Jean Claude, I really don't think that the Schrödinger equation isn't a wave equation, especially as it precisely has wave solutions of the type you gave (I would rather write them f(kx-wt)). Anyway, the Dirac equation is definitively a wave equation, and is very similar to the Maxwell equations.  The real difference between a matter wave and a quantal wave is that the latter has a phase velocity which is greater than light speed. It can't in any way be compared to a wave propagating in a medium, since it has not its local property.

I have reservations against wave theory as well. Could you please provide a reference to the sonic interaction you have mentioned?

If light were a wave in a medium, similar to sound or water waves, then it would be natural to think about describing the action of the medium itself rather than the photon. But since everything we know suggests that photons are not waves that exist as a disturbance in a medium, this approach would not seem to be relevant. It would take a major change in our fundamental understanding of what light is for this approach to be applicable here.

Dear Vikash,

there is no doubt that wave theory is very useful in predicting behaviour of many phenomena, my interests lie in the things it can't predict or where there is ambiguity, for example the particle / wave paradox with photons.

I am not aware of any papers in this area, hence my question: is anyone working here and if not, anyone interested in collaboration?

Dear John,

I am unsure what you mean by light being a wave in a medium. If light passes through glass, then its velocity decreases, the glass being the medium, in vacuum there is no conventional medium (although there are some theories looking at this). If you consider sound at the molecular level, there is no medium. Perhaps you are thinking conventionally with sound waves moving through a gas, for example helium, in this approach helium is the medium. What I am considering is the individual He molecules, they are not moving in any medium.

As for photons, there is the wave/particle duality paradox. for Photovoltaic effect they are considered particles (Einstein won the Nobel prize for this work), the twin slit experiment considers them waves. Photon energy is E=hc/lambda or E = hf; the equations that links the two views.

Paul

For the case of sound waves, the wave represents displacements of the molecules in the gas, so gas is the medium that I'm referring to.  A photon can propagate in vacuum, and is not simply a displacement of the atoms/molecules/whatever forming some medium.

Personally, I don't like thinking of wave-particle duality as meaning that sometimes lights acts like a wave and sometimes it acts like a particle. To me, it is *always* treated as a wave (though not a simple, classical wave). However, there are kinematic regions where the wave behavior is essentially the same as the particle behavior, and it's much easier to think of it as a particle, especially when dealing with single quanta (photons). But they aren't inconsistent or different pictures, there are simply limits where it looks like a classical wave, and limits where it looks like a classical particle.

@Paul Howard Riley . But if you considered that there was half a sentence wrong, in this famous paper from Einstein ? It would be a paradise if each of us only put a half sentence wrong in his/her papers.

Along these 110 years, a corpuscular nature of light was never, never, never experimentally confirmed. Only remains the initial statement from Max Planck : light can be bought or selled only by quanta h of action per cycle.

Period.

Excepted however with Bremsstrahlung and synchrotron radiation.

UNIFICATION of GRAVITATION and ELECTROMAGNETISM 

I unified at Quantum level, Electromagnetism and Gravitation with Ferent equation for the energy of a photon:

E = h × f + a × f

Ferent equation for photon – graviton interaction:

E = h × f + a × f - a × ν

where - a × ν is the negative energy of the graviton

ν is the frequency of the graviton

“I am the first who understood and explained Gravitation with high speed gravitons   v = 1.001762 × 10^17 m/s, with Negative Momentum, Negative Mass and Negative Energy” Adrian Ferent

https://www.researchgate.net/publication/299135595_Ferent_Gravitation_theory

Article Ferent Gravitation theory

maybe this might be of your interest: 

SECRETS BEHIND THE MACH-ZEHNDER PHENOMENON (1)

Research SECRETS BEHIND THE MACH-ZEHNDER PHENOMENON

Dear Prof. Riley,

                            I would like to point out that in an earlier answer one of the discussants pointed out that in Quantum Physics we are not interested in the Statistical properties of waves but the physical properties. I am not complaining about that it is certainly true. But in terms of your question I would like to add that when we consider waves in gases say and consider the statistical model of classical Brownian motion by which we calculate the value of the random variable at any point in the x-y plane we are also describing the position of the particle which that random variable can very well represent. When we derive the dynamics of such a random variable therefrom we consider the various  paths and determine the location of the saddle path by using data and probabilities based on the information we have. So I would like to say that this is very much wave mechanics in my understanding and it helps the physicist to solve k at the problem as much as the computer scientist as well as the statistician. So nonwave theories are not needed because we do not know how to handle physical wave theories. Earl Chair Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE

Experiments to test Ferent Gravitation theory!

