# Wave-Particle Duality (explained?)

1. May 22, 2010

### nshrikhande

Hi All,

I'm nk and new to your forum!
Pl., excuse my editing skills just this post. I'll catch on fast-promise!

It is known that "In physics and chemistry,
wave–particle duality is the concept that all matter exhibits
both wave-like and particle-like properties."-Reference,
http://en.wikipedia.org/wiki/Wave–particle_duality

However, I have not seen a math expression relating both
aspects of EM wave-particles. So, I deviced the following:

Let a photon be discrete represented by an ON-OFF beahaviour.
This is our typical periodic step function or even the Dirac-
delta function expressed periodically. Physically it means that
the photons energy OR sheer existence is periodic in this
fashion.

From the Fourier series (http://en.wikipedia.org/wiki/Fourier_series)
we know also that:

f(x) = A0 + Sum[n:0 to infinity] { An*cos(n*X) + Bn*sin(n*x) }

Erwin Kreyszig, Fourier series, Integrals and Transforms
Chapter 10
, Edition 8th.

where, f(x) (photon) is a peicewise continuous function which is discrete in
it's totality.

Thus LHS = discrete entity and RHS = continuous enity !

LHS = RHS simultaneously, ontologically speaking.

Any opinions on this approach to quantify, with assumptions, the QM problem
of wave-particle duality?

NK.

2. May 23, 2010

### filegraphy

a good example of a wavicle is an electron. It has the properties of a wave and a particle. An experiment that will be amazing to you is the double slit experiment. Sorry I could not help that much, but it is the best I can do.

3. May 23, 2010

### glengarry

There are no particles in quantum mechanics. Only wavefunctions, Hermitian operators, and vectors in state space. No duality, no problem.

4. May 24, 2010

### tom.stoer

Please correct your sentence: "there are no classical particles in the formalism of quantum mechanics".

Of course there are effects in nature (e.g. the famous photoelectric effect) which cannot be explained by continuous wavelike processes. In that specific case it's the phton that must be described as a quantum particle, not as an electromagnetic wave.

Of course we are leaving the realm of quantum mechanics here since photons must be described by quantum electrodynamics. But if we do that we see that the above mentioned states in an Hilbert space are exactly the entities carrying something like particle properties (e.g. momentum, spin).

5. May 24, 2010

### glengarry

Here's the way that I would describe the "intuitionistic" aspects of QM such as you are describing.

As little children, when we learn how to count, we typically do so by way of quantifying "real things" like apples or pennies. But, the purely abstract quantification procedure is the only thing that is meant by the "theory of counting." The apples and pennies are just heuristic devices that allow the actual theory to be learned.

Likewise, all of the talk about phenomena that are experienced that "seem" to have a so-called "localistic" causes are nothing other than convenient metaphors that allows one to learn how to successfully manipulate the symbolic machinery of a theory that describes the necessarily statistical nature of all experimentation.

This is why I advocate renaming quantum mechanics to something like, "the probabilistic theory of experimental measurement" and relocating it from physics departments into mathematics departments. If this happens, then all of the confusion that results from "interpretations" of quantum mechanics will come to an end.

6. May 25, 2010

### tom.stoer

Not really. I agree that on a formal level you are absolutely right. I tend to agree that some difficulties disappear if you restrict yourself to a probabilistic interpretation. And you can try to get rid of a realistic interpretation (whatever this means).

But what you are talking about is only formalism and interpretation. The problem is that if you look at the famous double slit experiment with only one single electron sent though the slits you will observe one single black spot on the screen. This is a strong hint that there is something like a particle, not necessarily in your formalism but in the experimental setup. I think you can't deny in general that you observe particle-like phenomena. That's why it's a blind spot of any interpretation of quantum mechanics if it omits to talk about particles.

7. May 25, 2010

### Dmitry67

What you are describing is stochastic interpretation of QM, which is one of the remaining 3 interpretations: SM, BM and MWI (after the death of CI and TI)

8. May 25, 2010

### tom.stoer

what are SM, BM and MWI, CI and TI?

MWI = many-worlds interpretation?
CI = Copenhagen interpretation?
...

