Particle-Wave duality and Hamilton-Jacobi equation

[Anonym]

Don't use the Marlon versus everybody else in this thread analogy because it is wrong.

I know that. But you use the provocative stile instead relevant discussion.
I pay you in your own coin.

At least, take the effort of defining what you mean to say ...

And you did not read what I already said:”I am working on that now.”

 

[dextercioby]

An electron's "position" violates the HUP.

I don't think so.

Daniel.

 

[vanesch]

I did not understand your argument. In my discussion with Quantum River I have in mind Ch.10, eqs. 10-3,10-4 and demonstration 10-13 that F2=S,S action (?) in H.Goldstein CM (second edition). 10-3 look like real valued SE. See also discussion in the paragraph 10-8.

I don't have the book handy, but have some souvenirs of that chapter. The point I was simply making, is that the H-J equation is a partial differential equation of a function S(q1,q2,...qn,t). If the system consists of more than 1 point particle, then this S does NOT correspond to any field over real space, which should be a function F(x,y,z,t). It is only in the case of a single point particle that the configuration space is 3-dim, and can be parametrized by x,y, and z. As such, S is not a "wave" or a "field" in real space (the x,y,z space), but only over configuration space (the q1,q2,...qn space).

 

[Anonym]

I don't have the book handy, but have some souvenirs of that chapter. The point I was simply making, is that the H-J equation is a partial differential equation of a function S(q1,q2,...qn,t). If the system consists of more than 1 point particle, then this S does NOT correspond to any field over real space, which should be a function F(x,y,z,t). It is only in the case of a single point particle that the configuration space is 3-dim, and can be parametrized by x,y, and z. As such, S is not a "wave" or a "field" in real space (the x,y,z space), but only over configuration space (the q1,q2,...qn space).

What I have in mind is entire reformulation of the classical physics in terms of functional analysis. Then it will be a functional.

 

[marlon]

I know that. But you use the provocative stile instead relevant discussion.
I pay you in your own coin.



And you did not read what I already said:”I am working on that now.”

This is all very nice but how about talking QM, ok ? Let's go back to the electron's position stuff.

marlon

 

[Anonym]

how about talking QM, ok ? Let's go back to the electron's position stuff.

Sorry, Marlon. It is just personal feature. I am not able to talk when I have no idea what I am talking about. You invite me to discuss the relativistic QM, not it's non-relativistic limit. If and when I will have an answer, I will talk.However, I am optimist. " Raffinert ist der Herr Gott, aber boshaft ist Er nicht"

 

[marlon]

Sorry, Marlon. It is just personal feature. I am not able to talk when I have no idea what I am talking about. You invite me to discuss the relativistic QM, not it's non-relativistic limit. If and when I will have an answer, I will talk.However, I am optimist. " Raffinert ist der Herr Gott, aber boshaft ist Er nicht"

It's not going to be THAT easy. I asked you a specific question in post 38 concerning your claim "that a particle is always accompanied by a wave in QED". Why won't you answer that question.

Also, i explained to you why we cannot be talking about stuff like "a photon's size" because it violates the ONLY existing QM formalism.

marlon

 

[vanesch]

What I have in mind is entire reformulation of the classical physics in terms of functional analysis. Then it will be a functional.

In post #23, I was just making the objection that quantum river seemed to imply that in classical physics, particles are guided by a wave (I took it: a field in real space, of the F(x,y,z,t) kind) which was given by the function S in the H-J equation. This is only possible in the case of a single particle, because only there, S corresponds potentially to a function of the kind S(x,y,z,t) which can be a genuine field in space.
The S in the H-J equation *can* be seen as a "field", but over configuration space, in which the entire world is just one point, and then this field guides the "world point" in configuration space.

 

[Anonym]

The S in the H-J equation *can* be seen as a "field", but over configuration space, in which the entire world is just one point, and then this field guides the "world point" in configuration space.

That sounds similar to what we are doing when we introduce Fock space and second quantization. What do you think?

 

[vanesch]

That sounds similar to what we are doing when we introduce Fock space and second quantization. What do you think?

Of course, that's a basic postulate of quantum theory: superposition ! A quantum state being a superposition of all non-quantum states (points in configuration space), this automatically defines a "field" (wavefunction) over the previous configuration space, where the "field values" are nothing else but the complex weights in the superposition. The configuration space in the non-quantum version becomes the index space of a basis for quantum state space (= hilbert space). That's exactly what happens in the Fock space description: to each individual non-quantum E-M configuration corresponds a basis vector in Fock space - although these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states.