In the double slit experiment the interacting observer is an instrument, detector…

My experiment is: if you replace the detector with a piece of metal the wave will collapse into a particle because of my theory photon – graviton interaction:

Ferent equation for photon – graviton interaction:

E = h × f + a × f - a × ν

where - a × ν is the negative energy of the graviton

ν is the frequency of the graviton

Here we know the frequency of the photon f and the frequency of the graviton f. With different metals we have different frequencies of the graviton ν.

A lot of experiments can be done, and will result a lot of data!

Ferent gravitation theory explains the double slit experiment!

https://www.researchgate.net/publication/299135595_Ferent_Gravitation_theory

You can read on my theory:

Decoherence explained by my theory

The electromagnetic wave is the superposition of 3 sinusoids; this means the electromagnetic wave will be collapsed by the presence of an electric field, of a magnetic field, of a gravitational field, by another electromagnetic wave… 

In my electromagnetic theory, gravity does collapse quantum superpositions, gravity bends light because light has 3 sinusoids, has a gravitational sinusoid!

In Maxwell electromagnetic theory, gravity does not collapse quantum superpositions, gravity does not bend light, because light has only 2 sinusoids!

So decoherence is due to the gravitational field, for example to the gravitational waves generated by the observer in the double-slit experiment.

Article Ferent Gravitation theory

Dear John,

"For the case of sound waves, the wave represents displacements of the molecules in the gas"

you are mixing up the concept of a group of molecules, as defined by Swift:

http://www.abdi-ecommerce10.com/asa/p-259-thermoacoustics-electronic-copy.aspx

which do displace to give pressure and velocity as you say,

with the micro scale where interactions are collisions between molecules or the container wall. Wave theory breaks down at the molecule collision level.

I agree with your comment

 "I don't like thinking of wave-particle duality as meaning that sometimes lights acts like a wave and sometimes it acts like a particle. To me, it is *always* treated as a wave (though not a simple, classical wave)." 

most modern theories do treat the quantum world as waves. However, they also struggle to explain a number of things.

My question is whether anyone is researching considering quantum effects as only particle interactions (for example photons).  I would like to share insights with them, as the preliminary work I have done suggests this may be a better way to resolve the problems.

Dear  Soumitra,

you say " I would like to point out that in an earlier answer one of the discussants pointed out that in Quantum Physics we are not interested in the Statistical properties of waves but the physical properties."

Could you explain something that puzzles me. In all the quantum experiments I have read, all theorems are based on statistical measurements. The Schrödinger equations are based on probabilities, the photon dual slit experiment only works with many photons (even if they traverse the apparatus one at a time). So it looks to me as if current quantum theories are all statistical the opposite of what you said.

Best Regards

Paul

The paper "Relational measurement and the uncertainty relations" proposes a broader system view.  

Abstract: The dichotomy between quantum measurement formalisms and experimental results, referred to as the "measurement problem" or "measurement collapse" is resolved by including calibration and sampling processes in a relational measurement process. Both formal proofs and experimental verification (by others) are presented.

Dear Prof. Riley,

                              My objective is just to carry forward with the discussion by starting with a particular reference, so it is not really important that someone said it or so I thought, but it is what I would like to argue and use my research to substantiate against. Earl Chair Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE 

Dear Prof. Riley,

                            I think there is some research recently published by Smith in the European Journal of Physics in which the skin effect in a round wire is found to be providing a cross section of a field for the quantum flow of electrons. By this experimental observation in the Econophysics lower dimensional system where the charge on particles is not always necessary to be considered for e.g. and in one of his publications from what the author says it can be inferred that the reflections or absorptions follow the uniformities of a meanfield. Hence the relation between multidimensional stochastic processes and the physical meanfield is established from these experiments. Of course infinite dimensional stochastic processes when integrated by efficient or min energy or maximum entropy methods, as I have had the opportunity to show in some of my research, please see the Haag's Theorem paper on my RG webpage for example, hence in the Shroedinger sense i.e. physically within a box, may shed almost all dimensions and will not necessarily generate an infinite dimensional meanfield. So the roundwire with skin experiment is perfectly suited I think to linear and fixed wires. This is an example in terms of what you said. I hope it makes sense. Earl Chair Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE

I think, sooner or later, we will be able to formulate a physical theory that does not need to be quantised, therefore, it would be a non-wave theory. I've been working on this problem for sometime and have developed, although scratchily, a temporal dynamics that could be used to explain quantum mechanics in terms of classical dynamics. Please refer to my works on RG for more details.