9. May 25, 2010

### unusualname

Last edited: May 25, 2010
10. May 25, 2010

### glengarry

The entire post-Born formal apparatus of QM is based upon the probability that a given result will ensue after an infinite number of trials. There is nothing interpretive about this. The entire significance of "interpretation" as regards QM is the all-too-human desire for certain statements that are classified under the heading of "physics" to have some kind of ontological significance.

It is important to realize that tremendous difference between the term "particle-like", as it is used in verbal discussions about empirical phenomena and the term "particle", as it is used as a purely heuristic aid when learning how the formal machinery of QM works.

In the first case, there is necessarily a definite volume of three-dimensional space whose constitution has somehow been altered as a result of some spatio-temporal chain of events. Our only concept of integral volumes of space is called a "field" (or "continuum"), and the concept of the "wave" is simply that of a modulation of a field.

The term "particle", however, is one and the same with the Newtonian center of mass, which is a [purely non-dimensional] point. Such an object cannot possibly have any material significance in a three-dimensional universe like ours. Points are simply geometric idealizations that allow mathematical formulations to attain trivial (i.e. analytic) solutions.

The discipline of mathematics, however, is continuing to evolve even as theoretical physics seems to remain stagnant. For instance, there is currently tremendous excitement in mathematical circles about Grigori Perelman's recent proof of the Poincare conjecture. This proof deals in the idea of the Ricci flow, which is a way to explicitly understand how n-dimensional manifolds evolve through time.

And once we realize that this kind of mathematics is nothing but a dynamical version of general relativity, we can start to appreciate that there truly is cutting-edge physical thought going on, but that the logic that is used to express it is far more rigorous than those people who we typically call "physicists" are able to handle.

My own ideas about how the universe "really" works are precisely in line with the thought that we must necessarily think of elemental physical bodies in terms of being modifications of a continuous, three-dimensional manifold. There is nothing new about this idea; perhaps the first "official" explication of it was William Kingdon Clifford's 1870 communication to the Cambridge Philosophical Society called, "On the Space-Theory of Matter." And this was, of course, inspired by the investigations of Lobachevsky and Riemann earlier in the century. Of course, leave it to Einstein to come along much later and receive all the credit for having "completely revolutionized" our understanding of how space can be understood to be structured.

Last edited: May 25, 2010
11. May 25, 2010

### tom.stoer

I agree. The answers depend on the question you are asking. You question (and therefore your answers) remains on the formal level. Fine.

But the question of this thread was for an "explanation of wave-particle dualism". You can simply say that questions regarding explanation are forbidden, but this is your choice, not mine.

Btw.: I think we agree that "particle" is a no-no. I prefer quantum object as it should be free of prejudices according to classical physics.That's why I don't like "wave-particle dualism" either.

yes
no - why?

Physical progress was often inspired by ontological reasoning. In rare case the result was ontological progress - simply because philosophy was not able to follow what happens in physics and therefore stagnated - but nevertheless it was physical progress. Look at Newton, Leibniz, Planck, Einstein, Bohr, Heisenberg - they all had their own "philosophy" or "ontology". Maybe these were wrong, ill-defined, etc., but they helped to develop new physical theories. That's why I think that ontological questions are still helpful in science.

In addition I still believe that there is some ontological "reality" behind pure empirism, even if it is different from what we are used to call "reality".

12. May 25, 2010

### glengarry

I feel that it is vital for people to realize that a statistical theory of measurement is nothing other than "statistics," and that the addendum, "of measurement" is simply an unnecessary appendage that adds nothing of value to the main concept in question. That is, what kind of statistical theory could it be that does not deal in some kind of quantification procedure (i.e. "measurement")?

Furthermore, the sense in which I use the phrase, "formal quantum mechanics" is the same sense in which I would use the phrase, "dog-like dogs." QM can only rightly be understood as a rigorous mathematical system, and to say that it has anything whatever to do with intuitive descriptions of reality is equivalent to uttering the phrase, "cat-like dogs."

And the only reason why I say this is so we can get beyond the neverending cycle of interpretating that which is necessarily uninterpretable, and we can start developing new and interesting mathematical models such as I linked to in my previous post.

In other words, I feel that it is necessary for theoretical physicists to finally take the "mathematical leap," and stop spending so much time and energy worrying over what any particular experimental results seem to indicate. This just means that theoreticians should start becoming unapologetic Platonists; that is, getting over the thought that empirical results can somehow "prove" perfectly well established mathematical theorems.

