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1-dimensional and even 2-dimensional models are deceptive. If you try to find out explicit 3-dimensional description of some wave process, infinite in space, not inside of conditional "pit with vertical walls", it will not be easy. With finite integrals of energy and charge density. It seems Schrödinger equation is not useful at all in this case (directional motion of compact soliton). Only for some spheric waves in 3-dimensional space. And I do not understand how it can be adapted to describe "free particle". Statistical and diffusive by its nature Schrödinger equation works only in presence of attraction centres (atomic nuclei). Otherways electron cloud will expand to the sizes of whole Universe. Theoretically this is "correct" also, but unusable practically, especially in computer simulation.

I meant free electromagnetic field equations that we can write like this: ∂E/∂t = 1/ε0 · curl H ∂H/∂t = - 1/μ0 · curl E F = sqrt(ε0) · E - i · sqrt(μ0) · H - i · ∂F/∂t = c · curl F Thank you for you answer. I am still wondered why in quantum mechanics functions have prefix "wave". If we get double time derivative, it will be proportional to "nabla in the fourth power", when in typical de Broglie equation only second power (div grad or curl curl, depending on the type of waves, longitudinal or transverse). Has it to be so?

It is not clear what complex wave function means in Schrödinger, Pauli, Dirac equations. Is it always two-component (complex), or can it be real, or are both variants possible in different situations? For example, how to understand: -i · h/(2·π) · ∂ψ/∂t = h2/(8·π2·m) · div grad ψ (for simplicity in absence of potential multiplied by function). The imaginary unit “i” simply shows that quantum operator is used instead of classical derivative, or function must be divided into two components: ψ = ψ1 + i · ψ2 and then in reality there are two equations ∂ψ1/∂t ~ div grad ψ2 ∂ψ2/∂t ~ div grad ψ1 (~ symbol means is proportional with a constant multiplier). In this case, the question arises how this relates to de Broglie equation, because it turns out to be ∂2ψ1/∂t2 ~ div grad (div grad ψ1) ∂2ψ2/∂t2 ~ div grad (div grad ψ2) instead of traditional ∂2ψ/∂t2 ~ div grad ψ or ∂2ψ/∂t2 ~ rot rot ψ for different kinds of waves. Or is function real (should be, or can be)? If Maxwell's equation is written as one formula, there are two components, electric field and magnetic, but instead of squared nabla single nabla (curl) is used, and this is consistent as de Broglie wave. Do Pauli and Dirac equations follow the same principle as Schrödinger equation with respect to the complexity of function, or there are differences?

Wherever the practical use of the magnetic moments of atoms is carried out, only the electrons own spins appear. For example, Wikipedia gives the following rule for calculating the moments of transition metals with a large number of unpaired electrons. Many transition metal complexes are magnetic. The spin-only formula is a good first approximation for high-spin complexes of first-row transition metals. Number of unpaired electrons, Spin-only moment (μB) 1 1.73 2 2.83 3 3.87 4 4.90 5 5.92 The relationship is almost linear, although it is obvious that electrons occupy d-orbitals with different "magnetic numbers" M at the same L and N. The type of electron cloud does not affect magnetic phenomena, at least at relatively large distances from the atom. It seems that the images of electrons spinning around a nucleus in books for schoolchildren and students are fiction and are of purely historical interest. Except for "Rydberg atoms," where an entire electron cloud an make coordinated movements. Which is not surprising, since the solutions of the Schrödinger or Pauli equations give the probabilities of finding an electron, respectively, the distribution of charge density and proper magnetic moment (spin), but do not indicate the prevailing direction of velocity at that point. Consequently, the movements are either completely chaotic, with equal probability in either direction, or mutually compensated so that no resulting magnetic moment is formed. For example, if the prevailing direction of velocity coincides with the gradient of the wave function or its square, and since the vector potential would be directed so, and the magnetic field represents its curl, and the curl of any gradient is zero.

After ionization of helium neutral atom. It might be simpler from technical viewpoint than creation of monoatomic hydrogen.

