Particle-Wave duality and Hamilton-Jacobi equation

Particle-Wave duality and Hamilton-Jacobi equation

According to Particle-Wave duality, an observer can't describe a natural object just from its particle-nature or wave-nature, because a particle is always accompanied by a wave and vice versa.

This reminded me some interesting aspects of the classical Hamilton-Jacobi Equation. The Hamilton-Jacobi Equation is also the only formalism of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 17th century) of finding an analogy between the propagation of light and the motion of a particle. [1]

Yes, to unite the particle equation and wave equation in one formalism is the original goal of Hamilton when he devised the HJE first. It is apparent from the name/theme from his two original papers [1].

I don't know whether Hamilton really did achieve his original goal even in the case of classical physics. I am interested in the Quantum Mechanics case. Is the Hamilton-Jacobi formalism the natural framework to describe the Particle-Wave duality?

In the one-order linear (or quasi-linear) partial differential equation, the wave of the equation could be buildup by the infinite orbits of particle motion. Could this hold in the two-order partial differential equation? I am curious why Hamilton thought he had achieved his goal, because the Hamiltonian of HJE is obvious not linear (there is the kinetic energy).

[1]:http://en.wikipedia.org/wiki/Hamilton-Jacobi_equation
 
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ZapperZ

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Particle-Wave duality and Hamilton-Jacobi equation

According to Particle-Wave duality, an observer can't describe a natural object just from its particle-nature or wave-nature, because a particle is always accompanied by a wave and vice versa.
Could you show me where exactly in the QM formalism that actually have this
"particle is always accompanied by a wave and vice versa"?

Zz.
 
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The wave/partical paradox is a real paradox that has te be solved.
This "duality" is an inherent property of the BASIS of QM. QM gives us perfect predictions and descriptions of atomic scaled fenomena. Actually, the fact that you were able write your post on a PC is thanks to QM and thus, thanks to the particle wave duality. In light of that, tell me, what exactly needs to be solved ? What is the problem ? What suggests to you that this duality IS a paradox that needs to be solved ?

marlon
 

ZapperZ

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I know. But may I also don't believe it. I have made an argument against it. Please look at that argument and make it wrong. The Duality is not possible in reality. You can't jump from a spread field to a local partical. Try to convince me with arguments. Don't think please that I am stupid.
However, you seem to think physicists are for blindly following something that you think you have falsified.

Also, we don't send our work to "big universities" for it to be well-known. We sent it for publishing in peer-reviewed journals, something you would have known if you have looked at where new things in physics are scrutinized.

There are no "duality" in QM. All particle-like and wave-like observations can be consistently described using one single formulation, not two. There are no paradox here to be solved, at least, not for this particular case.

I would also like to point out to you the PF Guidelines on overly-speculative posting. If your idea has not appeared in peer-reviewed journals, please do not post it here. If you still insist on discussing it, the Independent Research inside the General Physics section is the only place for it, per our rules.

Zz.
 

ZapperZ

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Yes you may think that. I listen also to other phycists who are also making really practical receach.
That's funny. I'm an experimental physicist. I assume that's what you mean by "practical receach<sp>". So how come you don't listen to me?

Fare well I go away here
Sayonara!

Zz.
 
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Particle-Wave duality and Hamilton-Jacobi equation

According to Particle-Wave duality, an observer can't describe a natural object just from its particle-nature or wave-nature, because a particle is always accompanied by a wave and vice versa.
Could you show me where exactly in the QM formalism that actually have this
"particle is always accompanied by a wave and vice versa"?
Considering that I don't want to discuss what "a particle is always accompanied by a wave and vice versa" means, can you show me where exactly Quantum River wrote that QM formalism have that?
Or, discussing about quantum physics = only discussing about QM formalism, for you?
 

ZapperZ

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Considering that I don't want to discuss what "a particle is always accompanied by a wave and vice versa" means, can you show me where exactly Quantum River wrote that QM formalism have that?
Or, discussing about quantum physics = only discussing about QM formalism, for you?
Er.. if "quantum physics" isn't "QM formalism", what is it "quantum physics" then? If you don't know what I am refering to as "QM formalism", open a standard QM text. That is what I am asking for. Where, in a standard QM, is there such a thing as "a particle is always accompanied by a wave and vice versa".

You don't believe that the OP is refering to some exotic QM, do you? Because if you do, then you would have asked what version of this exotica he is refering to.

Zz.
 

Demystifier

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In my opinion, yes!
In fact, the HJ formalism represents the basis for the Bohmian interpretation of QM, in which the wave-particle duality has a clear meaning. (See e.g.
http://xxx.lanl.gov/abs/quant-ph/0609163
especially Secs. II and IV.)
In Bohmian interpretation, the action S doesn't obey the usual Hamilton-Jacobi equation. The part of Quantum Potential is the soul of Bohmian formalism. Actually, I feel somewhat sick about the Quantum Potential. I tried lots of times to remove it and just failed all the times.
 
