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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.