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Derivation of the Schrödinger Equation from First Principles

Patrick Driessen

Published Apr 1, 2021• 0 citations
OpenAlex

Abstract

A particle model developed elsewhere is presented very briefly, in which particles are seen as vortex structures in a kind of fluid. The particle core is tied by vortex lines to the cores of other particles in the vicinity. The model is shown to explain spin ½ as a topological effect. It also sheds some light on the process of intrication. But vortex lines also induce velocities on themselves and on other vortex lines. When the Hasimoto transformation is used, the velocity induction equation is shown to give rise to the Schrodinger equation for the particle.

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In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Using the linear local induction approximation, we investigate the self-induced motion of a vortex-line that corresponds to the motion of a particle in quantum mechanics. Provided Kelvin waves, the effective Schrödinger equation, physical quantity operators, and the corresponding path-integral formula can be obtained. In particular, the effective Planck constant defined by parameters of vortex-line motion shows the mathematical relation between the two fields. The vortexline–particle mapping may help in understanding particle motion in quantum mechanics.

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There has been a recent tendency to apply Schroedinger's wave equation to macroscopic domains, from Bose-Einstein condensates in neutron stars to planetary orbits. In these applications a hydrodynamical interpretation, involving vortices in some 'fluid' medium is often given. The vortex picture appears surprising in light of more traditional interpretations of Schroedinger's equation, and indeed often appears to rely on ad hoc analogies. The purpose of this letter is to examine the vortex hypothesis in light of a simple, transparent mathematical framework. We find that Schroedinger's equation implies waves can be vortices also for a class of wave functions. Vortices occur in pairs that perform a sort of quantum computation by collapsing into either 0 or 1. Vortices collapsing to 0 ('0-vortices') are longer-lived, and their ratio to 1-vortices is discussed.

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Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…

•Jan 1, 2018•0 citations

Another derivation of the Schrödinger equation

Pierre Pelcé

We derive the Schrödinger equation, starting from principles other than that of superposition principle and of correspondance between physical quantities and linear operators as commonly used. We assume that the particle is carried by a compressible flow and that its density of momentum flux is only dependant on the spatial derivatives of the density . Then the condition that the flow can be derived from a potential is sufficient to show that satisfies the Schrödinger equation. This implies that the superposition principle is a property of some potential compressible fluids.

European Journal of Physics•May 1, 1996•5 citations
A particle model developed elsewhere is presented very briefly, in which particles are seen as vortex structures in a kind of fluid. The particle core is tied by vortex lines to the cores of other particles in the vicinity. The model is shown to explain spin ½ as a topological effect. It also sheds some light on the process of intrication. But vortex lines also induce velocities on themselves and on other vortex lines. When the Hasimoto transformation is used, the velocity induction equation is shown to give rise to the Schrodinger equation for the particle.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Using the linear local induction approximation, we investigate the self-induced motion of a vortex-line that corresponds to the motion of a particle in quantum mechanics. Provided Kelvin waves, the effective Schrödinger equation, physical quantity operators, and the corresponding path-integral formula can be obtained. In particular, the effective Planck constant defined by parameters of vortex-line motion shows the mathematical relation between the two fields. The vortexline–particle mapping may help in understanding particle motion in quantum mechanics.
Owing to three conditions (namely: (a) the velocity is represented by sum of irrotational and solenoidal components; (b) the fluid is barotropic; (c) a bath with the fluid undergoes vertical vibrations) the Navier-Stokes equation admits reduction to the modified Hamilton-Jacobi equation. The modification term is the Bohmian(quantum) potential. This reduction opens possibility to define a complex-valued function, named the wave function, which is a solution of the Schrödinger equation. The solenoidal component being added to the momentum operator poses itself as a vector potential by analogy with the magnetic vector potential. The vector potential is represented by the solenoidal velocity multiplied by mass of the fluid element. Vortex tubes, rings, and balls along with the wave function guiding these objects are solutions of this equation. Motion of the vortex balls along the Bohmian trajectories gives a model of droplets moving on the fluid surface. A peculiar fluid is the superfluid physical vacuum. It contains Bose particle-antiparticle pairs. Vortex lines presented by electron-positron pairs are main torque objects. Bundles of the vortex lines can transmit a torque from one rotating disk to other unmoved disk.
There has been a recent tendency to apply Schroedinger's wave equation to macroscopic domains, from Bose-Einstein condensates in neutron stars to planetary orbits. In these applications a hydrodynamical interpretation, involving vortices in some 'fluid' medium is often given. The vortex picture appears surprising in light of more traditional interpretations of Schroedinger's equation, and indeed often appears to rely on ad hoc analogies. The purpose of this letter is to examine the vortex hypothesis in light of a simple, transparent mathematical framework. We find that Schroedinger's equation implies waves can be vortices also for a class of wave functions. Vortices occur in pairs that perform a sort of quantum computation by collapsing into either 0 or 1. Vortices collapsing to 0 ('0-vortices') are longer-lived, and their ratio to 1-vortices is discussed.
Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…
We derive the Schrödinger equation, starting from principles other than that of superposition principle and of correspondance between physical quantities and linear operators as commonly used. We assume that the particle is carried by a compressible flow and that its density of momentum flux is only dependant on the spatial derivatives of the density . Then the condition that the flow can be derived from a potential is sufficient to show that satisfies the Schrödinger equation. This implies that the superposition principle is a property of some potential compressible fluids.