# And if everything was a wave?

1. Oct 8, 2009

### eoghan

Hi! A bullet follows a parabolic trajectory when fired in a gravitational field. I've read a text where this trajectory was derived supposing the bullet as a wave. So, can't the classical mechanics be rewritten supposing that everything is a wave? Or, in other words, is there a phenomenon which requires a corpuscular view to be explained?

2. Oct 8, 2009

### Staff: Mentor

Yes, classical mechanics is recovered in the appropriate limit of quantum mechanics.

3. Oct 8, 2009

### DrChinese

Yes there is: electron position can be ascertained with unlimited precision. This would not be possible in a pure wave view.

However, this should not be used as a proof that particles are not waves. The controlling rule is the Heisenberg Uncertainty Principle. An electron can act as a particle, wave, or a combination of the two. (Interference is something that waves do that particles cannot. An electron can be made to demonstrate wave behavior by exhibiting interference.)

4. Oct 8, 2009

### Halcyon-on

Yes, I agree with you that everything can be described in terms of classical waves (hopefully even QM).

But a wave of what?

5. Oct 8, 2009

### Dmitry67

Fundamental things do not consist of something.
It is like asking 'what numbers consist of'?

6. Oct 8, 2009

### zenith8

Are you referring to the Hamilton-Jacobi version of classical mechanics? As far as I know this is the only known way to write CM in terms of waves.

Essentially the problem of dynamics as defined by Hamilton's equations can be formulated in terms of a partial differential equation determining the evolution of a field $$S(q,t)$$. The role of this $$S$$ function is to generate a momentum (vector) field on configuration space. Integral curves along the field are possible trajectories of the $$N$$-particle system. The 'wave' therefore refers to an ensemble of identical systems rather than a single trajectory.

7. Oct 8, 2009

### zenith8

Not in orthodox QM - that's one of the main problems with it.

However, you can easily do it with the Broglie-Bohm interpretation of QM (how did you know I was going to say that?).

8. Oct 8, 2009

### conway

I don't know what you mean by this.

9. Oct 8, 2009

### Halcyon-on

Fundamental things are such as long as a more fundamental description is found. I mean that could not exist fundamental things in physics, that is an indivisible and elementary atom of nature. On the contrary if fundamental things really exist you are right.

Concerning the number I don't think we can speak about fundamental things in a physical way. The numbers, as every mathematical objects, are abstractions that we associate sometimes to apples and sometimes to stars. The fundamental things in physics are the things that we describe with some mathematical abstraction. If the world would be described in terms of elementary waves, these waves would be the fundamental things that we describes in terms of sin and cos.

10. Oct 8, 2009

### Staff: Mentor

Different interpretations of a theory, by definition, do not make different predictions (otherwise they would be different theories instead of different interpretations). If you can recover the predictions of classical mechanics with one interpretation of QM then you can do so with all interpretations.

That said, I have no problem with the Broglie-Bohm interpretation, and in fact recommend that people learn all interpretations and use each when convenient instead of restricting themselves to only one.

11. Oct 8, 2009

### zenith8

De Broglie-Bohm is indeed a different theory; it is based on a different set of axioms to orthodox QM and it makes different predictions which in principle - though with great difficulty - could be observed (in the so-called quantum non-equilibrium case). I apologize for my loose language.

Leaving the language issue aside, it is an unfortunate fact that the recovery of the classical limit require you to (a) specify your ontology (what exists), and (b) to include particles in that ontology. Orthodox QM does neither; one cannot recover the classical limit using that approach. It is (relatively) trivial in de Broglie-Bohm.

12. Oct 8, 2009

### Staff: Mentor

That doesn't matter much. It is always possible to choose a different set of axioms for the same theory.
That is very interesting, and I was not aware of it. In that case De Broglie-Bohm is not an interpretation of QM but a different quantized theory. However, if the differences are so difficult to predict then obviously they must not be relevant in the classical limit.
This is most certainly not the case. Questions of ontology are irrelevant to questions of predictions. The quintessential example is Special Relativity and Lorentz Aether Theory, which are two different interpretations of the same theory. They have decidedly different ontologies, but both produce the same experimental predictions in all cases and both recover classical mechanics in the appropriate limit. Different theories need not have any ontological similarities whatsoever in order to arrive to the same prediction in some limit.

In any case, nothing you have said so far lends any credence whatsoever to your assertion that classical mechanics is not recovered from QM in the appropriate limit. Momentum is conserved as is energy, the expectation values behave exactly as you would expect the corresponding classical variables to behave in the classical limit. Hamilton's equations work at both scales and lead directly to Newton's laws which define classical mechanics.

Last edited: Oct 8, 2009
13. Oct 8, 2009

### Phrak

The quesion was about 'everything', including measuring equipment and observers.