I can hardly imagine what experiments can confirm the presence of magnetic moments in hydrogen-like ions. It is probably necessary to maintain a certain concentration of monatomic hydrogen or ionized helium with the help of radiation, taking as a basis diatomic gas or neutral helium atoms. Then to maintain a stable concentration of excited states, again with radiation, for example, 2s or 2p, to measure magnetic moment and compare with obtained without action of radiation?

I suspect an electron in a hydrogen atom doesn't make any "orbital" motions. Just electron's own magnetic moment is statically smeared over probability cloud during chaotic throwing, like its electric charge. But if in excited states, that differ from the ground one, much greater moments are experimentally detected, then my assumption is incorrect.

Thanks for link to the article. I just wanted to separate the basic physical essence of equation from the mathematics involved in transitions between coordinate systems.

Can you cite experiments where, in some excited states of a hydrogen atom, magnetic moment significantly differs from Bohr's magneton was detected? Correction for magnetic moment of nucleus is insignificant. Only experimental data, not theoretical forecasts. Starting from the experiments of Stern and Gerlach, it seems that only moment of one magneton was detected, I could not find other information. But maybe I'm wrong and didn't search well?

Is it possible to write the Dirac equation without spinors? Something similar to Maxwell or Schrödinger's equations with simpler tensor algebra elements. Let values not be the same when going from the left coordinate system to the right, they could be valid only in the only system, for example, associated with the geometric center of hydrogen atom. But simple and obvious equation, or several equations.

I have doubts. But if neutrino has very little "mass", it can have some littlest moment also. Really this suggestion is inspired by joigus, and I think he will explain better if neutrino has magnetic moment. My suggestions are only the rudiments of possible theories. To explain something better, we need nonlinear equations. And I want to emphasize that my assumptions relate rather to the internal structure of elementary particles and have nothing to do with electron clouds in atoms built on the principles of long-action and statistics.

If neutrinos have magnetic dipole moment, evidently, they "partially" consist of magnetic field also. But there can be other fields like "weak", still never described and explained well in theories. And that's right, even to explain photon structure and behaviour non-linearity of equations is required.

Only with zero divergence electric field is impossible to construct something realistic, like particle. Except electric or magnetic dipole spheric radiation. Example of plain 1-dimensional wave, often encountered in books, does not exist in real world, it is infinite in space and has infinite energy.

It seems neutrino sequence is not limited of 3 types (e, mu, tau). Japanese scientist had proposed the law, how to calculate mass of next lepton via masses of previous ones. And there is no evident limit, after lepton-3 (tau) can be lepton-4 and so on. But masses become so great, that it is practically impossible to get such particles in experiment. And heavier leptons may have corresponding neutrinos. If neutrinos are left-handed, I think antineutrinos are right-handed. Obviously, it can be related to electric and magnetic field orientation.

It is very interesting if this might have anything to do with neutrinos. I think the nonlinearity of equations is also important, since neutrino types clearly correspond to leptons (electron, muon, tauon). For photons maybe it is possible that nonlinear effects have almost no influence. I only wished to say photon-like objects can be of two types with opposite fields orientation. How it correlates with "spin", additional investigations are required. Probably it is relation between magnetic moments and mechanical ones, created by energy flux or velocity vector V. In a photon all fields are transferred synchronously in one direction at the speed of light, so spin is recognized as +1 or -1. In an electron maybe other picture, magnetic moment is generated twice more effective than mechanical, after all-space integration of densities.

joigus, You gave good link to Arxiv article, confirming than on electric and magnetic field only, with zero divergences, we cannot build something realistic (except magnetic dipole spheric radiation), with behaviour like photon or electron for example. Only some kinds of exotic math games. Thanks for reference. In any case, electrons do not fly along "orbits" at almost constant speed. Rather they abruptly change their speed and direction of motion. If You try to simulate movement of electrons in an atom, soon there will be complete chaos. Only after great time possibly it can be noticed that in some places an electron is more frequent, others do not visit at all. But it is impossible to wait so long to gather good statistics. Only Schrödinger equation helps.