 Dear Prof. Ho,

                           I have also developed a String Theoretic Meanfield Theory for mining which makes use of factorisation by String perturbation in coal mining and other such types of mining and is listed on my RG page and is published in Journal of Physics A being a theoretical breakthrough as you have pointed out above. I have also published the String Theoretic Differential and Integral Heterotic Dbranes String coupled technology in Journal of Physics D. I only wish if Her Majesty and the British Government would help me share my innovation with a foreign multinational company like British Gas and set up manufacturing/mining companies all over the world including in  India and China where there are large mineral reserves as well as in the Scandinavian countries where there are also large potentials of Glacial mining for water for household and industrial use India with its tremedous talent pool and structural shift to Financial Capitalism can benefit from this home grown technology which has been developed partially at New York University, New York, USA and Indian Institute of Social Welfare & Business Mangement and Indian Statistical Institute, Kolkata also, can be of tremendous use to mankind and help me and my son Sandipan use this technology and theory developed for building our family Trust and heirloom, we being the First family in India which we share with the Nehrus and the Gandhis and the Singhs and the Senas being our blood relatives and we being the Earl family and the heir descendant of the Shahs and the Moghuls. I apologise to the Research Gate for using this open forum to develop this contract and the patent sharing arrangement in such a way, but with Her Majesty's blessings and your help we can tide over what has been a pathbrwaking structural experiment and structural change for India and the subcontinent and also South-South East Asia and now it requires new investments and new companies to integrate with this tremendous stock market boom. I would be obliged if this collaboration is assignable as British Gas and British Petroleum Mallick "new mining" technology collaboration. I don't want to use my Earl Fitzpatrick of York name in this to enable it to be further collaborative. Thanks. Earl Chair Professor Dr. S. K. Mallick QC EPS Fellow (In) MES MRES MAICTE

Paul,  you asked:   “My question is whether anyone is researching considering quantum effects as only particle interactions (for example photons).”    In our view,  quantum effects appear to be non-classical effects because we don’t have an adequate understanding of the makeup and dynamic structures of the photon and subatomic particles.  With such understanding,  quantum (wave) behavior could then be understood classically as particle interactions,  although the mathematical formalism of qm, qed, and qcd,  which use wave mechanics in the calculation of such behavior,  will still generally be applicable.

Attached is an article,  which gives a view for each of the photon (Fig. 1) and electron (Fig. 2).  This introductory article is excerpted from a larger body of work,  which gives derivations of the dynamic structures of the subatomic particles,  an atom,  a molecule,  and  gas structures (H2 & He).  As shown,  each modeled particle is a tiny bound field,  which were derived by employing Maxwell’s calculus.  Based on these models,  quantum effects can be understood classically.  For example,  the interference pattern in the double slit experiment can be understood as resulting from particle interactions.  This can be substantiated by building upon the Compton Effect,  which itself can be understood classically as interactions of the proposed particles.  These processes can be described in another post if you wish.

Research Derivation of the Electron (Post #2 Updated)

Dear Dr. Riley,

it seems the Bohmian approach to quantum mechanics (a book by Teufel, Durr: 'Bohmian quantum mechanics') could be appropriate here.

 @Marek Pietrow. For de Broglie and D. Bohm, the wave is not the electron, and the electron is not the wave. Just another dead end, so.

Dear Marek,

does that also include pilot wave theory or Broglie-Bohm ?

Regards

Paul

Yes, the theory is called (by Bohm) a causal interpretation and it is linked with de Brogile theory called 'the pilot wave' interpretation. The wave there has another meaning than in Copenhagen interpretation. Please see https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_(De Broglie–Bohm theory) and reference [1] therein to make an overview.

Actually you address 3 different issues in your question

1. the meaning of QM. The common interpretations look for physical interpretations (the 2 physics) : QM laws would be the manifestations of some bizarre properties of physical objects at the atomic and sub-atomic level. The other 2 interpretations show that the so called QM axioms come from the way measures are done and used. This is done either by starting from the measures (Jauch, Haag, Charles Francis) or by starting from the mathematical formalism of models used in Physics (myself).

2. How particles can behave as waves ? There is no physical interpretation (only that "this is so"). Actually the problem comes from the way we represent (through measures or models) the behavior of a family of identical particles submitted to the same experiment. It is intuitive that we do not deal with each particle individually, we represent a family of particles which have similar characteristics by a set of measures (or variables) which depend on a finite number of parameters, and this leads to a wave-like formalism.

3. How fields can behave like particles ? The common answer is the QTF which replaces the classic concept of fields, as physical entity present everywhere and acting locally, by the interaction of fermions and bosons (particles representing the field). My answer is that classic fields can show discontinuities (like a shock wave in a fluid) and it can be shown that these discontinuities must behave as particles : they are necessarily located on a line, move lke the field, and can be endowed with momentum, charge and mass.

Actually particles alone can do it.

You can think that the electron is just a particle.

The wave is something different. It just has to do with the probability of finding the particle, which is a different thing.

It is not a physical wave, as in enm theory.

I have seen some nice japanese experiments, where the wave builds up merely as a 

shadow formed by the points. The points are where the electrons landed.

