In  a course of Modern Physics or the Quantum Theory, the dual behaviour of particles

such as the electron, or the photon, is that in some instances they behave as particles and sometimes as waves. There is vast literatura on this mysterious behaviour.

It is also acomon term found in mathematics, such as dual space.

There is deBroglie wavelength, but more often now wavelet?

Dual space in mathematics, is not directly related to physics. The physical laws are often in Cartesian coordinates,  for example compatibility of strains. 

Curvature is physics ofcourse(?)

Rewriting equations into curvilinear if they work in Cartesian is not to understand. 

 @Juan Weisz. The Bohr "dualism" is rejected in Transactional Physics.

And the Afshar experiment proved that the Bohr's dualism is a feery tale just for the children. Never a photon becomes corpuscular.

Article Fifteen surreptitious, copenhaguist and corpuscularist postu...

De Broglie who concluded that electrons have both particle and wave behavior also had a hypothesis about wave-particle dualism for localized photons.

Here is how this can be implemented:

https://www.omicsonline.com/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.pdf

Louis de Broglie believed in coordinates, all the geometry learned at school. So he maintained the believing in corpuscles, and remained stuck the two feet in the same concrete sock for the remaining of his life.

Discovered in 1921, the Ramsauer-Townsend transparency effect is explained only if the electron is its de Broglie wave. So it is hidden to the students in every handbook of QM except the Sivoukhin's one. Too embarrassing !

The double particle photon model, needs extension in space, but not mass. It appear useful, to describe phenomena, and modern compared with recent thinking.

Dear Lena,

This particular double particle photon model provides both extension in space for inner electromagnetic oscillation behavior coupled with permanent localization in space for massive (omnidirectional inertia) behavior, This can be understood however only in the frame of the new 3-spaces geometry described in the paper, which for the first time allows integrating the magnetic behavior into the Maxwell compliant electromagnetic mutual induction mechanics.

Quite recent thinking in all regards.

In the context of string theory where so called elementary particles are not elementary objects, but they are various excitation modes of a one dimensional string, one sees a new type of duality which was unseen previously. Here fundamental particles of a given string theory gets mapped to a composite particle of another theory (and vice versa) under the duality map. Then the classification scheme of elementary and composite particles looses it's significance as it depends on which particular theory one uses to describe them.

Well, when electrons actually hit a screen and show up as points

they certainly are particles. The wave is something else, related to the shadow formed from  these points, related to the probability of finding them. There are experiments showing how an interference pattern builds up on a screen. Even single electrons at a time form these interference patterns in the equivalent of a two slit experiment.

Therefore we are not saying that the same thing is both a wave and a particle.

What I said is that in some experiments the wave property mostly shows up, and in others the particle property.

In Ramsauer Townsend the wave property shows up, but that does not rule out the particle aspect.

 Well, it is so "doubleplusgood duckspeaking", in the copenhaguist newspeak !

However, the interferences discovered by Thomas Young prove that lengths of coherence of individual waves are several decimeters long in the visible range. Lots of optical phenomena I have collected prove they are several wavelengths wide.

Sure an individual wave has only one emitter, and only one absorber. But what could be the physics that could transmute such individual waves into something corpuscular ?

Article Fifteen surreptitious, copenhaguist and corpuscularist postu...

Dear Jacques,

From what I have concluded, what we directly observe is made of sort of a "crowd effect" due to the incalculable number of individual submicroscopic photons that are involved. I suspect that resonance effects causes them to appear to us as continuous at our level. just like the picture on the new 4K tv-screens appear as clear as analog pictures when seen from even a few meters away. We know however that these pictures are made of very small individual pixels.

I think something similar explain what you conclude as being "continuous waves" at our macroscopic level as observed from our macroscopic instruments.

These photons are as small with respect to our macroscopic level as our planet is with respect to our galaxy. So whatever our macroscopic  instruments (as in the Young experiment) observe from the sub microscopic level generally involves practically astronomical numbers of individual submicroscopic photons, each moving at the speed of light, each individually interfering with one electron (itself electromagnetic in nature) of the electronic escorts of one of the atoms of which the instrument is made of, after having been deflected and its polarization orientation having been determined by the other atoms of the instruments that it did not directly hit but only grazed on its way to absorption.

Also, the EM oscillation is transverse to the direction  of motion at the speed of light which is an added complexity, which limits their width (their frontal cross-section) considerably with respect to the longitudinal wave length that we use to relate to their frequencies.

At the submicroscopic level, it is possible to come up with a logical electromagnetic  mechanics that provides an explanation of why elementary photons can have at the same time wave characteristics (electromagnetic oscillation) and corpuscular characteristics (omnidirectional inertia).

Of course, I may be wrong, but it can be made to mathematically pan out in total conformity with Maxwell.

Hope some of this makes sense to you;

They never have "corpuscular characteristics". But absorbers do exist. And each individual wave has one emitter and one absorber.

Transactionnal physics deals with

1. the synchrone transfer of power from emitter to absorber (for a photon), and the geometry of this Fermat's spindle.

