# Particle wave duality

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Before, you quoted Dirac on a topic where he was shown to be plain wrong. Now you cite Bohr in another topic where he was wrong. As Steven Weinberg remarks in Physics Today, November 2005, page 31:

All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe.

Our understanding of the world has advanced a lot of since Bohr. In last 100 years, we have developed advanced formulations which are very far from the original work by Bohr. We know today that particles always behave as particles with independence of the existence or absence of an observer.

I what wrote is standard quantum physics, whereas what you write is unrelated.

Our modern picture of the physical world as made of particles is based in QM. This is the standard vision of the physical world and you are who do not still get it (http://public.web.ce...rdModel-en.html):

You're moving the goal posts, you never addressed my well thought arguments.

Now you pick and choose quotes from the critics of Copenhagen Interpretation and argue as though it is a scientific fact.

The philosophy underlying this way of thinking was largely developed by Niels Bohr and others working in Denmark in the 1920's and 1930's and for this reason has become known as the 'Copenhagen interpretation' of quantum physics. It was heavily criticized by Albert Einstein, among others, and is not without its critics today, as we shall see later in this chapter. However, it is still the orthodox interpretation accepted by most working physicists and we shall spend some more time developing it further before explaining what some percieve to be its weakness and discussing alternative approaches.

...... Whether or not we find this satisfactory depends strongly on our own ideas and prejudices. It certainly did not satisfy Einstein, whose reaction was that Bohr's positivism was logically possible, but 'so very contrary to my scientific instinct that I cannot forego my search for a more complete conception'. No such 'complete conception' has yet emerged to command a consensus in the scientific community.

- A beginner's guide to quantum physics, 2007 edition.

Alastair Rae, his Quantum Mechanics, now in its fourth edition, has become a standard undergraduate text on the subject.

Either you're deliberately lying to everyone to desperately hold on to your extreme claims or these physicist's are lying to the students of the world. Let people decide who is doing crackpottery in this thread.

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! Moderator Note pmb, Please stop derailing the thread. If you wish to discuss it, you know what to do.

! Moderator Note pmb,   You were not ignored. Your reports were received and are being discussed. We don't immediately act on all reports for a number of reasons:   1. We're not all here all t

"Wave-particle duality" is often a misleading name. Electrons and photons are quantum mechanical particles. They do not behave in the same way that classical particles do, and indeed can't really be e

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Forgive me, pmb but I thought the wave equation under discussion here was the schrodinger wave equation

$i\hbar \left( {\frac{{\partial \psi }}{{\partial t}}} \right) = - \frac{{{\hbar ^2}}}{{2m}}\left( {\frac{{{\partial ^2}\psi }}{{\partial {z^2}}}} \right) + V\psi$

Which has solution the De Broglie waves given by

$\psi = D{e^{i\left( {kz - \omega t} \right)}}$

Which is periodic in omega.

You construct wave packets mathematically from two waves connected by fourier methods (series or transform)

As for particle in the box and potential wells etc the mathematical solution function does indeed exist beyond the walls, but we just say it has no physical significance, since it coincides with what we want inside the box.

Some figures.

An electron accelerated through 1 volt has a De Broglie wavelength of 1.2 nm. This is ridiculously smaller than the dimensions of the smallest object in a CRT (the mask holes) so the passage is particle like.

There is no such limiting obstruction in lecher lines so the wavelike behaviour can become apparent.

Juanrga was challenged to calculate these several times, not just by myself and failed to do so.

I also think at least some tunneling can be shown to be a characteristic of waves, not available to particulate behaviour.

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Forgive me, pmb but I thought the wave equation under discussion here was the schrodinger wave equation

Yeah. That's true. It was just easier describing waves in general.