Well, the mystery is still there, why probability in waves?

I believe your question rests on a mistaken assumption, namely that QM treats systems as waves. That is really not the case. It describes systems by a state, which is a complex valued function in the space describing all they system's possible configurations. This has some mathematical similarities with a wave, but has in fact nothing in common with it. The Bohmian theory attempts to obtain a particle picture, but does so at the cost of introducing interactions between arbitrarily distant particles, the intensity of which does not decay as the particles' distances go to infinity.

There are  indeed problems in QM, but these are not so easy to solve, and the so called wave particle duality, which is  a remnant of a bygone age, is thoroughly unhelpful in this respect. A good modern textbook would be a good place to start, maybe ``Quantum mechanics: The theoretical minimum'' by Leonard Susskind, though many more exist.

@ Leyvraz and all,

Thanks for raising such comments! Let me tell you I had severe problem in understanding quantum mechanics lectures during my university education. I had failed twice and I am quite proud of it since this is also a record in my department!

One of the approaches to QM is statistical which I find it hard to accept; could be due to the fact that somewhere in my head, I am inherently deterministic! :)

The way I have understood wave function is as follows and I will appreciate any comments on it.

I prefer a more physical approach (cannot rule out uncertainty completely though!). From wave-particle duality, we have lambda = h / momentum; so the localized particle can be approximated as a wave which is spread out in space! This wave is termed as deBroglie wave.

Now, as in the case of EM wave, the quantity which varies in space and time is the electrical and magnetic field; for acoustic waves it is the pressure; and for water waves it is the height of the water surface. Similarly, in the case of deBroglie wave it is the wave function which varies in space and time. And the square of this amplitude gives the intensity, the probability of finding the particle at a  given space and time.

To Vikash,

The word "wave" is used everywhere, and quite often not in an appropriate manner. Wave comes from the "wave equation", which is a family of partial differential equations whose solution is of the type f(t-x.v) that is the function takes the same values at points which are located with the same phase (t-x.v).  It happens that some physical phenomena are represented by such equations, in particular Maxwell equations (in special relativity) are "wave equations". But the Schrodinger equation of QM is not a wave equation and its solutions have totally different properties. So actually the concept of wave function of QM has nothing to do with a "wave", and the name is misleading.

@ Jean

Glad to know that. I had never heard of that; for me wave is more of a physical term. Could you please mention any relevant literature where I can find the whole class of wave equations?

Dear Professor Leyvraz,

My questions makes no assumptions, I merely ask if there are any research people looking at a particle (in particular photon) description of QM. I have many books on Quantum theory, simple to expert, old to modern.  

Jean makes some good points about the term "waves". In the context of this question, it is just a simplification to shorten explanations.

Jean Claude, I really don't think that the Schrödinger equation isn't a wave equation, especially as it precisely has wave solutions of the type you gave (I would rather write them f(kx-wt)). Anyway, the Dirac equation is definitively a wave equation, and is very similar to the Maxwell equations.  The real difference between a matter wave and a quantal wave is that the latter has a phase velocity which is greater than light speed. It can't in any way be compared to a wave propagating in a medium, since it has not its local property.

``In the context of this question, it  (the term ``waves'') is just a simplification to shorten explanations.''

OK. My point then is an emphasis that the word wave, with its connotation of classical waves, as observed, for example, in sound waves, does not at all correspond to the way in which quantum mechanics describes systems. I was under the impression that your description of ``non-wave'' theories, implied that you thought of ordinary quantum mechanics as a ``wave'' theory, something I believe to be profoundly misleading whenever you go beyond one-particle ssytems.

@ Vikash: ``Similarly, in the case of de Broglie wave it is the wave function which varies in space and time.''

The deep problem that is implicit in this remark is the following: you assume that the wave function varies *in space* (and time, but that I agree with).

This is simply incorrect. It is true enough for one particle, but for two particles 1 and 2, the wave function depends, not on space, but on *both* coordinates x1 and x2. So it does not vary in space, it varies as a function of the *configurations* of the two-particle system. That is the essential difference between physical wave phenomena and quantum mechanics. The pressure field does indeed depend on position (and time) alone. For a two-particle system, no such description exists.

There is one thing I do not understand in the question, so please bear with me. The question states  Quote...the interference patterns are generated by a very large number of individual molecular collisions; wave theory is just an easy way to describe the observations Unquote.

But in a slit experiment setup an interference pattern can be generated by a single particle. Not sure what the quoted sentence means? 

Dear Chris,

The quote describes what happens in sound waves.

For the Photon case, there have been no experiments where a single photon generates an interference pattern. What has been observed is that an interference pattern is generated when only one photon at a time passes through the slit(s). Many photons are needed to generate the interference pattern.

Theory says that the wave function passes through both slits and collapses at one place with a given probability.

