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

Progress of Theoretical Physics

The First Order Schrodinger Equation, the Symmetrization of the Stress Tensor and the Spin-Orbit Coupling

Yasushi Takahashi, C. Ropchan

Published Dec 1, 1986• 3 citations
Publisher

Abstract

The consequences of reducing the Schrödinger equation to two first order equations are examined in detail. Explicit calculations of the symmetrization of the stress tensors are given. It is pointed out that the spin-orbit coupling emerges in this formalism without stepping into the relativistic regime.
DOI
10.1143/ptp.76.1187

Related papers

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

Quantum stress and torsion distributions in the deuteron

Wim Cosyn, Adam Freese, Alan Sosa

Stress distributions in the deuteron are related to form factors of the asymmetric energy-momentum tensor through three-dimensional Fourier transforms. There are eleven such form factors, which we calculate in an impulse approximation. We compare the obtained form factors to prior results for the six form factors that have been previously calculated. We then elaborate on the formalism for relating the form factors to internal distributions of mass, mass flux, momentum, stresses, and forces, and obtain results for all of these distributions. We obtain the principal stresses for the symmetric part of the stress tensor, and show that the antisymmetric part describes reorientation of fermion spin by torsion stress when the nucleon moves between the S- and D-waves. Force distributions in the nucleons depend on the so-called non-conserved form factors through the Cauchy momentum equation, and are non-radial owing to the presence of tensor forces and spin-orbit coupling.

nucl-thCross-listedarXiv:2602.18298•Feb 23, 2026

On the linear forms of the Schrodinger equation

Yazid Kasri, Y. Gr, L. Chétouani

Generalizing the linearisation procedure used by Dirac and later by L\'evy-Leblond, we derive the first-order non-relativistic wave equations for particles of spin 1 and spin 3/2 starting from the Schrodinger equation.

•Aug 17, 2016•1 citation

On aq-analogue of the spin-orbit coupling

M. Micu, Fl. Stancu

Based on the tensor method, a q -analogue of the spin-orbit coupling is introduced in a q -deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = ℓ±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.

arXiv:nucl-th/0006081•Journal of Physics A Mathematical and General•Aug 3, 2000•0 citations

Relativized Wave Function: a First Step in a Unifying Approach to General Relativity and Bohmian Mechanics

Paulo Guilherme Santos

We propose a new way to unify General Relativity (GR) with Bohmian Mechanics (BM) that remains faithful to GR’s tensorial and deterministic philosophy. In our approach, the conventional quantum wave function emerges as the scalar contraction of a tensorial field $\psi^\mu$, namely $\Psi=\psi_\nu\psi^\nu$, thereby encoding quantum dynamics in a geometric object. Crucially, the tensor $\psi^\nu$ interacts directly with spacetime via the stress-energy tensor, with an auxiliary vector field $a^\mu$ serving as the mediator between the quantum and gravitational sectors.We analyze the equations for one single particle without accounting for spin. As an illustrative example, we show that under suitable approximations our formulation reproduces the non-relativistic Schr\"odinger equation with a Coulomb potential\textemdash as encountered in the hydrogen atom. We further establish a dynamical equation for $\psi^\nu$ and a corresponding volume-like equation for $a^\mu$.

•Jan 1, 2025•0 citations

Note On The Schrödinger Equation

C. Bizdadea, M. M. Bârcan, Iulian Negru +1 more

Abstract A second-order formalism leading to an equation describing the same dynamics as the Schrödinger one is developed under some compatible initial conditions.

Annals of West University of Timisoara - Physics•Dec 1, 2012•0 citations
Stress distributions in the deuteron are related to form factors of the asymmetric energy-momentum tensor through three-dimensional Fourier transforms. There are eleven such form factors, which we calculate in an impulse approximation. We compare the obtained form factors to prior results for the six form factors that have been previously calculated. We then elaborate on the formalism for relating the form factors to internal distributions of mass, mass flux, momentum, stresses, and forces, and obtain results for all of these distributions. We obtain the principal stresses for the symmetric part of the stress tensor, and show that the antisymmetric part describes reorientation of fermion spin by torsion stress when the nucleon moves between the S- and D-waves. Force distributions in the nucleons depend on the so-called non-conserved form factors through the Cauchy momentum equation, and are non-radial owing to the presence of tensor forces and spin-orbit coupling.
Based on the tensor method, a q -analogue of the spin-orbit coupling is introduced in a q -deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = ℓ±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.
We propose a new way to unify General Relativity (GR) with Bohmian Mechanics (BM) that remains faithful to GR’s tensorial and deterministic philosophy. In our approach, the conventional quantum wave function emerges as the scalar contraction of a tensorial field ψμ\psi^\muψμ, namely Ψ=ψνψν\Psi=\psi_\nu\psi^\nuΨ=ψν​ψν, thereby encoding quantum dynamics in a geometric object. Crucially, the tensor ψν\psi^\nuψν interacts directly with spacetime via the stress-energy tensor, with an auxiliary vector field aμa^\muaμ serving as the mediator between the quantum and gravitational sectors.We analyze the equations for one single particle without accounting for spin. As an illustrative example, we show that under suitable approximations our formulation reproduces the non-relativistic Schr\"odinger equation with a Coulomb potential\textemdash as encountered in the hydrogen atom. We further establish a dynamical equation for ψν\psi^\nuψν and a corresponding volume-like equation for aμa^\muaμ.