Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

A Modification of Gordon's Equations

H. Bateman

Published Jul 1, 1927• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.30.55

Related papers

Lagrangian Functions and Schrödinger's Rule

H. Bateman

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)

Proceedings of the National Academy of Sciences•May 1, 1927•4 citations

The Symmetry of the Stress-Tensor Obtained by Schroedinger's Rule

H. Bateman

In the recent developments of the calculus of variations used in the new quantum theory the problem arises of finding a general expression for a world-function such that a symmetrical stress-energy tensor may be derived from it by means of Schroedinger's rule.(1)

Proceedings of the National Academy of Sciences•Nov 1, 1927•3 citations

Derivation of the Schroedinger Equation and the Klein-Gordon Equation from First Principles

Gerhard Grössing

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.

arXiv:quant-ph/0205047•May 9, 2002•3 citations

Derivation of both Klein-Gordon equation and Time Dependent Schrödinger Equation form classical wave function

Mobarak Tag, Wail Hessen ALawad, Adil Elrayah +1 more

In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).

Journal of Karary University for Engineering and Science•Oct 14, 2022•0 citations

Solution of the Schrödinger Equation for Some Electric Field Problems

Lawrence H. Buch, Harry H. Denman

An alternate treatment of the Schrödinger equation for a charged particle in a uniform, constant electric field is given. The time-varying field is then treated. Next, using a Hamiltonian which gives the classical equations of motion for a particle in a viscous medium subject to a uniform constant, or time-varying field, the corresponding Schrödinger equation is solved. This procedure yields the classical electrical conductivity expressions for both dc and ac fields.

American Journal of Physics•Apr 1, 1974•24 citations

Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics

Donald H. Kobe

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.

arXiv:0712.1608•Dec 10, 2007•2 citations
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.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.