@ Leyvraz,

I am sorry to say but I felt your statement

"it varies as a function of the *configurations* of the two-particle system."

is not convincing enough! The simple explanation and the essence I gave in terms of the deBroglie matter waves still holds. Lets assume for just one particle case. Do you have any reservations against the idea I mentioned above?

@ Paul Howard Riley : When you write : "Theory says that the wave function passes through both slits and collapses at one place with a given probability", you assume that the only theory under the sun should be the Göttingen-København one. In transactionnal physics, there is no "collapse" of any wide "wave function". Real waves always remain thin and stiff from the emitter to the absorber. Wavelength and intrinsic frequencies are real things, as we assume the relativistic frame.

What is definetely out of human-held determinism is the permanent and unshieldible noise of Dirac-de-Broglie waves before a transaction succeds, and after its end ; counted in the laboratory time.

@ Vikash: For one particle I have no problems. But mechanics can never be purely one particle. In particular, measurement will always imply a *large* number of particles. So a one-particle wave analogy will never explain what you aactually *see*, that is, measure. And since QM is a theory that can and does explain many-particle systems, a wave theory can never be quantum mechanics.

I should insist that this is entirely standard. The problems caused by the ``wave-particle'' duality correspond to a stage of quantum theory that was historically important, but that is not usually considered in modern treatments.Hence my recommendation to read a modern textbook: QM is not something that was created by the founding fathers without undergoing any change.

Finally, I do not know how I should convince you that a wave that ``propagates'' in a space that has six dimensions (for two particles propagating in both in three dimensions) is not an ordinary wave. All other waves propagate in ordinary space. Did you not  think that ``probability waves'' did so as well? It appeared that way from your remarks This is a common misunderstanding, but it is definitely not true.

Historically, it is true that Schroedinger looked for ways of having such a description, but he gave it up, precisely because of the difficulties with several particles. Certainly standard Quantum mechanics does not identify the state function with a wave, bu rather uses it to compute the probabilities of different outcomes of possible measurements, something which can be done unambiguously, and gives results in agreement with experiment

An afterthought: among the best studied systems, for which QM gives incredibly accurate results, is the helium atom. This is, of course, a typical instance of a many-body (in effect two-body, since the center-of mass can be removed) system. It is also a case in which we cannot use waves to describe the state, since the ``waves'' are in six dimensions in the best of cases.

@ Leyvraz,

I admire your replies!

Before I put forward my understanding, may i ask which approach do you find the best to study quantum mechanics; more accurate and also comprehensible: mathematical, statistical (dominant), physical or anything else?

@ Vikash: Thank you! My belief is that the best approach consists in working out several example exactly in a mathematically rigorous way, and only then, once you have acquired a ``feel'' for the behaviour of quantum systems, ask yourself what it means.

As a comparison, could you explain to someone in simple terms how a top in classical mechanics behaves? It can be done, but only after having done a lot of explicit computations.

My attitude to de Broglie waves in general is quite negative. The point is, they allow for an easy visualisation of the quantisation conditions in ``old'' quantum mechanics, in particular for the oscillator and the hydrogen atom. But that visualisation is wrong in an essential manner and does not correspond to real QM. So one should start QM with simple quantum systems, such as a spin 1/2 particle. One should not try to teell the student: ``the de Broglie wave is an approximate description and QM just generalises that''. Rather say: QM is about systems like the spin 1/2 particle, and many particle systems are similar to a system of 2 spin 1/2 particles. This is, I believe, partly Feynman's approach.

 @F. Leyvraz. The waves or the calculus of the waves under certain hypothesis ?

@ Leyvraz,

Do you believe that the notion of deBroglie wave was mandatory to develop the idea of "Wave function" in particular, as I understand other initial features of QM say, Uncertainty principle and blackbody radiation have more of a statistical origin or at least can be developed from there.

If I have to teach Schrodinger equation to students, I cannot simply throw that equation on the blackboard. I guess, thats why many students find it hard to pass QM exam, if not grasp it!

@ Vikash: I agree you should not start with Schroedinger's equation. Again, look at Feynman's approach. It starts out by making the student familiar with the different aspects of QM, without trying to do the impossible, viz justify QM in classical terms. Once some familiarity is gained, you can do calculations with Schroedinger equation, and finally, via a semiclassical analysis, get back to classical mechanics, using Ehrenfest's theorem, possibly decoherence etc...

But I do not think you can hope for an honest understanding of the classical limit at the undergraduate level.

@ Jacques: I have no idea to what part of my replies you refer. I have reiterated that the word ``wave''  is, in my opinion, quite inappropriate to describe quantum mechanical states. That, of course, refers both to the physical concept of waves and to ``the calculus of waves''. But if you are  mathematically inclined, you might just conceivably view the Schroedinger equation as a ``calculus of waves''.