From this more canonical investigation without "alternative" velocity field V we get, that even for description of electric dipole radiation three "separate" fundamental field required, not two. Original Maxwell's equations anyway have to be extended, and this is not a religious dogma but a reason to think.

And an electron in an atom has no reason to move at speeds close to of light.

Of course. But without external source of attraction or directing field electron cloud dissipates. So I wonder how KG equation can be used by itself and extremely simplified. In Schrödinger's equation also often present potential from other electron clouds, specific member of additional repulsion of clouds with similar spin. One can add magnetic interactions. In KG are nothing similar.

Thanks for reference to Arxiv article. Anyway Maxwell equations for "pure field" are insufficient to describe even a photon. They are prone to deal only with E (D) and H (B) vectors, but at least four fundamental fields required: a, A, E, V. Maxwell's equations were often written before (specially in 19 century) with a term as product of velocity by charge density instead of current. But it seems that this term has come to be seen only as being associated with point charges forming some kind of cloud. Not as a fundamental field existing in the zero state everywhere on the continuum.

It is too simple to describe something like particle. But maybe useful for some specific cases, like 1-dimensional or 2-dimensional current in frozen conductor. Anyway, stationary point-like solution gives infinite integrals over all the space. Relativistic theories arose on the principle of long-range action. Of classical and quantum physics, only small fraction of equations work on the locality principle, so that first time derivative depends on the spatial state of some fields. Maxwell's equation for independent field without sources, Schrödinger's equation. The rest usually implies presence of material points, hamiltonians and long-range action. In no way KG equation is "controversial", but it came from long-action area. The most prominent feature of short-action (locality) is explicit first time derivative for any field (or at least for fundamental ones). Schrödinger's statistical equation is built so, and Maxwell's (for pure field without charges). It seems relativity is completely incompatible with motion of particles and even "big" charged bodies, because of problems with field internal energy conservation laws. Relativity is useful for such objects as rockets with clocks and stations with observers, nothing more. Schrödinger's equation works only in the presence of powerful centers of attraction, like atomic nuclei. Each member is important, and only the nuclear attraction collects an electron cloud, other members dissipate. My work is not designed to believe, but to think and calculate. "Very wide range" does mean if field intensity decreases with distance as 1 / r, 1 / r2, 1 / r3 and so on. "Simplest geometric shapes" does mean namely with minimal number of local extrema. If a photon emitted by electron cloud with statistical background, there is no cause to be very complex in shape. For example, visualization of function 1 / s looks much simpler than of r2 / s3 having minimum at r = 0 and other at f'(r) = 0 with other root of quadratic equation 2 · r / s3 - 3 r3 / s5 I have not found any word "classic" or "classical" in my last article about objects moving at the speed of light.