2. The physics of the Dirac-de-Broglie noise, where the successful transactions emerge, eventually. 

For an electron, replace "transfer of power" by transfer of charge, spin and so on.

So works an Ahoronov-Bohm experiment. Each electron pass simultaneously above and under the micro-solenoid, and converge to its absorber (of small size).

Jaques

for electrons

It is because the wave is not something phyisical. It is merely a probability amplitude.

The particle is physical and hits the screen.

I think now you switched the topic to photons. The coherence length depends on

if it is a laser (long) or a comon source (short).

The problema in this case is what is a photon. Here the question is still open,,

there are long discussions on research gate, whitout much agreement.

when the light is emitted or absorbed it is by means of photons

Dont know what you mean by width,the electric field magnitude, or beam width?

or slit width?

Lena

Dual space is both mathematics and used in physics as k space.(or wavector space.)

the magnitude of wavevector is in inverse lenth,, 2 pi divided by lamda, lambda is the wave length.

the direction of wavector is in the direction of motion of the light or particle.

k space results from three not colinear wavectors

One of them, k1, is r2 cross r3 divided by( r2 cross r3) dot  r1 in terms of real space vectors r1,r2,r3

An electron does not become an abstract and unphysical thing, just because it has an intrinsic frequency, mc²/h. And an electron is not the probability to find an electron-corpuscle.

When the Göttingen-København sect deals with their  "wave function", they borrow the macro-time of the laboratory, and extrapolate its properties where they were never validated.

Of course the electron is not a probability

The macro time of the lab is the only one I know of,precluding relativistic effects.

 generally I do not know enough about the transactional stuff to argue

The interpretations of QM are so diverse, cannot comunícate without common

background.

Each object is both an emitter and absorber, sending a forward going and recieving

backward going waves in time? How does that all come together?. The forward

going is psi and the backward going is psi*? Do both obey schrodinger?

So was the paper of Yakir Aharonov, Peter Bergmann and Joel Lebowitz in 1964. However they draw only statistical conclusions.

In the first state of the noise description, unconsciously given by de Broglie in 1923, each electron was not an emitter nor receiver, but rather synchrone with itself on all its spatial extension. This changes with Dirac in 1928, as now the electron (wave) has four components, two are orthochrone in microtime, and two are retrochrone in microtime, and all on the light cone. So now we know that the background noise, Dirac-de-Broglie's is bidirectionnal in all the microtimes. There is no way to shield from this background noise.

There are no ways to monitor the noise, where most attempts of transactions are unsuccessful, but some emerge. There are no ways to predict which successful transaction will emerge, nor when. But we can rate the admittances so the probabilities.

Greetings and Salutations Lena,

"Stationary Waves? ... Can it be described with a density, or will that be a third order effect?"

I believe the photon and electron wave-particles can be formulated as 4D gravitoelectromagnetic stress energy density tensor wavetrain integrations--and have posted on ResearchGate the thesis with Mathematica code for such, postulating a universal wavefunction stationary domain and fluid range:

Abstract

Natural evolution of ATP cognitive systems is to meet AI peer review standards. ATP process of axiom selection from Mizar to prove a conjecture would be further refined, as in all human and machine learning, by solving the real world problem of the proposed AI peer review challenge: Determine which conjecture forms the higher confidence level constructive proof between Standard Model of Physics SU(n) lattice gauge group operation vs. present non-standard 4D GEM EOS SU(n) lattice gauge group spatially extended operation in which the photon and electron are the first two trace angular momentum invariants of a gravitoelectromagnetic (GEM) energy momentum density tensor wavetrain integration spin-stress pressure-volume equation of state (EOS), initiated via 32 lines of Mathematica code. Resulting gravitoelectromagnetic spectrum ranges from compressive through rarefactive of the central cosmological constant vacuum energy density in units of pascals. Said self-adjoint group operation exclusively operates on the stress energy momentum tensor of the Einstein field equations, introducing quantization directly on the 4D spacetime level, essentially reformulating the compounded, and still diverging, Yang-Mills virtual superpositioned particle lattice gauge groups quantization of the vacuum—into an optimized single hyper-complex multi-valued GEM U(1)×SO(3) lattice gauge group Planck spacetime mesh quantization of the vacuum. Thus the Mizar corpus already contains all of the axioms required for relevant DeepMath premise selection and unambiguous formal natural language parsing in context deep learning.

Thesis AI Peer Review Challenge: Standard Model of Physics vs 4D GEM EOS

When waves are refered to, it is meant the usual form, say

sin(kx-omega t) used in physics, or in Fourier analysis.

In wavelet theory an analysis is made similar to Fourier analysis, but uses waves which may have discontinuities, such as rectangular waves, or of sawtooth form.(periodic structures with these shapes). I have not seen this in physics context of electrons yet, might be interesting.

Curvature is both in Physics and Mathematics. The math preceded Einstein in the Works of Gauss and Riemann.

General relativity uses general transformations adapted to curved space time.

Juan, thnx for summary.