 @F. Leyvraz. I have no more to be a devotee, as I am retired, and now free of my writing. I am not eager to "describe quantum mechanical states", but to describe reality. Maybe these two goals are not fully compatible.

As you surely know, when the interaction is electromagnetic, the wavelength to consider is no more the broglian one, but the Dirac-Schrödinger's : h/(2p).

Proved in the Compton scattering and in the experiment of Gouanère & al.

The difficulty with the Schrödinger equation is that, contrary to a usual wave equation, it is of first order in time, and complex. That is often presented as a particularity of quantum mechanics, followed by a rant about the Hamiltonan as an evolution operator whose eigenvalues are the energy levels blah blah blah. Actually, both features are linked, the function is complex because, as it is in first order in time, the intial value of the time derivative must be coded in the function itself. That's exactly the same thing as when a differential equation of second order is replaced by two of first order using an auxiliary variable, and analogous to the Hamilton equations of first order for the two variables p and q, or p+iq. The Schrödinger equation has this particular form because it is a non relativistic equation, not because it is quantum mechanical. It is the approximation of the Klein-Gordon equation for a scalar, which is real and of second order in time and space. The Dirac equation is a step further, and has several component for the same reason. The Maxwell equations are multi component and of first order too, but they don't trigger such a fuss.

What "states of knowledge" have to do with reality and eventual description of reality ?

Surreptiously, this "knowledge" begins by postulating that the things to know are corpuscles, and that the evolution in time respects the newtonian postulates.

Charles, all physical waves evolve unitarily in time, because of energy conservation. Mathematically and physically, there is no difference, apart from the phase velocity, and above all nonlocality that is not coded into the equation, but requires an additional postulate. Only because of the projection postulate does the propabilistic interpretation appear, in which the probability density is proportional to the energy density. Quantum mechanics can't be contained in a wave equation or in a Hilbert space alone. Unitarity is just a consequence of symmetry, and underlies the conservation laws, including the one of probability. Finally, the reason of the form of the Schrödinger equation is that is must be the nonrelativistic approximation of a relativistically invariant equation.

As for the first order nature of the Schroedinger equation, I would simply have said it follows from the requirement of determinism. We want a state to determine, by its initial value, the ulterior evolution of the state. This is only possible if the equation is first order.

F. Leyvraz, that is also true for a second order equation, if both the function and its time derivative are given initially. 

 @Charles Francis.

You have not explicited the physical properties of the so-called "observer". We are big animals, with slow senses, slow perceptions. We live extremely far from the involved intrinsic frequencies, that play the major roles : 1020 Hz minimum, and no maximum. No ceiling : think to neutrons stars.

So the theories these big animals can elaborate may be extremely odd. More odd : some are big-endians and other may be small-endians, depending on the locally reigning prince.

The photon or particle description can be used even for the basic two slit

interference phenomenon.

In the double slit, let x1 be the shorts distance to the screen, and x2 the long distance to the screen, at the same screen point.

The condition for constructive interference is x2-x1 = n lambda. In general we allow n

to be positive or negative integer. lambda is the wavelength of light.

Let us multiply both sides by the linear photon momentum hf /c. The result for the left hand side, is the closed line integral of pdx. , for the triangle x1,x2, d  , where d is the separation of the two slits.

The result for the right hand side is simply nh.

Thus the requirement of stable orbits, normally used for particles in the old quantum theory, gives the condition for constructive interference in the two slit Thomas Young experiment.

In the sonic case simply use acoustic phonons instead of photons.

@Juan Weisz. You just do not have the physics that could yield your wizardry.

Without any magical transmutation of light into corpuscles, nor any magical transmutation of electronic waves into corpuscles, Augustin Fresnel gave the physics of the interferences in 1821. It is still valid. Planck in 1900 and Einstein in 1905 proved that any individual wave has a beginning, and an end when a quantum h is transferred. Because from a stationnary state to another stationnary state, the emitter can only emit one quantum of cycling h (very rarely two), and the same for the absorber : from a stationnary state to another stationnary state it can only absorb one quantum of cycling h.

jACQUES,

As usual I cannot understand you. There is of course a purely wave explanation for interference.

From one stationary state to the next in energy, you emit or absorb a quantum of energy hf, where f is the frequency. This is one mechanism that underlies the emmission or absortion of  electromagnetic energy from solids, and the key to Black Body Radiation. You mignt want to go through the Black Body radiation and the photoelectric effect.

@ Claude Massé: ``that is also true for a second order equation, if both the function and its time derivative are given initially.''

Agreed, of course. But if you argue that the *state* itself, in other words psi, fully characterises the system, or yet in other words, that psi is the quantum mechanical equivalent of a *phase space point* (p, q), then psi needs to satisfy a first order differential equation. I am not clear as to what the state is, in the case of the Klein-Gordon equation.