$i\hbar \left( {\frac{{\partial \psi }}{{\partial t}}} \right) = - \frac{{{\hbar ^2}}}{{2m}}\left( {\frac{{{\partial ^2}\psi }}{{\partial {z^2}}}} \right) + V\psi$

Which has solution the De Broglie waves given by

$\psi = D{e^{i\left( {kz - \omega t} \right)}}$

Which is periodic in omega.

Note on jargon: The typical way to refer to that is to say that is that its periodic in time. By the way, I only meant to imply that waves aren't periodic by definition. And not all waves in quantum mechanics are periodic.

The expression you just gave is the solution for a free particle with a well defined momentum. It isn't a general solution. There really isn't a geneal solution to Schrodinger's equation since the equation has the potential as a variable in it.

Let's take a look at the situation of an infinite step potential well where V(0) = + infinitity and V(L) = infinity and V(x) = 0 for 0 < x < L . The solution is of the form

$\psi(x) = C_1e^{ikx} + C_1e^{-ikx}$

where C1 and C2 are constants. Putting $\psi(0) = 0$ we get

$\psi(x) = C_1(e^{ikx} - e^{-ikx})$

or

$\psi(x) = A sin kx$

Applying $\psi(L) = 0$ requires sin kL = 0 or $kL = n\pi$

This means that k is restricted to the following values

$k_n = \frac{n\pi}{L}$

The energy eigenvalues are

$E_n = \frac{n^2h^2}{8mL^2}$

The complete solution has then the following eigenfunctions

$\Psi_n(x, t) = A_n sin \frac{n\pi x}{L} e^{-i(E_n/\hbar)t}$ 0 < x < L

$\Psi_n(x, t) = 0$ (x < 0 and x > L)

These are standing waves of course. The most general solution is a sumation of these eigenfunctions. When that is done the result no longer specifies a particle with well defined momentum.

You construct wave packets mathematically from two waves connected by fourier methods (series or transform)

A wave packet is locaized in space. To create one you have to have a superposition of a continuum of wave numbers to form a Fourier integral.

Wave packets aren't always periodic either.

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You're moving the goal posts, you never addressed my well thought arguments.

Now you pick and choose quotes from the critics of Copenhagen Interpretation and argue as though it is a scientific fact.

You are who introduced here Bohr and the Copenhagen Interpretation as justification for your ideas, without being aware that (as SW remarks) "Bohr's version of quantum mechanics was deeply flawed" and that the Copenhagen interpretation "is surely wrong".

Either you're deliberately lying to everyone to desperately hold on to your extreme claims or these physicist's are lying to the students of the world. Let people decide who is doing crackpottery in this thread.

I repeat again, in advanced and modern treatments of quantum physics we do not use wave-particle duality, because it plays absolutely no role in a fundamental description of our physical world.

Quotes from introductory and outdated textbooks do not count as valid argument against advanced and modern sources.

Your quotation from "A beginner's guide to quantum physics" is not an argument about what Steven Weinberg (a Nobel Prize for Physics who is currently considered one of the most important living physicists) wrote in Physics Today.

Your appeals to Dirac and Bohr ancient quotes will not change the definitions and statements made by CERN and IUPAC (two leading academic bodies).

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and that the Copenhagen interpretation "is surely wrong".

That comment flies in the face of mainstream quantum mechanics. The Copenhagen interpretation is one of the postulates of quantum mechanics.

To claim otherwise is personal conjecture and as such it doesn't belong in the ordinary physics forums but belongs in the speculation forum.

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pmb, in your original classical wave equation a is equivalent to my omega a period. The arbitrary functions F and G do not need of themselves to be periodic.

In your later exposition of Schrodinger, n is periodic.

As soon as you introduce some integer that can be 1..2..3..4.... you have introduced periodicity.

And yes standing waves treat it differently from travelling waves. There would be no point in having the distinction otherwise.

Of course a wave packet is localised. That is what the Fourier transform is about.

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pmb, ...

I prefer Pete, even in open forum.

in your original classical wave equation a is equivalent to my omega a period. The arbitrary functions F and G do not need of themselves to be periodic.