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Quantum River:” the action S doesn't obey the usual Hamilton-Jacobi equation. Actually, I feel somewhat sick about the Quantum Potential.”

I feel the same.

Quantum River:” The Hamilton-Jacobi Equation is also the only formalism of mechanics in which the motion of a particle can be represented as a wave.”

Classical mechanics is the Wave Mechanics.
 
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Demystifier, thanks!
I think the quantum action correction function is its main result. I must say it is very good.
I have got some of its results (in the paper you provided) too. But there are still lots of fundamental problems here.
The part of Quantum potential is like a ghost. It is always there.
 
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Demystifier

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The quantum potential is, of course, necessary in order to reproduce the standard predictions of QM with particle trajectories.
However, it is possible that there is even more about quantum potential.
For example, in
http://arxiv.org/abs/gr-qc/0611037
it is suggested that quantum potential could explain dark energy. Thus, it could be more than a ghost.
 
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Er.. if "quantum physics" isn't "QM formalism", what is it "quantum physics" then? If you don't know what I am refering to as "QM formalism", open a standard QM text. That is what I am asking for. Where, in a standard QM, is there such a thing as "a particle is always accompanied by a wave and vice versa".

You don't believe that the OP is refering to some exotic QM, do you? Because if you do, then you would have asked what version of this exotica he is refering to.
A lot of questions on this forum cannot be related to QM formalism only. If what you say were true, then tens of threads should be canceled. Furthermore, the title of the Forum should be changed from "quantum physics" to " QM formalism", to make sure people know that asking things which don't belong to that formalism will have the simple answer: "not defined in the present theory".
But this Forum is intended as a "discussion about physics" or something else?
 

ZapperZ

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A lot of questions on this forum cannot be related to QM formalism only. If what you say were true, then tens of threads should be canceled. Furthermore, the title of the Forum should be changed from "quantum physics" to " QM formalism", to make sure people know that asking things which don't belong to that formalism will have the simple answer: "not defined in the present theory".
But this Forum is intended as a "discussion about physics" or something else?
I noticed that you never did address my question. All you gave is a statement with no support.

So which part of "quantum physics" isn't part of what I consider as the "quantum formalism"?

Zz.
 
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Quantum River:” The Hamilton-Jacobi Equation is the only formalism of mechanics in which the motion of a particle can be represented as a wave.

CarlB:”The rules of the game are never specified, but they are known to be so sophisticated that they are difficult to imagine.”

Eventually Quantum River discuss formulation of the complete set of principal physical postulates (Rules of the Game), which is the only possible absolute result in Physics.

Let remember a “history”: A. Einstein presentation on 5-th Solvay Congress. A. Einstein opened the discussion with the statement that he personally did not contribute essentially in the development of Q.M. I know exactly what he meant: A. Einstein did not write the Equations of Motion. But in his presentation he discuss the other problem: Theory of Measurements. It’s history is very similar to the problem of mass: initially it was self-obvious that the inertial and gravitational mass are identical. GR is all about that this is not self-obvious. The measurements in the classical physics is something self-obvious, since all dynamical variables then known were what we call today self-adjoint operators with the continous spectrum. The QM formalism (W. Heisenberg UR) demonstrated that the measurements of the same variables in QM behave differently. The theoretical physics do not exist without measurement theory. Everybody agreed with that, everybody contributed in attempt to solve a problem: A. Einstein, E. Schrödinger, W. Heisenberg, J. von Neumann, E.P.Wigner, F. London, E. Bauer,N.Bohr, M. Born, B.Podolsky, N.Rosen. It is much more than duality, but duality express it in the most clear way (without clear definition what is a particle and/or wave in CM as well as in QM). I do not want to enter in the classifications, but in my view first four principal physical postulates may be regarded as dealt with the measurements. And only the Fifth is the dynamical postulate.

What I have in mind may be considered as an additional step towards realization of a program initiated by E. Schrödinger (perhaps even by W.R. Hamilton) to treat all of the physics as wave mechanics:

The universal mathematical architecture of the physics is given in terms of ten functional - analytical frameworks, suitable to incorporate the results of the physical measurements.

Real, complex, quaternion and octonion states with real scalar product should be equivalent to the theory of classical fields. Unification of electromagnetism with gravitation should occur already in the classical field theory.

Complex, quaternion and octonion states with complex scalar product should allow realization of present unification schemes. Notice, that pure relativistic quantum electrodynamics does not exist, because there are no elementary sources of pure electromagnetic radiation. Neutrino is an elementary source of pure weak radiation.