Let us consider what field " blobs" can be, moving at the speed of light in a certain direction while maintaining their shape. That is, compact formations capable of traveling long distances compared to their size without significant changes in structure. Unlike dipole radiation, which propagates spherically in all directions. Perhaps such structure have emissions of atoms during the transitions of electron clouds to less energetic levels. Discussion of how justified use of the term "photon" in relation to such objects is beyond the scope of this article. Let us take as basis the equations, existence of which in the real world is justified in the topic on dipole radiation: ? The following symbols are used: Scalar potential = a Vector potential = A Electrical field = E Speed of light in vacuum = c Time derivatives are denoted by singlequote ' a' = - c2 · div A A' = - E - grad a E' = c2 · rot rot A The formulas are given in cylindrical coordinate system (ρ,φ,z), associated with the point of space where the geometric center of field blob is located at the time of observation. Let us put r2 = ρ2 + z2 Motion occurs along z-axis at the speed of light and structure of field object remains unchanged, that is, ∂/∂t = - c · ∂/∂z for all physical quantities. Also, integral of internal energy throughout all the space must be finite, density of which is expressed by the law: u = ε0/2 · E2 + μ0/2 · H2 where E 2 = Eρ2 + Eφ2 + Ez2, H2 = Hρ2 + Hφ2 + Hz2 H = 1/μ0 · rot A, B = rot A = μ0 · H Let us put J = rot B = rot rot A Let us start with the mathematically simplest descriptions possible from the point of view of field laws mentioned above. In cylindrically symmetric case, when ∂/∂φ = 0 for all physical quantities. Basic equations are divided into two independent systems: 1. With circular electric field. Aφ' = - c · ∂Aφ/∂z = - Eφ → Eφ = c · ∂Aφ/∂z → ∂Eφ/∂z = c · ∂2Aφ/∂z2 Eφ' = - c · ∂Eφ/∂z = c2 · Jφ = c2 · (- ∂2Aφ/∂z2 - ∂2Aφ/∂ρ2 - ∂Aφ/∂ρ / ρ + Aφ / ρ2) → ∂Eφ/∂z = c · (∂2Aφ/∂z2 + ∂2Aφ/∂ρ2 + ∂Aφ/∂ρ / ρ - Aφ / ρ2) Equating ∂Eφ/∂z from two equations, we get ∂2Aφ/∂ρ2 + ∂Aφ/∂ρ / ρ - Aφ / ρ2 = 0 → ∂/∂ρ (∂Aφ/∂ρ + Aφ / ρ) = 0 If Aφ is not zero in all the space, so ∂Aφ/∂ρ + Aφ / ρ = 0, and Aφ is proportional to 1 / ρ, that gives infinite energy integral. Hence, such non-zero components of compact radiations can not exist. After artificial creation or computer modeling such structures will diverge in waves in all directions, instead of moving in one direction at the speed of light. 2. With circular magnetic field. a' = - c · ∂a/∂z = - c2 · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z) → ∂a/∂z = c · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z) Aρ' = - c · ∂Aρ/∂z = - Eρ - ∂a/∂ρ → Eρ = c · ∂Aρ/∂z - ∂a/∂ρ ∂Eρ/∂z = c · ∂2Aρ/∂z2 - ∂2a/∂ρ/∂z Az' = - c · ∂Az/∂z = - Ez - ∂a/∂z → Ez = c · ∂Az/∂z - ∂a/∂z ∂Ez/∂z = c · ∂2Az/∂z2 - ∂2a/∂z2 Eρ' = - c · ∂Eρ/∂z = c2 · Jρ → ∂Eρ/∂z = c · (∂2Aρ/∂z2 - ∂2Az/∂ρ/∂z) Ez' = - c · ∂Ez/∂z = c2 · Jz → ∂Ez/∂z = c · (∂2Az/∂ρ2 - ∂2Aρ/∂ρ/∂z - ∂Aρ/∂z / ρ + ∂Az/∂ρ / ρ) Equating ∂Eρ/∂z from the equations for Aρ' и Eρ', we get c · ∂2Aρ/∂z2 - ∂2a/∂ρ/∂z = c · (∂2Aρ/∂z2 - ∂2Az/∂ρ/∂z) and conclude that a = c · Az if we are talking about quantities decreasing to zero with distance from the center goes to infinity. From the equation for a' then follows ∂Aρ/∂ρ + Aρ / ρ = 0, which means Aρ = 0 if Aρ is not proportional to 1 / ρ with infinite energy integral. From the equation for Az' follows Ez = 0 at a = c · Az The following equations remain valid: Eρ = - ∂a/∂ρ = - c · ∂Az/∂ρ whereas from ∂Ez/∂z = c · (∂2Az/∂ρ2 + ∂Az/∂ρ / ρ) = 0 it follows that with non-zero Az must be Az proportional to ln(ρ) and energy integral is infinite. Thus, no valid expressions for field formations were found. The situation changes if we assume that div E ≠ 0 (non-zero charge density) and introduce additional terms into formulas for E' using the velocity field: E′ = c2 · J - grad (E · V) - V · div E where div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z in case of circular magnetic field, whereas case of circular electric field remains within previous calculations, since there div E = 0 Assuming that Vz = c is in the entire space around isolated field object, whereas Vρ = 0 and Vφ = 0, and since E · V = Ez · c, we get Eρ' = - c · ∂Eρ/∂z = c2 · Jρ - c · ∂Ez/∂ρ - 0 · div E → ∂Eρ/∂z = ∂Ez/∂ρ - c · Jρ → ∂Eρ/∂z = ∂Ez/∂ρ - c · (∂2Az/∂ρ/∂z - ∂2Aρ/∂z2) Ez' = - c · ∂Ez/∂z = c2 · Jz - c · ∂Ez/∂z - c · div E → ∂Ez/∂z = - c · Jz + ∂Ez/∂z + div E → c · Jz = div E → c · (∂2Aρ/∂ρ/∂z - ∂2Az/∂ρ2 + ∂Aρ/∂z / ρ - ∂Az/∂ρ / ρ) = div E The following equations