# I Wave particle duality and phonons

1. Oct 4, 2016

### Collin237

I was reading the thread about wave particle duality linked from the newsletter, and I noticed it said (to use my own words) that the conflict between wave and particle dynamics can be avoided by using operator dynamics instead. Unfortunately, in the case of phonons, I've never seen a description that takes advantage of this. It always sounds to me like essays about phonons start out by talking about atoms vibrating -- apparently using the word "vibrating" in the classical sense -- and the waves of vibration are metaphorically particles, and then presto the metaphor becomes an equation.

What would make more sense to me is:
1. What are the ladder operators of a phonon, in terms of those of the electrons and nuclei in the material? (That is, actually using operator algebra, not just handwaving.)
2. Does the canonical relation between the two ladder operators follow from algebra, or is it inferred from experimental evidence? (or both?)
3. What puts these operators on a par with those of actual particles in terms of physical existence, rather than just pulled out of a hat?

2. Oct 5, 2016

### drvrm

recently i saw a 'course material' on the area of your interest in mit education site
i think following reference is useful to get at the formalism of ladder operators for 'vibrating' one dim. chain of oscillators...i will try to search and provide you proper references..

3. Oct 5, 2016

### Simon Bridge

The question seems to be confused, or conflating models, or making an equivocation... can't tell. Consider:
1. how would deriving operators for phonons from the operators of atomic electrons or nucleons not a case of deferring the handwaving to another situation? how is deriving operators from "the algebra" not a result of whatever handwaving made the algebra in the first place? (Why would you expect the ladder operators for phonons in a material to map to ladder operators for electrons and nucleons anyway? Does this make sense in terms of what phonons are?)
2. see: "scientific method" and "philosophy of science";
3. nobody is suggesting operators are real in the same way that real particles are real.

Since phonons are an emergent phenomena that only loosely map to the idea of a quantum mechanical particle, it may be that the usual operator models need to be modified to handle them. You will need to map out the rules for states and excitations for example. But how is a phonon an example of a "particle" in the sense of "wave particle duality"?

However - it is probably possible to produce an operator formalism for phonons ... it is just unclear if it will meet the conditions you want.

Which thread are you talking about? Any "conflict" between wave and particle dynamics is an artifact of the models that rely on just one or the other way of thinking about Nature. afaik these models are all outdated - though they may be used with care in some situations.
Do you know of any area of the standard model where there is a conflict?

4. Oct 5, 2016

### Collin237

I am confused, and asking for clarification in what I thought was a reasonably precise way.

By grounding the theory or model on fundamentals.

I was under the impression that operator algebra is one of the well-established tools of physics, warranted by its ability to described experimentally observed results, isn't it?

I was under the impression that phonons have been successfully used to describe phenomena actually observed in real materials. I was also under the impression that there is a theory of them in reasonably good standing. Thirdly, I would assume that the physics community has something more substantial than wishy-washy supervenience to show for their work on this subject.

How am I the one being unscientific here?

Nobody is suggesting that they aren't either. I was under the impression that such questions don't apply on physicsforums.com.

It may also be that someone wrote about how to go about such a modification, and that I'd like to read about it. Which I thought was clear from my post.

That's the same question I was trying to ask you.

Yes it is clear that it would, because that's what I asked for.

No. Do you know of any area of it that doesn't use operator algebra?

5. Oct 5, 2016

### Demystifier

I think most of your questions about phonons are answered in "Concepts in Theoretical Physics" by Ben Simons
http://free-ebooks.gr/en/book/concepts-in-theoretical-physics

6. Oct 5, 2016

### Simon Bridge

I was trying to get a clarification on what you are asking ... ie.
... bit in bold needs defining.

... I did not accuse you of being unscientific - I was answering your question.

... oh then the short answer is likely that it is not the kind of particle that is usually considered in the usual wave-particle duality discussions.
But I cannot be sure because I'm not sure of what you are saying.

For instance:
You say you do not know of any area where there is a conflict between wave and particle dynamics in the standard model, and yet here you are saying:
... (my bf) when setting up the context for your questions.
Perhaps I am misreading this here ... what is the problem you are trying to fix/avoid, and how does it apply to phonons?
I think if you want to further clarify this issue, this is where you need to start.

The hardest part of scientific inquiry is the bit where you find a question to ask. This sort of process is not so much a criticism of you as it is an attempt to help you refine your question so the answer will mean something. Everybody goes through it, but maybe I'm being unfair: what is your education level?

... except the conditions are ill defined in post #1. I know you know what you mean.
You seem to be saying that some demonstration of an operator formalism will not be good enough for you, it needs something else as well.
But since you are happy to take the operator algebra of the standard model as basic?

I would agree, especially off your response, that the Ben Simmonds book (link post #5) will help you with phonons - and something like that is probably the best way forward for you.