Plato's idea was that there are two different "worlds" that should be understood as existing completely apart from one another. The one world is "ideal." It is the realm of geometric forms, and it is here that the most compelling "thought experiments" of today are being conducted. The other world is that which is immediately sensed, and keeping one's thoughts anchored within it only degrades one's ability to construct mathematical models that genuinely offer hope for the creation of an ontological system of thought that all humans can eventually come to agree with.

13. May 26, 2010

### tom.stoer

Again: I do not question that your perspictive is logically valid, but I question that
a) it is shared by the mainstream of physicists
b) it is the only position which is logically viable
c) not sharing it is problematic regarding scientific progress

For me quantum mechanics and especially mathematics are "only formal" in the sense that there is some "reality" or some "link to reality" which means that the formal system itself is not ontologically identically with reality; this link is still to be discovered. (I know that as soon as you identify reality with its formalization in a mathematical system all "dualities" disappear; but this is not my belief).

The problem why I think you position is problematic is that it forces one to stop about the interpretation of qm. But there is no logical reason why I should just stop here in qm, and why I should stop exactly now. The same idea of stopping regarding a formal system in physics applies to other systems as well and would therefore bring scientific progress to an end. Why should I stop thinking about the interpretation of qm but continue with the quest for an explanation of the special symmetry group U(1)*SU(2)*SU(3)? It's matter of taste. It's your personal focus of your research program.

Last but not least I think your interpretation of Plato's philosophy is not correct. If you read πολιτεία and Άπολογία you will find out that his philosophy is not regarding two worlds "existing completely apart from one another", but that he continuously links the idea with the phenomenon; he tries to understand the phenomenon based on the ideas. Platon's world of phenomena is ontologigally subordinated, but nevertheless present in his philosophy.

Restricting Plato's philosophy to the world of ideas and surpressing the phenomena is like cutting the strings of a marionette and limiting the whole scenario to the puppet player. For certain questions this is a rather useful approach, but it is not the whole approach.

14. May 26, 2010

### nshrikhande

Tom, thanks for your observation. But maybe I did not sufficiently clarify my main points.
What was meant, when it was mentioned that when LHS = RHS, (i.e., discrete photon = continuous wave, in energy/existence) is that both particle and wave are simultaneously existing; mathematically a natural consequence of Fourier series etc. So in the double slit experiment (D.S. Experiment) one shall get a single spot on the screen (discrete) as well as a wave distribution. What I predict is that there must be waves of all "colors" diminishing from the 1st brightest one quite fast. This really needs doing a fine experiment to check out.

Thus "my" reasoning says that in the D.S. Experiment both must manifest.

15. May 26, 2010

### nshrikhande

You have been helpful actually. You say "it has the properties of a wave and a particle." The and is fantastic! Can anyone think on this AND business? Logically and/or Ontologically? This is not mere formalism. It is the sheer physicality of this and aspect that is to be pondered upon.

Thanks much!

16. May 26, 2010

### filegraphy

Thanks. The experiment took me awhile to take in. If you can look up the two slit experiment. It changed my thinking.

17. May 26, 2010

### nshrikhande

I've read on double slit exp mainly from the three Roger Penrose's books and honestly that is not very enlightening. I need data from a modern perspective and time to verify my very theoretical results. Then there is R. Feynman's version of many paths from source to double slit target...I find that hugely artificial. Math is always correct in it's paradigm. But is the underlying reality continuous. If not how do we use differential Calculus? I have pondered much on these topics starting from Zeno's paradox to B. Russell. For example, Cantor's ""resolution" by redefining classes of sets, or was it done by B.R? seems non-physical.

Hey, I'm really digressing.

18. May 26, 2010

### Dmitry67

I dont see any difference between
(particles+sum over histories) and waves.

19. May 26, 2010

### tom.stoer

I understand perfectly what you mean; but unfortunately there is one comment in order: the electron is not both a particle and a wave, neither mathematical nor physically. It is neither a particle nor a wave. By this I mean that thinking about a quantum object as a particle can be shown to lead to contrradictions (classical particles do not interfere); the same applies to waves.

So I prefer to think about a quantum object as neither particle not wave instead of both particle or wave.

20. May 26, 2010

### filegraphy

Yes, I agree with you completely. Would you refer to this object as a wavicle? That is what I refer to an electron as.