@ Charles:`` The description of the "deterministic" evolution of a probability is not determinism.''

Here we disagree: knowing psi(0) I can calculate psi(t). Since I can (very much ``in principle'') measure an arbitrary self adjoint operator, I can choose, at any time, to measure the projector on psi(t). If this is well done, it should constantly yield one, and thus no randomness of any kind arises.

Another, somewhat less kooky, way to argue for quantum determinism, goes as follows: again, knowing psi(0) you predict psi(t). Then you can make a set of measurements which allow to determine psi(t), such as tomographic measurements. So if you have a large number of identically prepared systems, you will see that at each time t, the measured system provides the statistics uniquely characteristic of the state psi(t).

In any case, the state contains enough information to predict the future state and retrodict the past. I do not think it can be reduced to the probabilistic predictions it makes.

@Juan Weisz. When you rush so, "No no no ! No frequency ! Only energy !", you miss all the professional background accumulated in a century by radioelectricians, electronicians, spectroscopists, opticians and radiocrystallographists. You miss all the knowledge on antennas, resonators, coupling of resonators, cross-sections of resonating atoms or molecules, width of Laue or Debye-Scherrer reflex, so on.

The sect has exacted you with a huge ignorance. Exactly as does any endonoxious sect, like the Geotruc of the WatchTower...

 Worse. As phonons are sampled on dozens, more likely many thousands of atoms, there is no way to transmute them into something small, corpuscular. And in the metals, the conduction electrons rebounce on phonons. There is no way to transmute a conduction electron into something small, smaller than dozens of atoms. At definite Fermi level, they have definite Fermi speed, definite Fermi wavelength, and definite phase speed.

Lots of necessary knowledges can't cross the distance between two lecture rooms on the same campus.

@ Charles: Why can I not choose, at time t, to measure the projector on the calculated psi(t)? This then, if my computations are correct, should always give one, no probabilities involved. Of course, arguably, you can only measure certain observables, not all self-adjoint operators. But, in any case, there are other ways to confirm the computed psi(t).

As to understanding psi(t) as a probability distribution, the problem is, of course, with the fact that the different distributions it generates, say the momentum and position distributions, cannot be generated by joint distributions, which is, at least, strange.

The main word here is a “photon”. It was the photon which is responsible for the wave nature of quantum theory. We are compelled to explore the world around us through photons and this is the only possible way to make experiment. But we are able to create a theory in which all interactions could be described without wave functions.

Charles,

"you can no more choose a particular outcome of an experiment than you can choose which number you are going to throw on a die"

This is too simplistic a statement. We already know that if the experiment is set up to look for a particle (in the twin slit experiment) then there is no interference, i.e. it looks as if particles are being projected. Whereas if the distances and slit widths are optimum then interference patterns are produced. I think this is what  Leyvraz may be alluding to.

By changing the apparatus the experimenter can change the outcome.

Charles,

I can see both sides of the argument here.

Just to play devils advocate.

What changes would you allow for a different answer? If your argument is that one has to use the same apparatus, over the same time period, then it is obvious you will get the same answer. So one has to change something to get a different result. What "something" would you allow?

When the complex plane becomes explicitly defined plane in 3D, the quantum states, at least in the case of qubit states, get clear geometrically meaningful lifts in the 3D geometric algebra as operators acting on observables, also elements of that algebra. The quantum mechanically associated items like eigenvalues, eigenstates, self-adjoint operators ,etc., not possessing roots in the world of real physical observables, states, measurements, go away. Everything becomes as a stage of spinnable objects in 3D being transformed by operators (states). No non-physical "imaginary unit" in exponents, often considered as indicator of wave behavior - the unit becomes bivector in 3D, particularly defined by the Hamiltonian lift to 3D geometric algebra. Some details can be found on my website soiguine.com, the page "Technology", link "Lecture".

Well, it is easy to doubt the wave theory of electromagnetic waves, as soon as

there is absortion or emision from matter (especially the photoelectric effect)

Historically Newtons first thoughs about light was particles.

``I don't know what you mean, Paul. The fact is that you get a probabilistic result, given initial conditions, is fundamental.''

Suppose you start in an eigenstate of a system. After some time you make a measurement corresponding to the question? Are we still in this eigenstate? Teh answer wil laways be ``yes'' No probabilitites.

The same thing can be done, trivially, if you start with an arbitrary state, the dynamics specified by an arbitrary Hamiltonian. The measurement you must make at time t consists in asking the system whether it is indeed in state psi(t).

QM is really deterministic, until and unless you start asking the ``wrong'' questions.