You wrote

$\Psi(x, t) = Ae^{i(kx - \omega t)}$

We need to put this in a similar form so we write

$\Psi(x, t) = Ae^{ik[x - (\omega/k)t]}$

Let $v = \omega/k$. This is the phase velocity of the wave. We now have

$\Psi(x, t) = Ae^{ik(x - vt)}$

which is now in the form

$\Psi(x, t) = F(x - vt)$

In your later exposition of Schrodinger, n is periodic.

Numbers aren't periodic. n is an integer which labels the wave-parameter andenergy eigenvalues. The wave function is periodic in the temporl sense but not in the spatial sense. Note that the wave function goes to zero outside the walls of the potential. This is not a spatially perriodic function though.

As soon as you introduce some integer that can be 1..2..3..4.... you have introduced periodicity.

I disagee. Quantizing a variable does not make it periodic whatsoever. Bound particles do not have a spatially periodic wavefunction.

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You are who introduced here Bohr and the Copenhagen Interpretation as justification for your ideas, without being aware that (as SW remarks) "Bohr's version of quantum mechanics was deeply flawed" and that the Copenhagen interpretation "is surely wrong".

Those are not my ideas, that's the accepted scientific attitude in the absence of a well tested interpretation.

Are you aware that SW remaks are subjective opinions and not objective truths?

I repeat again, in advanced and modern treatments of quantum physics we do not use wave-particle duality, because it plays absolutely no role in a fundamental description of our physical world.

Quotes from introductory and outdated textbooks do not count as valid argument against advanced and modern sources.

Your quotation from "A beginner's guide to quantum physics" is not an argument about what Steven Weinberg (a Nobel Prize for Physics who is currently considered one of the most important living physicists) wrote in Physics Today.

Your appeals to Dirac and Bohr ancient quotes will not change the definitions and statements made by CERN and IUPAC (two leading academic bodies).

If my arguments are invalid then why don't you address them rather than quoting someone who is anonymous or the opinions of a Nobel laureate or accept that you were wrong.

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Pete, what do you understand the periodic table to mean?

What do you mean by a periodic structure when calculating Brillouin Zones?

By periodic I mean that if f(a) is a solution then f(2a) is a solutions as is f(na). Or that something repeats regularly along at least one cordinate axis.

Solitary travelling waves can be considered to repeat everywhere along an axis

Solitary stationary waves are the combination of a wave within the stationary envelope and an envelope function.

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Pete, what do you understand the periodic table to mean?

A test? Don't I get a chance to study for it?

The periodic table is a list of the known elements arranged according to their properties

What do you mean by a periodic structure when calculating Brillouin Zones?

Nothing, becase I never talk about the periodic table.

I forgot what those zones are. Its been decade since I've studied it.

By periodic I mean that if f(a) is a solution then f(2a) is a solutions as is f(na). Or that something repeats regularly along at least one cordinate axis.

Your definition does not conform to the standard definition which is as follows: the definition of a periodic function means that f(x) = f(x + L) where L is the period of the function. What you have doesn't define a periodic function. E.g. your function could meet that meaning "f(a) is a solution then f(2a) is" but have arbitrary values inbetween a and 2a and etc. and it'd still be periodic by your account.

Solitary travelling waves can be considered to repeat everywhere along an axis

Solitary stationary waves are the combination of a wave within the stationary envelope and an envelope function.

What do you mean by "solitary"?

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That comment flies in the face of mainstream quantum mechanics. The Copenhagen interpretation is one of the postulates of quantum mechanics.

To claim otherwise is personal conjecture and as such it doesn't belong in the ordinary physics forums but belongs in the speculation forum.

I would disagree with that statement. The Copenhagen interpretation is not needed for quantum mechanics. That is why it is an "interpretation". It is used by us to "philosophically make sense" of quantum mechanical predictions. But the predictions would stand without it.