14. Oct 9, 2009

### twofish-quant

Yes there is. Light has some behavior that is particle-like. This becomes obvious if you have light levels that are low enough so that you can see the invididual light particles (i.e. photons). If you every see the world through night vision googles or CCD's, you can actually see invididual light particles hitting the glasses.

Also solar photovoltic cells work when invididual light particles hit electrons. What this means is that if you shine red light on a solar panel it won't knock out electrons.

15. Oct 9, 2009

### zenith8

It's not difficult to predict, it's just difficult to measure. Because particles and waves (which are both supposed to exist in this theory) are logically independent entities, then the former need not be distributed as the square of the wave as in Born's rule. It can be shown that they just approach that distribution under a dynamical equilibrium process.

Interestingly, it is evidently a mathematical reformulation of QM rather than an interpretation because one can do unique calculations with it. A particular nice example is in calculating the propagator - i.e. the probability amplitude for a particle moving from one place to another after some time - by integrating Lagrangians along trajectories a la Feynmann's path integral version of QM. Professor Feynmann sums integrals of the classical Lagrangian along all possible paths in the entire universe. If you include the potential function of the deBB quantum force in the Lagrangian, then you can achieve exactly the same thing by summing integrals of the resulting quantum Lagrangian over er.. precisely one path - the one the electron actually follows - rather than an infinite number of them. Gotta be an improvement.

Yes - I seem to recall it was me who kept bugging you about Lorentz theory about a year ago. I don't think you convinced anyone then about the role of ontologies either.

Nope. I'm afraid you're not allowed to deduce a classical theory of matter from a statistical theory of observation, i.e. from any solution of the Schroedinger equation in any limit, even well-localized ones (packets) that remain so over time. You have to supplement the pure theory of linear fields by a physical ontological postulate (like in de Broglie-Bohm) or you can't claim that a material object is at definite x independent of measurement as in classical mechanics.

And anyway, you can only pretend to get away with it by presupposing what you want to define. Remember in standard QM, only the results of measurements and possibly the wave function exist. Orthodox QM presupposes the validity of classical concepts since it is only in this way that you can unambiguously communicate experimental results in the quantum domain. We then use the so-called correspondence principle which demonstrates the consistency of quantum theory with this presupposition.

The correspondence principle is a vague notion stating that, in effect the behaviour of quantum systems reproduces classical physics under suitable conditions (e.g. in limit of large size or large quantum numbers or - ludicrously - as $$\hbar \rightarrow 0$$). With the Schroedinger equation interpreted probabilistically, Ehrenfest's theorem shows that Newton's laws hold on average, in that the quantum statistical expectation values of the position and momentum operators obey Newton's laws (if V varies slowly over the wave packet). However, again this is a presupposition: which operators are chosen to correspond to physical quantities or measurements? Guess what - only those ones that reproduce classical mechanics in the limit.

This has ludicrous consequences for 'measurement' - as absolutely any interpretation which supplies the electrons with a trajectory will show you. One trajectory that quantum particles behaving in a quantum manner absolutely do not have is the Newtonian one; working out the expectation value of the momentum operator gives you the instantaneous momentum only in the classical limit. The very word 'measurement' implies that you are measuring some property of the particle that existed before the experiment but you are not. If you analyze it the momentum operator just gives you one particular component of the stress tensor of the wave field, and has nothing to do with the momentum of anything (which puts all those endless discussions of Heisenberg's uncertainty principle into perspective). It's all a comforting but false superconstruction. Remember Einstein: 'Your theory will one day get you into hot water.When it comes to observation, you behave as if everything can be left as it was, that is, as if you could use the old descriptive language.' Heisenberg et al. got away with labelling the old Einstein as a bumbling old fool back then, but many people have since realized how sharp Einstein was being about this.

To conclude the game, I need only quote Bohm himself. As standard QM only reproduces CM statistically and because the statistical interpretation only gives probabilities of different classical outcomes, Bohr argued that CM does not emerge from QM in the same way that e.g. CM emerges as approximation of special relativity at small v. He argued that CM exists independently of QM and cannot be derived from it - it is inappropriate to understand observer experiences using purely QM notions like wave functions as different states of experience of an observer are defined classically and do not have a QM analog.

So to sum up, you cannot logically deduce a model of substantial matter and its motion from an algorithm which has no such concepts at all (i.e. it makes no statements as to what matter is). This leads to a further problem: orthodox QM is practically successful but seemingly fundamentally ill-defined. Bohr postulates the existence of classical 'measuring devices' (in deBB theory, the instruments themselves emerge out of the limit). However, in the Bohrian view, there is no clear dividing line between 'microscopic indefiniteness' and the definite states of the classical macroscopic realm. What happens to the definite states of the everyday macroscopic world as one goes to smaller scales? Where does the macroscopic definiteness (where 'real' means something) give way to microscopic indefiniteness (where it doesn't)?. Does the transition occur somewhere between pollen grains and macromolecules, and if so, where? What side of the line is a virus on? In orthodox QM, there is no answer.