 @John Robert Arrington. What do you do of polarized light, nicols, Kerr effect, Faraday effect, magnetic circular dichroism, quarter-wave plates, absorption spectrography, radiocrystallography ? mmmh ?

Dear Dan S Correnti, Thank you for your valuable comment and your paper on hadronozation of scattered particles..

You write: "however, the photon needs to be understood first.". I totally agree with your statement. I guess the most advanced theory of photon was suggested in last papers of Einstein written from 1945 to 1955. Actually Einstein has obtained the non-linear Maxwell equations from the geometry. I am sure we should go this way. Of course there is a lot of work, but it is a good news.

@Jacques Lavau.  I am in favor of all of them.

There are at least two problems about thinking of photons as little particles; the first is

that their number is rather indefinite, dependent on temperature for example, they are bosons, created out of nothing.

A second is that a

constrained packet of something has to be described through

fourier analysis, as having a wide range of frequencies instead of just one.

Ideas about the second question?

A very recent article is related to the subject of this thread.  The conclusion of such 2016 HEP article by the PHENIX Collaboration at RHIC called  “Recent PHENIX Results on Hard Probes and Direct Photon Production”  states:   “Theoretical models need to assume very early emission of direct photons from the interacting system [to] describe large excessive yields of direct photons. Large values of flow coefficients point to late emission of direct photons when temperature is lower but collective flow has time to develop. These two observations are difficult to reconcile within the same models and there is still no satisfactory description yet available.”

The above issue is usually referred to as the “direct photon flow puzzle” in the QCD/HEP fields.  Mathematical models that describe the resulting dynamics of  p+p,  p+A, or  A+A high-energy collisions shouldn’t be derivable without having a physical understanding of how subatomic particles are constructed and how their internal dynamics work.

Any physical model of the photon (thus, hadron) is incomplete until it can be integrated with physical models of subatomic particles since all of the many types of photons are emitted and absorbed by these particles.  Therefore,  photons must be the constituents of the dynamic structure of each subatomic particle and they must also be consistent with and generate the particles’ properties,  which include energy, mass, gravitational potential, a cylindrical B-field that is mobilized while a particle translates, magnetic dipole field, Coulomb’s electric force potential, and Lenz’s law. 

If it is possible to ‘completely’ scatter all the photons making up the pair of projectiles at RHIC,  and to also measure the individual energy of each such scattered photon before such photons re-combine to form hadrons at the PHENIX,   the collaboration should find that the sum of all the scattered photon energies will equal the sum of the energies of the two colliding projectiles.  This would enable an explanation for the “direct photon flow puzzle”.

In addition to integrating the photon model with models of subatomic particles,  the dynamic structures and models of subatomic particles must be consistent with the dynamic structure and model of the atom and its properties,  which include the nuclear force,  stability of the atom under the electric force, and Pauli’s exclusion principle. 

In addition,  the models for the photon, electron, nucleons, and nucleus have to be consistent with experimental results from high-energy particle accelerators, such as the RHIC.  For example,  enhanced photon production,  enhanced collective photon flow,  and  hadronization of multitude of ad hoc particles  occur in such high-energy particle collisions.  Here is a link to a recent study on such phenomena,  which is based on recent experimental results at the RHIC and LHC: 

https://www.researchgate.net/publication/298213589_Hadronization_of_Scattered_Particles_in_High-Energy_Impacts

The link also gives general descriptions of derived models for the photon, subatomic particles, and the atom that were referred to above.  It also examines the inter-relationship between these models and the high-energy study.

As shown in the link,  the dynamic structures of hadrons consist of photon formations.  Such photons are scattered upon collision of hadrons,  then hadronization of the photon plasma occurs instantly giving new mostly unstable ad hoc formations of photons (hadrons),  these hadrons then decay due to their random ad hoc origination,  and from this,  successive instant re-hadronization to other particles from the re-emitted photons occurs,  and/or re-emitted photons do not re-form and leave the interaction area. 

Hadronization of photon plasma is similar to that,  which occurs in laser-particle interactions.  Photons can assemble into particles, such as the electron or hadron, when there is sufficient resonance, density, and mixture of them. For example, environments that have a sufficient resonant and dense mixture of photons are those that immediately follow laser-particle interactions and collisions of particles in particle accelerators. The interactions that enable particle formation result from the electric and magnetic properties of photons, as given by Maxwell’s calculus and EM equations.  See the link for derivation of this process.

Article Hadronization of Scattered Particles in High-Energy Impacts

A lot of confusion comes from wrong interpretation what is "state". When properly formalized as operator its "wave" sense is just a mathematical expression containing exponent. When the recent is generalized to explicit geometrical object, namely element of even geometric algebra in three dimensions, it becomes absolutely clear that such state as wave has nothing to do with the object itself it acts on. Many information can be found in texts available from my account at Research Gate.