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I would disagree with that statement. The Copenhagen interpretation is not needed for quantum mechanics. That is why it is an "interpretation". It is used by us to "philosophically make sense" of quantum mechanical predictions. But the predictions would stand without it.

I think what pmb was saying was that it was the most accepted theory to date in the context of quantum field theory.

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Before, you quoted Dirac on a topic where he was shown to be plain wrong. Now you cite Bohr in another topic where he was wrong. As Steven Weinberg remarks in Physics Today, November 2005, page 31:

All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe.

You're such a liar, now I know why you don't give us links when you quote someone.

The complete quote says this and can be found here- Copenhagen Interpreation - wiki

Steven Weinberg in "Einstein's Mistakes", Physics Today, November 2005, page 31, said:

All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from? Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics.
The Copenhagen rules clearly work, so they have to be accepted.
But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus.

No one is arguing that the Copenhagen Interpretation is a complete conception and not without its problems today, there is a real problem and we all know it. In the absence of a well tested theory of the Interpretation of quantum mechanics Copenhagen Interpretation must be accepted.

You're deliberately posting misinformation here and creating havoc and confusion among the well coherent thought of all the scientists in the scientific community.

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I would disagree with that statement. The Copenhagen interpretation is not needed for quantum mechanics. That is why it is an "interpretation". It is used by us to "philosophically make sense" of quantum mechanical predictions. But the predictions would stand without it.

Here's the way I see it. From Principles of Quantum Mechanics - 2nd Ed. by R. Shankar, page 116. The autrhor writes

Postulate III

If the particle is in a state $|\psi>$, a measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $P(\omega) = |<\omega|\psi>|^2$. The state of the system will change from $|\psi>$ to $|\omega>$ as a result of the measurement.

To my thinking this is the Copenhagen interpretation. Without it we'd be without this postulate and not know what the wavefunction means. I.e. we'd have the prediction but we wouldn't know what it meant or what to do with it. We can't make a prediction without this interpretation to tell us what it is we're looking to measure. Taking away the Copenhagen interpretation takes away our understaning of the meaning of the wavefunction. In that sense I disagree with you.

I think what pmb was saying was that it was the most accepted theory to date in the context of quantum field theory.

I was referring to quantum mechanics, not quantum field theory.

No one is arguing that the Copenhagen Interpretation is a complete conception and not without its problems today, there is a real problem and we all know it.

I don't understand. What are you saying that its problems are? There is the concept that the observer making the measurements are a quantum system. Is this what you're referring to?

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I don't understand. What are you saying that its problems are? There is the concept that the observer making the measurements are a quantum system. Is this what you're referring to?

Read this paper - Can wave-particle duality be based on the uncertainty relation? - Stephan Durr and Gerhard Rempe from the Max Planck Institute, Germany.

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Read this paper - Can wave-particle duality be based on the uncertainty relation? - Stephan Durr and Gerhard Rempe from the Max Planck Institute, Germany.

I'm going to bed right now. Tommorow when I wake up I'll go cash my check and buy an ink cartridge for my printer so that I can print that out and read it. At that time I'll get back to you on this.

Severian - Do you believe that there is a wave-particle duality? If so then can you please post a definition of it? Thanks.

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There is no such limiting obstruction in lecher lines so the wavelike behaviour can become apparent.

Juanrga was challenged to calculate these several times, not just by myself and failed to do so.

That is not true. As I said to you several times, the EM wave theory used to describe lecher lines is an approximation derived from the underlying quantum theory of particles (photons).

Those are not my ideas, that's the accepted scientific attitude in the absence of a well tested interpretation.

Are you aware that SW remaks are subjective opinions and not objective truths?

If my arguments are invalid then why don't you address them rather than quoting someone who is anonymous or the opinions of a Nobel laureate or accept that you were wrong.