Quaternion and octonion states with quaternion scalar product should describe wave mechanics of space-time continuum.

Octonion states with octonion scalar product should allow ultimate realization of idea of elementary particles picture of natural phenomena.

Please excuse me for something irrelevant and personal. First time I presented that at the seminar given at Technion, Haifa, 1983, Israel. On the last row in the conference room sat old man, alone. He did not ask any question during my lecture. I did not know who he was. After the lecture I remained to organize my notes. Everybody gone. He remained also. He came to me and said:” Young man, may I invite you for cup of tea?” Sure. We went to his office. Then he said:” I am Natan Rosen. You can not imagine how I sad that A. Einstein had no opportunity to listen what you told us today. He would be very pleased with it.”

Quantum River:” Actually, I feel somewhat sick about the Quantum Potential. I tried lots of times to remove it and just failed all the times.”

I wrote: “Get rid of the Quantum Potential”. I am not sure. This is only my guess. I guess that the solution of that problem will allow to justify the validity of the Principle of Least Action in CM, the physical analog of the Fifth Postulate of the Geometry and thus the Rules of the Game will be known.

But the Game will never be over, it is not possible to finish it, there is no Final Theory and never will be. The Final Theory means death of human civilization. I leave that rubbish dreams to S.Weinberg and S.Hawking.

ZapperZ:” if "quantum physics" isn't "QM formalism", what is it "quantum physics" then? If you don't know what I am refering to as "QM formalism", open a standard QM text.”

I do not understand words. I understand only the translation of the natural phenomena on the language of mathematics.

Daniel Gleekstein.
 
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I noticed that you never did address my question. All you gave is a statement with no support.

So which part of "quantum physics" isn't part of what I consider as the "quantum formalism"?
A photon's position, an electron's size, time as an operator...and many other concepts that can be discussed in a "quantum physics Forum" even if they don't exist in QM formalism.
 
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Quantum River:” The Hamilton-Jacobi Equation is the only formalism of mechanics in which the motion of a particle can be represented as a wave.

CarlB:”The rules of the game are never specified, but they are known to be so sophisticated that they are difficult to imagine.”

Eventually Quantum River discuss formulation of the complete set of principal physical postulates (Rules of the Game), which is the only possible absolute result in Physics.

Let remember a “history”: A. Einstein presentation on 5-th Solvay Congress. A. Einstein opened the discussion with the statement that he personally did not contribute essentially in the development of Q.M. I know exactly what he meant: A. Einstein did not write the Equations of Motion. But in his presentation he discuss the other problem: Theory of Measurements. It’s history is very similar to the problem of mass: initially it was self-obvious that the inertial and gravitational mass are identical. GR is all about that this is not self-obvious. The measurements in the classical physics is something self-obvious, since all dynamical variables then known were what we call today self-adjoint operators with the continous spectrum. The QM formalism (W. Heisenberg UR) demonstrated that the measurements of the same variables in QM behave differently. The theoretical physics do not exist without measurement theory. Everybody agreed with that, everybody contributed in attempt to solve a problem: A. Einstein, E. Schrödinger, W. Heisenberg, J. von Neumann, E.P.Wigner, F. London, E. Bauer,N.Bohr, M. Born, B.Podolsky, N.Rosen. It is much more than duality, but duality express it in the most clear way (without clear definition what is a particle and/or wave in CM as well as in QM). I do not want to enter in the classifications, but in my view first four principal physical postulates may be regarded as dealt with the measurements. And only the Fifth is the dynamical postulate.

What I have in mind may be considered as an additional step towards realization of a program initiated by E. Schrödinger (perhaps even by W.R. Hamilton) to treat all of the physics as wave mechanics:

The universal mathematical architecture of the physics is given in terms of ten functional - analytical frameworks, suitable to incorporate the results of the physical measurements.

Real, complex, quaternion and octonion states with real scalar product should be equivalent to the theory of classical fields. Unification of electromagnetism with gravitation should occur already in the classical field theory.

Complex, quaternion and octonion states with complex scalar product should allow realization of present unification schemes. Notice, that pure relativistic quantum electrodynamics does not exist, because there are no elementary sources of pure electromagnetic radiation. Neutrino is an elementary source of pure weak radiation.

Quaternion and octonion states with quaternion scalar product should describe wave mechanics of space-time continuum.

Octonion states with octonion scalar product should allow ultimate realization of idea of elementary particles picture of natural phenomena.

