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Proof of Magnetic Flux Quantization by Quantum Mechanics

Daehyeon Kang

Published Nov 1, 2017• 0 citations
OpenAlex

Abstract

At present, By the requirement that the phase factor of the wave function has unity, Quantum values of magnetic flux are calculated, which seems to lack quantum logic. Thus, in this paper, we apply the quantum mechanics logic strictly to this problem,The unit value is derived and the result is (hc/2e). It is exactly the same as that obtained by Faraday's law of electromagnetic induction and classical quantum theory.

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arXiv:1609.04446•Sep 15, 2016•1 citation
At present, By the requirement that the phase factor of the wave function has unity, Quantum values of magnetic flux are calculated, which seems to lack quantum logic. Thus, in this paper, we apply the quantum mechanics logic strictly to this problem,The unit value is derived and the result is (hc/2e). It is exactly the same as that obtained by Faraday's law of electromagnetic induction and classical quantum theory.
In a ring of s-wave superconducting material the magnetic flux is quantized in units of Φ0=h2eΦ_0 = \frac{h}{2e}Φ0​=2eh​. It is well known from the theory of Josephson junctions that if the ring is interrupted with a piece of d-wave material, then the flux is quantized in one-half of those units due to a additional phase shift of πππ. We reinterpret this phenomenon in terms of geometric phase. We consider an idealized hetero-junction superconductor with pure s-wave and pure d-wave electron pairs. We find, for this idealized configuration, that the phase shift of πππ follows from the discontinuity in the geometric phase and is thus a fundamental consequence of quantum mechanics.
We give a complete quantum analysis of the Aharonov-Bohm (AB) magnetic phase shift involving three entities: the electron, the charges constituting the solenoid current, and the vector potential. The usual calculation supposes that the solenoid's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized electron moving in this vector potential. Recently, Vaidman presented a semiclassical calculation [L. Vaidman, Phys. Rev. A 86, 040101 (2012)], later confirmed by a fully quantum calculation of Pearle and Rizzi [preceding paper, Phys. Rev. A 95, 052123 (2017)], where it is supposed that the electron's vector potential may be well approximated as classical. The AB shift is then acquired by the quantized solenoid charges moving in this vector potential. Here we present a third calculation, which supposes that the electron and solenoid currents may be well approximated as classical sources. The AB phase shift is then shown to be acquired by the quantized vector potential. We next show these are three equivalent alternative ways of calculating the AB shift. We consider the exact problem where all three entities are quantized. We approximate the wave function as the product of three wave functions: a vector potential wave function, an electron wave function, and a solenoid wave function. We apply the variational principle for the exact Schr\"odinger equation to this approximate form of solution. This leads to three Schr\"odinger equations, one each for vector potential, electron, and solenoid, each with classical sources for the other two entities. However, each Schr\"odinger equation contains an additional real ccc-number term, the time derivative of an extra phase. We show that these extra phases are such that the phase of the total wave function produces the AB shift. Since none of the three entities requires different treatment from any of the others, this leads to three alternative views of the physical cause of the AB magnetic effect.
We have calculated the changes in the quantized magnetic fluxes corresponding to the electronic orbits of Dirac hydrogen atom within the framework of Dirac theory and found that the flux change between the final state and the initial state is equal to (±Φ 0 or zero).We have also calculated the non-zero matrix elements for dipole transitions (such as 1s↔2p and 2p↔3d) for the solutions of Dirac hydrogen atom. We find a complete agreement among all the above results. The combined result is that photon should carry an intrinsic flux quantum of (Φ 0 =hc/e).
The orbit equation for the motion of a quantum in a potential is proposed by defining uniquely the quantum mechanical momentum from the Schrodinger wave equation of a one-particle system and using the Hamilton-Jacobi formalism. On this occasion it is necessary to introduce a new dynamical postulate, “initial condition”, which is independent of the quantization condition as well as of the Schrodinger wave equation. The momentum is determined in such a way that it is approximately equal to the corresponding classical mechanical momentum in the classical region of validity of the coordinates. The momentum in the non-classical region is in general very small in comparison with that in the classical region. The quantum-mechanical motion well corresponds to the classical motion. If a system is macroscopic, the motion of the particle is described by Newtonian mechanics and the quantization condition is of no validity. The “quantization of magnetic flux” derives from the quantization condition. It is closely related to the energy level structure of the atomic system. The above argument is applied to the scalar electromagnetic and sound wave equations to give the dynamical motion of a photon and a phonon without recourse to the so-called second quantization method.
It is shown that a wave mechanical quantum theory can be derived from relativistic classical electrodynamics, as a feature of the magnetic interaction of Dirac particles modeled as relativistically circulating point charges. The magnetic force between two classical point charges, each undergoing relativistic circulatory motion of small radius compared to the separation between their centers of circulation, and assuming a time-symmetric electromagnetic interaction, is modulated by a factor that behaves similarly to the Schrödinger wavefunction. The magnetic force between relativistically-circulating charges has been shown previously to have a radially-directed inverse-square part of similar strength to the Coulomb force, and sinusoidally modulated by the phase difference of the charges' circulatory motions. The magnetic force modulation in the case of relatively moving centers of charge circulation solves an equation formally identical to the time-dependent free-particle Schrödinger equation, apart from a factor of two on the partial time derivative term. Considering motion in a time-independent potential obtains that the modulation also satisfies an equation formally similar to the time-independent Schrodinger equation. Using a formula for relativistic rest energy advanced by Osiak, the time-independent Schrödinger equation is solved exactly by the resulting modulation function. The significance of the quantum mechanical wavefunction follows straightforwardly from these observations. After considering the modification of Wheeler-Feynman absorber theory required by the adoption of Minkowski-Osiak relativity, the model is extended to obtain the full complex Schrödinger wavefunction.