Your ideas about that universe is not made of particles confronts with overwhelm standard consensus among scientists. I repeat everything in the universe is made of particles:

http://www.particlephysics.ac.uk/explore/atoms-and-particles.html

http://public.web.cern.ch/public/en/science/StandardModel-en.html

http://www.pbs.org/wgbh/nova/teachers/activities/3012_elegant_09.html

...

Steven Weinberg remarks are not "subjective opinions" but scientific remarks that reflect our more modern and advanced understanding of quantum theory.

I already stated why all your arguments are invalid. I have explained in this thread what is an elementary particle, what are its properties, how we experimentally describe them, why particles are never waves, under what limits we can associate a wavefunction to the state of a particle and when we cannot, I have emphasized that in advanced and modern textbooks duality is not even mentioned because it plays no role in QM (I have cited some of those advanced textbooks)...

You have not addressed any of that.

I would disagree with that statement. The Copenhagen interpretation is not needed for quantum mechanics. That is why it is an "interpretation". It is used by us to "philosophically make sense" of quantum mechanical predictions. But the predictions would stand without it.

You and everyone else who has studied a bit of quantum mechanics.

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Your ideas about that universe is not made of particles confronts with overwhelm standard consensus among scientists. I repeat everything in the universe is made of particles

Repeating doesn't answer the question, because that wasn't the question that was asked. Do quantum particles ever exhibit what one would classically call wave-like behavior? Do they ever exhibit classical particle behavior?

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The complete quote says this and can be found here- Copenhagen Interpreation - wiki

Steven Weinberg in "Einstein's Mistakes", Physics Today, November 2005, page 31, said:

All this familiar story is true, but it leaves out an irony. Bohr's version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules that govern everything else in the universe. But these rules are expressed in terms of a wave function (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from? Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wave function, the Schrödinger equation, to observers and their apparatus.

And, contrary to your claims, that quote from Physics Today states that Bohr's version of quantum mechanics was deeply flawed and states that The Copenhagen interpretation is wrong.

Now you confound "The Copenhagen interpretation" with "the probabilistic rules of the Copenhagen interpretation". You are confound the Copenhagen rules and the Copenhagen interpretation.

What will be the next?

Your ideas about that universe is not made of particles confronts with overwhelm standard consensus among scientists. I repeat everything in the universe is made of particles

Repeating doesn't answer the question, because that wasn't the question that was asked. Do quantum particles ever exhibit what one would classically call wave-like behavior? Do they ever exhibit classical particle behavior?

Repeating questions that were answered before, will not change the response.

I have a new question for you. Do you believe that a "quantum particle" (your own words) is a particle? Or do you believe that this is a radical claim?

Edited by juanrga
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Now you confound "The Copenhagen interpretation" with "the probabilistic rules of the Copenhagen interpretation". You are confound the Copenhagen rules and the Copenhagen interpretation.

He's not confused. They refer to exactly the same thing. I.e.

http://en.wikipedia.org/wiki/Copenhagen_interpretation

The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta, entities which fit neither the classical idea of particles nor the classical idea of waves. According to the interpretation, the act of measurement causes the set of probabilities to immediately and randomly assume only one of the possible values. This feature of the mathematics is known as wavefunction collapse. The essential concepts of the interpretation were devised by Niels Bohr, Werner Heisenberg and others in the years 1924–27.

.......

The term 'Copenhagen interpretation' suggests something more than just a spirit, such as some definite set of rules for interpreting the mathematical formalism of quantum mechanics, presumably dating back to the 1920s.

immortal: Please take note that juanrga has never provided nor recognized the definition for the terms that he's using. In this way he keeps his straw argument alive and kicking. I recommend ignoring him unless and until he states the definitions of wave-particle duality and Copenhagen interpretation. Also, did you notice that juanrga never provided a reason why the 'Copenhagen interpretation' is supposed to be wrong?

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Juanrga.

That is not true. As I said to you several times, the EM wave theory used to describe lecher lines is an approximation derived from the underlying quantum theory of particles (photons).