The trouble is that phonons are not particles in the sense that a real electron or a photon is a particle, they are usually labelled quasi-particles to distinguish them. Similarly we have perfectly respectable pseudo-forces, and virtual particles ... kinda like forces, kinda like particles, but not actually forces or particles.

7. Oct 5, 2016

### vanhees71

The phonon is an excellent example, why wave-particle dualism is nonsense. So first get an idea what a phonon is by looking at the concrete problem described by it: You have a crystal. Start from the classical picture (it's always good to do so to get an intuitive idea what's going on). So you have effectively a many-body system consisting of bound molecules that forms some lattice, which can be described by one of the symmetry classes developed by Schoenfliess in the 19th century, but that's of minor importance at this point. So you have a regular lattice bound together by electromagnetic forces, but even that its electromagnetic forces binding it together is irrelevant at this level. So it's just a bound system of molecules forming a regular lattice at rest. Now consider small perturbations to this "ground state". Say, you are hitting a crystal with a hammer. Then it will start to vibrate, and to lowest order each molecule just makes a harmonic oscillation around its ground-state position. So the crystal as a hole whill vibrate. If your crystal consists of $N$ particles you'll have of order $3N$ modes of vibration.

Now think about the quantum theory of such vibrations. You can consider each mode as an independent harmonic oscillator, and you know from QM 1 how to quantize the harmonic oscillator. Using the annihilation and creation operators for the excitations of the field modes, you'll see that this is pretty much the same mathematics that occurs in the quantum-field theoretical formulation of free bosonic particles: The non-interacting quantum fields have a very similar desription with annihilation and creation operators for each momentum mode. The only difference is that the momenta a continuous for free particles, while you have discrete modes for the vibrating crystal. This mathematical analogy is however of great intuitive and calculational value, because now you can use a kind of "particle picture" analogy to solve problems about the collective vibrational modes of the crystal! Of course there are no particles, but the quantum description of this model looks very similar to it. This is an example for a quasiparticle (invented by Landau in his groundbreaking work on liquid hydrogen). Since the lattice vibrations of a crystal are just sound waves propagating in a crystal these quasiparticles are named "sound particles" or, because Greek sounds a bit more intellectual "phonons".

As you see phonons are not something like a classical particle ("billard ball") but a collective excitation of a crystal. It's only the math underlying the QT description that makes these collective excitations in this approximation look like "particles". So there is no wave-particle duality whatsoever.

The great thing with this particle analogy is however, that now all the techniques learnt to approximately solve quantum field theories (in solid-state/condensed-matter physics it's usually non-relativistic quantum field theory you start with) are now applicable in condensed-matter physics, including Feynman diagrams for the perturbative evaluation of the interacting case. E.g., you can also consider the electrons within the crystal, and you have interactions of the electrons with the lattice vibrations, which can be described by effective interactions between "phonons" and "electrons" in the same way as you describe elementary particles in QFT.

For a very good introduction into the techniques involved, see Landau&Lifshitz vol. 9.

8. Oct 5, 2016

### Collin237

I meant that a phenomenon as well-confirmed as phonons doesn't just transpire by folly. I trust that those who study it have a rigorous development filed away somewhere accessible.

I was saying that I haven't found a more apt view of phonons and I was looking for one.

I was saying that discussions of phonons don't seem to be "up to the league" of the standard model.

Sophomore algebra and calculus, and various tidbits I've picked up from math and physics books. Basically, I'm an eclectic math geek. You might ask me specific questions to test my understanding.

I accept the standard model as warranted by evidence, in the sense of Scientific Realism. But I'm wary of the temptation to just put functions together like Scrabble tiles, beyond the reach of applicability. There's a lot of sloppy math out there -- I need not name the usual suspects! The something extra I'm looking for is a development clear enough to show that the logic is above board.

Last edited: Oct 5, 2016
9. Oct 5, 2016

### vanhees71

There are very many condensed-matter physics books around, among them also some very good ones. As a non-expert of this field, I'm sure I miss some good new ones, but the one we had at the university I consider still excellent:

N. W. Ashcroft, N. D. Mermin, Solid State Physics, Harcourt Inc. 1976

A more modern treatment I also like very much, because it's using QFT and path integrals

A. Altland, B. Simons, Condensed Matter Field Theory, Cambridge University Press, 2010

Another standard text with a broader range of applications is

A. L. Fetter, D. Walecka, Quantum Theory of Many-Particle Systems, McGraw Hill

I don't think that this is all only "sloppy math" ;-)).

10. Oct 5, 2016

### ZapperZ

Staff Emeritus
11. Oct 8, 2016

### Collin237

How low frequency, and how short lasting, can a vibration be that can be approximated like this?