remain true ∂a/∂z = c · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z) Eρ = c · ∂Aρ/∂z - ∂a/∂ρ Ez = c · ∂Az/∂z - ∂a/∂z From the expression for Ez' after substitutions it follows: c · (∂2Aρ/∂ρ/∂z - ∂2Az/∂ρ2 + ∂Aρ/∂z / ρ - ∂Az/∂ρ / ρ) = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z = c · ∂2Aρ/∂ρ/∂z - ∂2a/∂ρ2 + c · ∂Aρ/∂z / ρ - ∂a/∂ρ / ρ + c · ∂2Az/∂z2 - ∂2a/∂z2 → ∂2a/∂ρ2 + ∂a/∂ρ / ρ + ∂2a/∂z2 = c · (∂2Az/∂ρ2 + ∂Az/∂ρ / ρ + ∂2Az/∂z2) Which leads to the conclusion a = c · Az Then Ez = 0, also ∂Aρ/∂ρ + Aρ / ρ = 0, hence Aρ = 0 to avoid infinity of energy integral. As result we get: a = c · Az, Aρ = 0, Ez = 0 Eρ = - ∂a/∂ρ = - c · ∂Az/∂ρ Which corresponds to the equation derived earlier from Eρ' ∂Eρ/∂z = ∂Ez/∂ρ - c · (∂2Az/∂ρ/∂z - ∂2Aρ/∂z2) Herewith Bφ = - ∂Az/∂ρ = Eρ/c Charge, spin and polarization If one looks in the direction of movement of field object, it is easy to notice that in the above version with annular magnetic field it is possible to orient this field clockwise or counterclockwise. Accordingly, radial electric field will be directed from z-axis outward or inward to this axis. To one type of field formations can be attributed conditional positive "spin", to the second negative. Let us try to find out how intensity of fields can decrease at distance from the geometric center of object. Let a = A0 / s, где A0 = amplitude constant, and s2 = R2 + ρ2 + z2, where R = object's scaling constant, possibly having an indirect relation to conditional "wavelength" in experiments. Note that ∂s/∂ρ = ρ / s, ∂s/∂z = z / s Then Az = A0 / c / s, Aρ = 0, Eρ = A0 · ρ / s3, Ez = 0 div E = ∂Eρ/∂ρ + Eρ / ρ = A0 · (2 / s3 - 3 · ρ2 / s5) The integral of charge density (divided by dielectric constant) over the entire space will be equal to ∫-∞+∞∫02·π∫0∞ (2 / s3 - 3 · ρ2 / s5) · ρ ∂ρ ∂φ ∂z = 0 That is, although charge density is not locally zero, the object as a whole is charged neutrally. This is natural, for example, for radiation arising from atoms and molecules, taking into account laws of conservation, since the particles located there will not give up part of their charge. In general, when E = Eρ = - ∂a/∂ρ, the subintegral expression ρ · div E = ρ · (∂Eρ/∂ρ + Eρ / ρ) = ρ · (- ∂2a/∂ρ2 - ∂a/∂ρ / ρ) = - ρ · ∂2a/∂ρ2 - ∂a/∂ρ = ∂/∂ρ (- ρ · ∂a/∂ρ) Computing the integral ∫0∞ ρ · div E ∂ρ we get for ρ = 0 the function - ρ · ∂a/∂ρ = 0, for ρ = ∞ the function - ρ · ∂a/∂ρ = 0 if ∂a/∂ρ decreases by absolute value with a distance faster than 1 / s Further computation of integrals by φ and z will not change zero result. The author of this article tested using MathCAD zero equality of the triple integral for a = A0 · ρ2 / s3 with Eρ = A0 · (2 · ρ / s3 - 3 · ρ3 / s5), also for a = A0 · ρ4 / s5 with Eρ = A0 · (4 · ρ3 / s5 - 5 · ρ5 / s7), for a = A0 · ρ / s2, a = A0 · z / s2, a = A0 / s2 Very wide range of such objects is neutrally charged in general, although it is likely that field formations are statistically inclined to take simplest geometric shapes, with minimum number of spatial extrema. It should be noted that when a = A0 / s2 or s appears with even higher degrees, field formation receives significantly greater ability to penetrate matter than with a = A0 / s or a = A0 · ρ2 / s3 Accordingly, the probability of registration of field object by measuring instruments is reduced. Which may be similar to the behavior of neutrinos in experiments. Polarized field object can be described as follows: s2 = R2 + X · x2 + Y · y2 + Z · z2 where R, X, Y, Z are scaling constants ∂s/∂x = X · x / s, ∂s/∂y = Y · y / s, ∂s/∂z = Z · z / s If a = A0 / s, where A0 is amplitude Az = A0 / c / s, Ax = 0, Ay = 0 Ex = A0 · X · x / s3, Ey = A0 · Y · y / s3, Ez = 0 Bx = - A0 / c · Y · y / s3, By = A0 / c · X · x / s3, Bz = 0 div E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z = A0 · (X / s3 - 3 · X · x2 / s5 + Y / s3 - 3 · Y · y2 / s5) At the same time, all the above formulas for case of circular magnetic field remain true, E′ = c2 · J - grad (E · V) - V · div E Ex' = c2 · (∂Bz/∂y - ∂By/∂z) - 0 - 0 = 3 · A0 · c · X · Z · x · z / s5 Ey' = c2 · (∂Bx/∂z - ∂Bz/∂x) - 0 - 0 = 3 · A0 · c · Y · Z · y · z / s5 Ez' = c2 · (∂By/∂x - ∂Bx/∂y) - 0 - c · div E = 0 That is, there may be no cylindrical symmetry, with different X and Y, the field object will be stretched or extended along x- axis or y-axis. Compression or extension along z-axis is determined by multiplier Z. With significant differences between coordinate multipliers, structures arise with predominant orientation of fields in one direction (and the opposite also) in areas with high field energy density.