The Fourier analysis is powerful in analysis, but I think the new notation FFT, i.e. FastFT, is chosen for a reason, to emphasize that there are other solutions, and that adding different terms in the expansion is not always the correct solution.

And wavelets is in the shape of one wave, in some applications, (not propagating wavefronts)

Lena

Sometimes you have to numerically implement a Fourier transform, and there are certain things that can be done

to make such codes run faster. An FFT incorporates these

tricks..

  For example in Fourier transform spectroscopy, you Fourier transform the output signal to get the spectrum of the source.

Yes, that is clear. In Modal analysis too.

It is the modal super position, that is not obvious, (?) but one has to look at every certain problem if many dof. Or if there is some general rule, i.e. the second divides one more time (violating the bc, or if many dof, creating a new boundary). (But then, that can continue in smaller levels, and undeterminate).

I have experience in piping, so I know how ASME did superposition in spatial direction. For frequency, in practise,  they said that there were not enough energy to excite higher modes.

Juan Weisz. There are no way to produce interferences with "amplitude of probabilities".

However interferences still occur when no "crowd effect".

Even with ultra-reduced intensities, when only one photon or only one electron travels through the optical device, the electron or the photon interfers with itself, and never finds its absorber where the two branches differ by half a period in optical time.

stationary waves do not travel. there is up and down motion only.

or no motion at all.

ie quantum bound states is one example.

Jackes

I could not disagree more strongly. The interference is at the level of amplitudes

or the electric fiels in  electromagnetism

Where do you get these strange ideas from?

the  probability amplidude of the electron interferes, when after a split, from a comon source, the beams rejoin.  Probability amplitude is the same as wave function.

How do you prove that one photon is NOT a propagating electromagnetic field ?

How do you prove that a photon escapes to the Fresnel physical optics, 1819 ?

We perform diffractograms with electrons too, and with neutrons if you have access to a neutronic source. So how do you prove that electrons and neutrons ceased to be waves ? Each one.

Why in any QM book, excepted the Sivoukhin, the Ramsauer-Townsend transparency effect is strongly censured ? Just incompatible with the Göttingen-København dogmas... This censure is a smoking gun.

I find Ramsauer- Townsend in my book Schiff, and I see no reason to censure it.

It goes quite along with all the rest that is known

The definition of the photon as a quanta should be enough to answer you

Diffraction patterns point to the action  of waves, I cannot see your problem

.Every book in optics I know of gives interference at the level of electric

field, which you then square to find intensity. The quantum interference at the level of wave function is also very basic stuff.

In the Feynman : censored.

 In the Greiner : censored.

In the Basdevant : censored.

In the Cohen-Tannoudji, Diu & Laloë : censored.

In the Messiah : censored.

In the Landau and Lifchitz : censored.

In the Pauling and Bright-Wilson : censored.

In the Atkins : censored.

In the Elbaz : censored.

In the Dirac : censored.

In the Ngo and Ngo : censored.

In the Lévy-Leblond and Balibar : censored.

 ...

The thing is this  particular experiment is a bit more complex to explain properly

That is why I think it is not always treated.

Simply the whole doctrine of the Göttingen-København pack and their heirs relies on corpuscular ideations.

For 1921, the Ramsauer-Townsend transparency proves that no corpuscules exist at all and no "corpuscular aspects" : the electron is its de Broglie wave.

Period.

In 1928, P.A.M. Dirac proved that there are four components in this wave.

In 1933 in his Nobel lecture, he mentionned he knew the corrected version - alas published nowhere - of the Schrödinger discovery that the Compton scattering is a Bragg diffraction, but on the Zitterbewegung spatial frequency of the rebouncing electron.

In 2011, I had to re-discover this fact, unknown everywhere.

"the compton scattering is a bragg diffraction"

Nuts, scattering off just one electron not the same as scattering off a whole bunch of atoms

we know dirac has 4 components.

we use both wave and particle idealization

Compton has proved that a photon is scattered on a quasi free electron, a conduction electron.

In his first paper in 1927, Schrödinger had not yet the right Bragg equidistance : he had not yet the complete wave equation.

Article The Zitterbewegung : key of the electron-photon scattering u...

"scattering off just one electron not the same as scattering off a whole bunch of atoms".

What is common in these Bragg diffractions is the equidistance of successive planes.

The Compton diffraction is the most simple in the frame of the center of inertia : only one direction of plane. But this plane cannot be predicted. No experimental angular preparation is possible.

In a crystal, there are several directions of planes, and several equidistances. Either we use the Laue method, on a monocrystal, but with polychromatic "white" illumination, either we use powder of the crystalline variety to investigate, and obtain a Debye-Scherrer record, under the K_alpha of a choosen anticathode, with frequential filters.

We had obtained Laue of carbide inclusions on the big electronic microscope, but we must wobulate the tension of the microscope.

Of course the goal of the radiocrystallographist is to obtain the crystalline structure, so he/she uses an intense source of X-rays or electrons. However the theory must be valid from the individual wave.