considerable interest. A stress-energy tensor is derived from the Lagrangian function by Schroedinger's rule and is used to determine one of the constants in Schroedinger's equation. The analysis seems to indicate that a stationary, isolated electron has a definite boundary and uniform volume density of electricity.
The Schroedinger- and Klein-Gordon equations are directly derived from classical Lagrangians. The only inputs are given by the discreteness of energy (E=hbar.w) and momentum (p=hbar.k), respectively, as well as the assumed existence of a space-pervading field of "zero-point energy" (E_0=hbar.w/2 per spatial dimension) associated to each particle of energy E. The latter leads to an additional kinetic energy term in the classical Lagrangian, which alone suffices to pass from classical to quantum mechanics. Moreover, Heisenberg's uncertainty relations are also derived within this framework, i.e., without referring to quantum mechanical or other complex-numbered functions.
Ordinarily, the stress tensor that one derives for a Madelung fluid is not regarded as being coupled to a strain tensor, which is consistent with the fluid hypothesis. However, based upon earlier work regarding the geometric nature of the quantum potential, one can, in fact, define a strain tensor, which is not, however, due to a deformation of a spatial region, but to a deformation of a frame field on that region. When one expresses the Madelung stress tensor as a function of the strain tensor and its derivatives, one then defines a constitutive law for the Madelung medium that might lead to a more detailed picture of its elementary structure. It is pointed out that the resulting constitutive law is strongly analogous to laws that were presented by Kelvin and Tait for the bending and torsion of elastic wires and plates, as well as the Einstein equations for gravitation if one takes the viewpoint of metric elasticity.