Klein–Gordon equation - Wikipedia All-space integral of square of the function sin(k · r) / r (spherically symmetric stationary solution) also is infinite. k is some constant, r is the distance from the center of coordinate system. My speculations have nothing to do with QED. The most similar analogy is like how stars and planets form from clouds of cosmic dust and gas, so I imagine the formation of elementary particles from fields. But this process requires nonlinear equations. In my calculations they are still linear. Some useful thoughts can be taken from here: Here is the .pdf version of original article. ehveng.pdf

Classical gravitoelectromagnetism is based on long-action. I try to avoid this and use only short-action differential equations. Without "sources" of potentials and other fields, like charges or masses in their mechanical representation as "points", or some density, also referring to "cloud" of points.

1. In my understanding, the structure, for example, of electron is detailed description of the fields that make up the central part with high energy density. An electron cannot be divided, it does not have many point-centers of energy compaction, although it can annihilate under certain conditions. The structure of it is described rather by the scaling constant R that I used in equations. 2. My calculations have nothing to do with quantum physics, this is an exit to the level of "pure fields" on continuum, where we can dig as deep as we want and there will be always something even smaller. Theories are designed mostly for computer simulation of processes at distances comparable to the dimensions of elemental particles. Using raster matrices, finite differences or finite elements methods. What will happen, for example, in the collision of particles. And such simulation is much cheaper than natural experiments.