I didn't describe a theory I described a real world practical situation. I can actually observe, measure and display the wavelike behaviour.

The behaviour is incontrovertibly there and real. It is not 'derived from any theory'.

The nodes and antinodes of the wave are not parts of a quantum waveform, but they exist.

The question is like asking 'why do I not see a standing voltage waveform between my house and the power station?'

One is a wave theory that offers this for the real periodic variation in voltage that I observe and measure.

The other is a theory of particles that, as far as I know, should lead to a different solution that of no periodic waveform, rather a constant voltage or constantly decreasing voltage with the only variations due to random motion of those particles.

If you think that taking a container of light (but not zero) weight plastic balls and blowing them down a tube can lead to a periodic distibution of balls I would be glad to read you mathematical derivation.

Edited by studiot
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The author of Quantum Mechanics Eugen Merzbacher hs a page online about the Copenhagen Interpretation. He concludes the artilce by saying

In recent times, the predictions of quantum mechanics, analyzed in terms of the Copenhagen interpretation, have been confirmed experimentally for ever more entangled states. This is the topic of Anton Zeilingerís subsequent talk in this symposium. The play Copenhagen has provided us once more with a memorable opportunity for examining these fundamental issues.

Case closed!

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Repeating questions that were answered before, will not change the response.

You didn't answer it before. You wrote something, but you didn't actually answer the question. I asked a question to which the answer would be a yes or a no.

I have a new question for you. Do you believe that a "quantum particle" (your own words) is a particle? Or do you believe that this is a radical claim?

Unlike what I asked, that's a loaded question. A simple yes or no does not suffice, because there is the nuance that we have classical and quantum particles. I know you know this. "Quantum particle" is also your words. Part of your standard terminology.

The bottom line is that you are arguing a different point than everybody else. That we have labeled everything as a particle does not address the question, and you acknowledge this when you add the disclaimer that particle doesn't mean classical/Newtonian particle or little sphere. Because they don't behave that way, do they? They do other things, like interfere and diffract, which are wave-like phenomena.

IOW, the question being asked is not what label we have attached to electrons, etc., which is the question you've been answering.

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You didn't answer it before. You wrote something, but you didn't actually answer the question. I asked a question to which the answer would be a yes or a no.

Yeah, but I'm sure that you know by now that juan hasn't been giving direct answers to direct questions. More often he simply won't respond at all.

Unlike what I asked, that's a loaded question. A simple yes or no does not suffice, because there is the nuance that we have classical and quantum particles. I know you know this. "Quantum particle" is also your words. Part of your standard terminology.

Perhaps you missed it but in an earlier post I pointed this out to him, i.e. that he's using a straw argument in which the heart of his straw is a different, yet undefined, understanding of the terms we're talking about.

I explained this to juan in the post # 92

http://www.scienceforums.net/topic/67437-particle-wave-duality/page__view__findpost__p__688843

which basically states de broglies hypothesis and the QM postulate whereby the squared magnitude is the probability density. Then I explained to him that de Broglie hypithesis epitomizes the wave-particle duality and explained that an ensemble of identical experiments with sinlge particles. He ignored it of course. He simply won't address counter arguments to his premises.

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Pete,

I said periodicity is fundamental to wave behaviour and have pointed out some uses of the word periodic that do not conform to your 'definition'.

Brillouin zones are periodic in space. The periodic table is periodic in atomic number.

There are indeed other structures which are periodic but are not waves.

However I still maintain all waves are periodic.

Back to waves, have you ever heard of the 'periodic equation' for a wave? You can write these for any wave (as far as I know). Sometimes they can be quite complicated, especially if there is a dispersion term in the original PDF as with Schrodinger.

Incidentally adding a dispersion term can lead to the solitary waves I mentioned earlier with appropriate boundary conditions.

Coulson does a fair job of deriving these periodic equations for many wave types in his book

Waves.

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