 

[Anonym]

Of course, that's a basic postulate of quantum theory: superposition ! A quantum state being a superposition of all non-quantum states (points in configuration space), this automatically defines a "field" (wavefunction) over the previous configuration space, where the "field values" are nothing else but the complex weights in the superposition. The configuration space in the non-quantum version becomes the index space of a basis for quantum state space (= hilbert space). That's exactly what happens in the Fock space description: to each individual non-quantum E-M configuration corresponds a basis vector in Fock space - although these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states.

First of all: thank you. But what you mean "these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states"?

 

[Anonym]

It's not going to be THAT easy. I asked you a specific question in post 38 concerning your claim "that a particle is always accompanied by a wave in QED". Why won't you answer that question.

Marlon, come on! Renormalization.

 

[Anonym]

If someone talks about the 'size of [insert favorite item here]', then one must always look at how that [insert favorite item here] is defined. If someone thinks there is a size of [insert favorite item here], then I'd like to hear how such a thing is measured, especially when in a standard formalism, we simplify our theory by adopting a point particle. We still see no deviation between that simplification and what we observe YET.

I did mistake above (#29, corrected). Everything consistent with delta x=0. Perhaps,it may be used as definition what point-like object (particle) means in wave mechanics. Bounded fermion is extended object. Free moving fermion/boson is point-like. If so, the gluons are not point-like. Excuse me that I am lazy. What we see in deep-inelastic? Only quarks, right? Why we do not see gluons? May be they are “blurred” over entire volume of the hadron?

 

[marlon]

Marlon, come on! Renormalization.

Sorry, that was post 30...Anyhow, no i don't get it...renormalization ? Besides, i asked for an equation of that wave that accompanies a particle in QED. Why won't you give me that equation ? Answering with renormalization is a bit hollow and basically useless.

marlon

 

[marlon]

If so, the gluons are not point-like. Excuse me that I am lazy. What we see in deep-inelastic? Only quarks, right? Why we do not see gluons? May be they are “blurred” over entire volume of the hadron?

But a gluon is defined based upon the colour quantumnumber. Why are you then talking about the gluon's "position" in a hadron ? Don't you see the flaw in that ? Same goes for what you say about photon size BTW.

marlon

 

[lightarrow]

I'm not sure how this has degenerated into the issue of a discussion in this forum.

Remember, your initial claim made it appear to be that there is a difference between "quantum physics" and "quantum formalism", the latter being the phrase I had used (and is often used in other instances and books). I continue to ask for examples to support your claim. Somehow, you have managed to change this into an issue of what is being discussed on here.

It is in what you write after:
"If someone talks about the 'size of [insert favorite item here]', then one must always look at how that [insert favorite item here] is defined. If someone thinks there is a size of an 'electron', then I'd like to hear how such a thing is measured, especially when in a standard formalism, we simplify our theory by adopting a point particle. We still see no deviation between that simplification and what we observe YET. Thus, anyone claiming that an electron has a "size" will have to redefine all the properties of an electron and how such a quantity should be measured."

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.
As you always point, and I thank you for this, we must always give precise definitions of concepts, especially in quantum physics. Are we well aware (I'm not) of what are the consequences of assuming the electron as point like?
I know this has worked well, up to now, however I don't believe the electron is point like; at the same time, I still don't know if and how would be possible to create a definition for an electron's size.

 

[ZapperZ]

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.

But you seem to have missed the point of that reply from me that you quoted. It is a matter of CONVENIENCE that we select these things to be "point particle" and that so far, this convenience WORK. However, nowhere in the formulation is the property of the width of it is DEFINED!

Secondly, we are now playing with semantics. You are adopting the "x=0 is not the same as x is undefined" concept. While I agree that technically this is correct, it doesn't fit into this scenario. Why? Because if you want to be absolutely technical about it, setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there! Yet, how could it have a mass, charge, and can collide with other particles and make itself known? So herein lies the contradiction.

So by calling an electron to be "point particle", we IGNORE the issue of its width till we know more to be able to address it. Till then, the concept of an electron being "point" particle works in the majority of the situation we deal with. But by doing that, does that mean the width of an electron is well-defined? Nope! Just because I can set the geometry of a cow on Alpha Centauri to be a sphere doesn't mean that it is. That is why you don't see anywhere in the Particle Data book the radius of an electron being set to zero.