Please excuse me for something irrelevant and personal. First time I presented that at the seminar given at Technion, Haifa, 1983, Israel. On the last row in the conference room sat old man, alone. He did not ask any question during my lecture. I did not know who he was. After the lecture I remained to organize my notes. Everybody gone. He remained also. He came to me and said:” Young man, may I invite you for cup of tea?” Sure. We went to his office. Then he said:” I am Natan Rosen. You can not imagine how I sad that A. Einstein had no opportunity to listen what you told us today. He would be very pleased with it.”

Quantum River:” Actually, I feel somewhat sick about the Quantum Potential. I tried lots of times to remove it and just failed all the times.”

I wrote: “Get rid of the Quantum Potential”. I am not sure. This is only my guess. I guess that the solution of that problem will allow to justify the validity of the Principle of Least Action in CM, the physical analog of the Fifth Postulate of the Geometry and thus the Rules of the Game will be known.

But the Game will never be over, it is not possible to finish it, there is no Final Theory and never will be. The Final Theory means death of human civilization. I leave that rubbish dreams to S.Weinberg and S.Hawking.

ZapperZ:” if "quantum physics" isn't "QM formalism", what is it "quantum physics" then? If you don't know what I am refering to as "QM formalism", open a standard QM text.”

I do not understand words. I understand only the translation of the natural phenomena on the language of mathematics.

Daniel Gleekstein.
Very, very interesting, Daniel. I sincerely wish you with my heart to accomplish your purpose.
 
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Lightarrow:” I sincerely wish you with my heart to accomplish your purpose.”

Lightarrow:” A photon's position, an electron's size, time as an operator...and many other concepts that can be discussed in a "quantum physics Forum" even if they don't exist in QM formalism.”

The purpose is to invite you to participate in the festival. It also will help you to understand what ZapperZ desperately try to explain to you:” A photon's position, an electron's size, time as an operator...and many other concepts” are inherent part of QM formalism as well as CM formalism (but Quantum world is not a Classical world).
 

ZapperZ

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A photon's position, an electron's size, time as an operator...and many other concepts that can be discussed in a "quantum physics Forum" even if they don't exist in QM formalism.
Er... come again?

In quantum formalism, a "position", "time", etc.. are all observables with definite mathematical definitions. We don't call them operators for nothing. So where are these things not existing in QM formalism?

And as for the "size of a photon", how are you able to "discuss" such a thing when it is not defined? I mean, this isn't just some pedestrian discussion where we can simply use anything we like and not care about what exactly we mean. YOU may have your own definition of it, but how confusing do you think it would be if we ALL use our own definition? Who's to say my definition is less valid than yours? What shall we do? Look at the "standard" definition from QM? What is it then? Does QM have a clear definition of the size of a photon? Please show me where.

Just because you can "discuss" it doesn't mean that it has a well-formulated concept in physics.

So show me where else are there concepts in quantum physics that have no QM formulations.

Zz.
 
Anonym, Thanks for sharing with us your personal experience. It makes me feel a little nearer to Einstein now. :wink:
Could you explain your ten functional-analytical frameworks a little more? For example, could your frameworks circumvent the measurement problem?
 
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vanesch

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This reminded me some interesting aspects of the classical Hamilton-Jacobi Equation. The Hamilton-Jacobi Equation is also the only formalism of mechanics in which the motion of a particle can be represented as a wave.

Let us not forget that the H-J equation defines a scalar "field" over configuration space, which can only be put in bijective relationship with 3-dim space for the single-particle system. As such, the H-J equation is not a genuine field equation in "real space".
 
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For example, could your frameworks circumvent the measurement problem?
I hope you mean solution, not circumvention. Yes, quant-ph/0606121.
Notice that tr(ei)=0 for i=1,2,3 also and for i=1,…,7 as well.
The measurement problem is a problem within classical physics, since the measurement instruments are macroscopic. Additional explanation is presented in post #115 , The wave packet description session, discussion with Reilly. If you need clarification about Hurwitz algebras and Hurwitz theorem, I will give you.
 
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Er... come again?

In quantum formalism, a "position", "time", etc.. are all observables with definite mathematical definitions. We don't call them operators for nothing. So where are these things not existing in QM formalism?

And as for the "size of a photon", how are you able to "discuss" such a thing when it is not defined? I mean, this isn't just some pedestrian discussion where we can simply use anything we like and not care about what exactly we mean. YOU may have your own definition of it, but how confusing do you think it would be if we ALL use our own definition? Who's to say my definition is less valid than yours? What shall we do? Look at the "standard" definition from QM? What is it then? Does QM have a clear definition of the size of a photon? Please show me where.

Just because you can "discuss" it doesn't mean that it has a well-formulated concept in physics.

So show me where else are there concepts in quantum physics that have no QM formulations.

Zz.
I wrote "an electron's size". Is it defined in QM formalism? If not, why on this Forum, people more clever than me sometimes discuss about it, about if an electron is pointlike or not?
 

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