This is already getting too long, so I will just briefly summarize the deBB approach to the classical limit. In this theory there are two forces (the classical one, and the 'quantum' one - the latter resulting from the wave field pushing the particles around). The classical domain is where the wave component of matter becomes passive and exerts no influence on the corpuscular component, i.e. the state of the particle is independent of the state of the field. So we ask how does the state or context dependence characteristic of the quantum domain turn into the state independence at the classical level? Answer: when the actual trajectories look Newtonian, which is when the quantum Hamilton-Jacobi equation (which arises - in addition to a simple continuity equation - from splitting the complex Schroedinger equation into two real ones) reduces directly to the classical Hamilton-Jacobi equation (a standard way of rewriting classical mechanics in terms of waves - see my response to the OP). Under the right circumstances the force on the particle then goes from $$-\nabla (V+Q)$$ to $$-\nabla V$$ (here $$Q$$ is the potential energy function of the quantum force) and the trajectories become Newtonian. Limits like m going to infinity or whatever are not enough; this must be in addition to a proper choice of state.

You're going to argue with me aren't you, because you always do. Go on then..

Last edited: Oct 9, 2009
16. Oct 9, 2009

### Hurkyl

Staff Emeritus
An ontology for quantum mechanics is irrelevant in the question of whether or not it captures the classical case in the limit. To answer that question, the thing that matters is that one can specify an ontology for classical mechanics in terms of the abstract mathematical theory of quantum mechanics.

17. Oct 9, 2009

### zenith8

Ah, God how could I not have seen that? [slaps forehead and hurls himself from the top of a high building].

18. Oct 9, 2009

### bjacoby

So it's obvious that light is a particle. But now let's ask if it's a wave? I see no wave properties! For example you say interference show that light is a wave. But I see no interference. If I shine photons at a slit or double slit I observe a classical interference pattern, right? Wrong! I observe a bunch of photon particles landing on my detector! Now it turns out if I collect a bunch of impact positions and add them all up, lo and behold the pattern that results is exactly the same mathematical function as a wave diffraction pattern. But this pattern is made up of particle impacts NOT wave intensities! It is NOT a wave property at all. Who knows why such trajectories form into wave solutions when averaged over large numbers?

And it's worse than that. Light transmits energy through the empty vacuum of "empty" space. Particles can transmit energy through "nothing at all" but waves cannot. Waves need a medium to transmit energy. And it gets still worse. If I shine light on a photcell, we find the energy is transferred almost instantaneously...as a particle would. A wave would take TIME to transfer an equal amount of energy and it goes on and on.

So I put it to you. How can light be a "wave only" phenomena? It can't even be a wave and particle phenomena! Where are the waves? Well sure, there are "probability waves", but what are they? What is the medium they travel in? What generates probability waves? How do they "collapse" into sudden manifestation of objects? What is going on? Obviously there is lots of hand-waving and nobody with a clue. The fact that a QM wave theory gives correct answers does not prove that such waves exist. The fact that the statistical trajectory landings we just talked about form wave solutions when averaged means that it is not surprising that wave equations give "correct" answers. But it all explains nothing.

19. Oct 9, 2009

### zenith8

Our very own Louis-Victor-Pierre-Raymond, 7th duc de Broglie and Professor David Bohm!

I thank you.

20. Oct 9, 2009

### Cthugha

21. Oct 9, 2009

### Staff: Mentor

Why should I argue? You clearly agree with me about my original point. You just think the standard interpretation is "ludicrous", and I'm fine with that.

22. Oct 9, 2009

### DrChinese

Because waves are spread out and do not exist in the limit of a point. So if an electron were a wave, then trying to isolate it to a progressively smaller volume should not work. (Again I am not asserting an electron is purely a particle either; simply stating that it is the HUP that describes wave vs. particle behavior.)

23. Oct 9, 2009

### DrChinese

You have ventured into the realm of the semantic at this point. Clearly, the standard view is that light has both particle *and* wave nature depending on how it is observed. But it is no more possible to single out the particle view as preferred as it is to single out the wave view as preferred. Such a decision will necessarily depend on your definitions - none of which will change the underlying character of the actual physics.

The physics, of course, being the language of QM. So your criticism is really a critique of the theory; that critique would be readily accepted if there were a better theory offered. Barring that, we simply return to where we were: we don't know why the math works as it does. I don't see that as much of a criticism; we have the same situation with General Relativity too. We have that with any fundamental theory in fact.

24. Oct 9, 2009

### conway

I wonder if you can come up with a specific experimental scenario which illustrates this difficulty.

25. Oct 9, 2009

### DrChinese

Well, I would say that any experiment in which the particle nature is observed, you will see the disappearance of wave effects. The quantum eraser experiments are a good example, as you go from wave to particle view and see interference appear/disappear.