But where in here is the example of "quantum physics" different than "quantum formalism"?

Zz.

 

[marlon]

If in theory is adopted a point particle for the electron, then the value of it's size is defined: it is zero. But, at the same time, the electron's size is not defined in QM formalism, so how can we know its value?. It's this that I don't understand.

First of all, a point does not have a size equal to zero. "Nothing" has a size which equals zero. You should have written "the size of a point particle has zero dimensions"

Now, a photon is a point particle in the energy base. Just like in a spatial coordinate base (like the Euclidean frame of reference) where the points denote coordinate values, the points of the energy base denote energy values. A photon is nothing more than such an energy value and thus a point in the energy base. It is with respect to this base that a photon is a point particle with zero dimensional size (which is NOT the same as saying that the size is zero). Again, a photon is NOT a point particle in a spatial base so it is NOT defined as an object with finite spatial boundaries.

marlon

 

[lightarrow]

..the DEFINITION of a photon. A size is a spatial coordinates thing whuile a photon is a point particle in an energy base.

Can you explain me better the exact definition of photon? I have never found it; the only one I have found is "quantum of energy in the electromagnetic field".

 

[marlon]

Can you explain me better the exact definition of photon? I have never found it; the only one I have found is "quantum of energy in the electromagnetic field".

Not a quantum of energy in the electromagnetic field but a quantum of energy ASSOCIATED to the electromagnetic field. The "association" is ofcourse the quantisation of the EM waves. That is all

marlon

 

[Anonym]

Lightarrow:”Are we well aware (I'm not) of what are the consequences of assuming the electron as point like?”

You should not required to assume anything. This is firmly established experimental evidence (as well as all fundamental fermions and bosons).
The consequence is that QT exactly as Classical Physics are elementary particles physics.

Lightarrow:”But, at the same time, the electron's size is not defined in QM formalism”

The size is well defined notion in the standard formulation of QM: dispersion, delta x. There is no room in the QM formalism for other definition. In the non-relativistic QM it is adequate (see for example the calculation of the first Bohr radius for H-atom). The point is whether or not delta x=0 is consistent with the high energy experimental evidence (relativistic QM).

Lightarrow:”however I don't believe the electron is point like”

Physics in general and PF in particular have nothing to do with religion.

ZapperZ:” setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there!”

ZapperZ, come on! You give classical example of the circular argument. In addition, your statement is in contradiction with everything that is known in Classical Physics. And a cow on Alpha Centauri also have nothing to do with that.

 

[Anonym]

Besides, i asked for an equation of that wave that accompanies a particle in QED. Why won't you give me that equation ? Answering with renormalization is a bit hollow and basically useless.

I do not know what you have in mind. To avoid any misinterpretation, if you mean physics/0504008, I have no doubt that the eqs. are adequate. In order to convince yourself you just substitute the presented solutions into the suggested eqs. It is a matter of few minutes. In addition, I can’t imagine more simple form of the solutions. Obviously, the presented set is the complete orthonormal set of the positive energy solutions. No sea and Zitterbewegung are required, they are the artificial consequences of the Dirac formulation. However, I do not understand the content of the obtained result. I hope that if I will find the basis in the real Hilbert space, I will understand better what I am doing.

 

[ZapperZ]

ZapperZ:” setting something to have a width of zero means that that something isn't there! We're not talking infinitesimal here, we're talking about ZERO. So that object isn't there!”

ZapperZ, come on! You give classical example of the circular argument. In addition, your statement is in contradiction with everything that is known in Classical Physics. And a cow on Alpha Centauri also have nothing to do with that.

You still don't know how to use the quote function?

The argument WAS intentionally circular to illustrate the absurdity in stating that a zero width is a valid and KNOWN size of an electron! You just stated MY point!

Zz.

 

[vanesch]

First of all: thank you. But what you mean "these are not the usual basis vectors in Fock space, which is usually spanned with the eigenvectors of the (E,P) observables, and not the EM configuration states"?

