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arXiv:1710.01256

Bohmian quantum mechanics revisited

Arbab I. Arbab

Published Sep 30, 2017• 0 citations
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Abstract

By expressing the Schrödinger wave function in the form $ψ=Re^{iS/\hbar}$, where $R$ and $S$ are real functions, we have shown that the expectation value of $S$ is conserved. The amplitude of the wave ($R$) is found to satisfy the Schrödinger equation while the phase ($S$) is related to the energy conservation. Besides the quantum potential that depends on $R$, \emph{viz.}, $V_Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}$\,, we have obtained a phase potential $V_S=-\frac{S\nabla^2S}{m}$ that depends on the phase $S$ derivative. The phase force is a dissipative force. The quantum potential may be attributed to the interaction between the two subfields $S$ and $R$ comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass $mc^2$ with an internal frequency of $2mc^2/\hbar$, satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz $R$ satisfies the Klein-Gordon equation, then $S$ must satisfies the wave equation. Conversely, if $R$ satisfies the wave equation, then $S$ yields the Einstein relativistic energy momentum equation.

DOI
10.48550/arxiv.1710.01256

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By expressing the Schrödinger wave function in the form ψ=ReiS/ℏψ=Re^{iS/\hbar}ψ=ReiS/ℏ, where RRR and SSS are real functions, we have shown that the expectation value of SSS is conserved. The amplitude of the wave (RRR) is found to satisfy the Schrödinger equation while the phase (SSS) is related to the energy conservation. Besides the quantum potential that depends on RRR, \emph{viz.}, VQ=−ℏ22m∇2RRV_Q=-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}VQ​=−2mℏ2​R∇2R​,, we have obtained a phase potential VS=−S∇2SmV_S=-\frac{S\nabla^2S}{m}VS​=−mS∇2S​ that depends on the phase SSS derivative. The phase force is a dissipative force. The quantum potential may be attributed to the interaction between the two subfields SSS and RRR comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass mc2mc^2mc2 with an internal frequency of 2mc2/ℏ2mc^2/\hbar2mc2/ℏ, satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz RRR satisfies the Klein-Gordon equation, then SSS must satisfies the wave equation. Conversely, if RRR satisfies the wave equation, then SSS yields the Einstein relativistic energy momentum equation.
By expressing the Schrödinger wave function in the form ψ = R e i S / ℏ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., V Q = - ℏ 2 2 m ∇ 2 R R , we have obtained a spin potential V S = - S ∇ 2 S m that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass m c 2 with an internal frequency of 2 m c 2 / ℏ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
By expressing the Schrödinger wave function in the form ψ = R e i S / ℏ , where R and S are real functions, we have shown that the expectation value of S is conserved. The amplitude of the wave (R) is found to satisfy the Schrödinger equation while the phase (S) is related to the energy conservation. Besides the quantum potential that depends on R, viz., V Q = - ℏ 2 2 m ∇ 2 R R , we have obtained a spin potential V S = - S ∇ 2 S m that depends on S which is attributed to the particle spin. The spin force is found to give rise to dissipative viscous force. The quantum potential may be attributed to the interaction between the two subfields S and R comprising the quantum particle. This results in splitting (creation/annihilation) of these subfields, each having a mass m c 2 with an internal frequency of 2 m c 2 / ℏ , satisfying the original wave equation and endowing the particle its quantum nature. The mass of one subfield reflects the interaction with the other subfield. If in Bohmian ansatz R satisfies the Klein-Gordon equation, then S must satisfies the wave equation. Conversely, if R satisfies the wave equation, then S yields the Einstein relativistic energy momentum equation.
A formulation is proposed for the behavior of a particle in a box based on two assumptions: (1) particles have an associated phase which changes periodically with time as defined by de Broglie, and (2) the transmission of phase information follows optical principles, including reflection at potential boundaries and relativistic Doppler effects. This approach defines optimal trajectories by matching the phase of a source and its reflections. The momentum of the particle is guided by a de Broglie wave which follows the equation R exp {i[w0 (1-v^2/c^2)^-1/2][(t - vz)/c^2]} cos {[w0(1-v^2/c^2)^-1/2/c](vt-z)} This wave is generated by reflections occurring an even number of times, the phase and amplitude of which correspond to Lorentz transforms relative to the time and position, respectively. Nodes are produced by reflections occurring an odd number of times, which can be conceived to generate a wave traveling in the opposite direction and shifted by a phase constant π. Application to an individual particle in an infinite potential well affords the same results for energies and nodes as derived by the Schrödinger equation.
A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
It has recently been shown that the classical electric and magnetic fields which satisfy the source-free Maxwell equations can be linearly mapped into the real and imaginary parts of a transverse-vector wave function which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator is physically appropriate to the free photon. The free-particle Klein-Gordon equation for scalar fields modestly extends the classical wave equation via a mass term. It is physically untenable for complex-valued wave functions, but has a sound nonnegative conserved-energy functional when it is restricted to real-valued classical fields. Canonical Hamiltonization and a further canonical transformation maps the real-valued classical Klein-Gordon field and its canonical conjugate into the real and imaginary parts of a scalar wave function (within a constant factor) which in consequence satisfies the time-dependent Schroedinger equation whose Hamiltonian operator has the natural correspondence-principle relativistic square-root form for a free particle, with a mass that matches the Klein-Gordon field theory's mass term. Quantization of the real-valued classical Klein-Gordon field is thus second quantization of this natural correspondence-principle first-quantized relativistic Schroedinger equation. Source-free electromagnetism is treated in a parallel manner, but with the classical scalar Klein-Gordon field replaced by a transverse vector potential that satisfies the classical wave equation. This reproduces the previous first-quantized results that were based on Maxwell's source-free electric and magnetic field equations.
We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which, in particular, ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines for the particles can begin and end.