For fields, this is a kind of tricky issue. Compare it to the NR single-particle quantum system. The configuration space of a featureless point particle is simply R^3 (generalized coordinates x,y,z). The hilbert space of this single-particle quantum system is then, by definition, spanned by basis vectors which are indexed upon the configuration space: each |x,y,z> is a (generalized) basis vector spanning the Hilbert space. But the "fock space basis" equivalent of this Hilbert space is spanned by the common eigenstates of the {E,Px,Py,Pz} set of operators, which turn out to be non-degenerate in this case and correspond to the |E,px,py,pz> vectors, with E = 1/(2m) (px^2 + py^2 + pz^2).
Of course, in this example, they are simply related by a Fourier transform.

For the EM case it is more tricky. The Fock space basis is again given by the eigenvectors of {E,Px,Py,Pz}, but this time, there is a lot of degeneracy, and it turns out that a tidy way of writing down all these eigenvectors is by using finite n-tuples of natural numbers, which correspond to "photon states" with n different modes. The relationship to a classical EM *configuration* is not so evident, and I'm kinda struggling myself to get a clear view on this. Mind you that a classical *mode* is not a configuration, but an entire solution to the classical EM field equation ; one should picture a classical configuration rather as a totally arbitrary vector potential distribution *at a single instant*. (if I understand this well, one shouldn't even take A but a related quantity). Note that a configuration doesn't determine the energy (we also need the derivatives of the potential for that) for instance. To each such configuration must correspond then also a basis vector, which should be able to be expressed as a function of the Fock elements.

This is (partially) treated in Mandel and Wolf, section 10.4 - but I have to say that this is some time ago that I studied this, and I remember vaguely not having understood everything clearly myself. But the idea goes more or less as follows: to each classical EM mode is assigned a harmonic oscillator, which has a single q and a single p variable assigned to it. The independent quantization of each of these harmonic oscillators corresponds to the canonical quantization procedure for the EM field and leads to the Fock basis. But the set of all these q variables of all these harmonic oscillators is nothing else but a set of generalized coordinates of the configuration space of the EM field. We can write the inproduct between a specific Fock basis state and a configuration eigenstate, using the position representation for the energy eigenstates of harmonic oscillators.

 

[Anonym]

You still don't know how to use the quote function?

I already learned “the quote” but still have problems with “the multi quote”. Any way I prepare post outside the PF and should keep quote text in front of my eyes. Then copy-paste do a job in one step. In addition, “each one has his small shortcomings”

The argument WAS intentionally circular to illustrate the absurdity in stating that a zero width is a valid and KNOWN size of an electron! You just stated MY point!

You may check that I consistently say that argue with the experimentalist is waste of time.

 

[Anonym]

I'm kinda struggling myself to get a clear view on this.

Thank you. I have Mandel and Wolf. Please give me ref. to your papers and others that you consider relevant.

 

[ZapperZ]

You may check that I consistently say that argue with the experimentalist is waste of time.

Well, that's funny. I know of a theorist here in our division that said the same thing about OTHER theorists.

Besides, if you truly believe that's the case, when why did you engage in an "argument" with me in the first place?

Zz.

 

[vanesch]

Thank you. I have Mandel and Wolf. Please give me ref. to your papers and others that you consider relevant.

Well, as I said, it is worked out in Mandel and Wolf, section 10.4: the so-called q-basis is close to what I called the "configuration basis".

(and BTW, I'm also an experimentalist )

 

[Anonym]

Well, that's funny. I know of a theorist here in our division that said the same thing about OTHER theorists.

“Each one has his small shortcomings”.

Besides, if you truly believe that's the case, when why did you engage in an "argument" with me in the first place?.

My intention was to contribute something that maintain “duality” or “complimentarity”. All that remind me old Jewish story. Three man came to Rabbi, two of them in sharp disagreement with each other. The first presented his point of view. Rabbi said:”you are right.” The second presented his point of view. Rabbi said:”you are right”. The third said :”Rabbi, it is impossible”. Rabbi said:”you are also right” (human realization of the linear superposition).

You said:This is a point, but I know it down to that bound.
I said: This is a point, but I do not know what internal angular momentum of the point means.
Marlon said: This is not a point, since the dispersion of dimensionality not =0.
And Vanesch said: you have some souvenirs of that chapter.

 

[ZapperZ]

You said:This is a point, but I know it down to that bound.

Really? All of the stuff where I said that the width of an electron really is not defined you just ignored?

Then my version of the story is:

You said "I only read what I care to".

Zz.