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I can't seem to understand the dual nature of matter completely.

If electrons have a wave nature, then if two electrons were to collide, wouldn't they undergo interference and form an electron wave of larger or smaller amplitude? Does that mean the electrons merge and turn into a "bigger" electron?

Then again, this would mean that the electrons would have a phase difference, so what determines the phase of matter?

Also, if we were to consider two photons colliding, would they also undergo interference and form a "bigger" photon?

Why don't normal everyday objects undergo interference? Just because the wavelength (according to the de Broglie equation) of something is small doesn't mean that it doesn't undergo interference (take for example gamma rays which have a very small wavelength but are the most powerful form of radiation).

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This is kind of a big question which takes a book to answer (in my opinion). There is a lot of math involved, and I don't think you can understand this before you learn all of it. I suggest you take a basic quantum mechanics course or read a good book about this, all of it (like the first volume of Cohen-Tannoudji, for example). –  Rafael Reiter Mar 6 '13 at 19:11
    
As Feynamn put it, don't worry if you don't understand QM, because no one does, not even Feynman. Meanwhile, master the math! –  Ethan Mar 6 '13 at 19:53
    
    
The phrase used is "diffractive scattering". –  dmckee Nov 4 '13 at 16:54
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3 Answers

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First one needs to understand that electrons, and nature in general, are what they are - neither waves nor particles. Whether they exhibit wave like or particle like behaviour depends on the experiments we do. An electron microscope, for instance, uses interference between many electrons to create an image of the object.

For a long time people doubted the ability of matter to change properties depending on what WE, the observer, did, but Alain Aspect and others showed that the result of an interference experiment does depend on our choice to measure interference, even of we set things up so that the choice is made after all the photons have travelled past the mirrors/slits in question.

For quantum mechanical waves, except in interesting topological circumstances, phases are usually unobservable. The amplitudes may be multiplied by any phase without affecting the outcome of the experiment, which is determined by the (real) squares of the amplitudes. The phases are certainly necessary for the description of waves, but it is not right to say 'the phase of matter'. There is also nothing 'bigger', because as a state/ensemble evolves the total of all amplitudes squared is conserved - this is the law that probabilities always sum to 1. If you want to think of intensity, as in the number density of photons in a beam, this might be changed by various means, but remember that you are then talking about many photons. If you integrate the interference pattern, the expected total intensity does not change from what it was.

Everyday objects do undergo interference, but on such large scales that we cannot separate the effects from the complex world around us, the quantum state of which defies description. Some people doubt this, but all evidence indicates that the world really is a quantum one.

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Oh.. so electrons only show wave-like properties when we try to measure those properties. That sounds a bit strange doesn't it.. Would this be related to Heisenberg's uncertainty principle? One can only measure either the position or the momentum but not both at the same time?? It sounds similar.. –  kaliaden Mar 7 '13 at 5:29
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This is a good question and the physics involved to answer it is interesting too. I will try to answer it without necessarily going into the details of quantum electrodynamics that is needed for a full account of the phenomenon.

IN SIMPLE TERMS:

Basically your question is about $e^-+e^-\rightarrow e^-+e^-$ interaction. Two electrons do interact with each other, their wave functions do interfere. But this interference is not direct like it is for light “waves” or water waves, in the sense to give more intense light or bigger water wave at a constructive interference point. The electron wave functions interact via the electric charges carried by the electrons, i.e. it is an electromagnetic interaction. So it is not a direct addition of wave functions. We calculate the cross-section of the interaction and other relevant quantities we might be interested in.

We do add all the components that make up the wave function of an electron, however, as we do when we solve the Schrodinger equation for an electron in a box for example, and then the wave function of the electron is a superposition of all solutions. But this is not addition of smaller electrons to get a bigger one. We add the probability amplitudes so that we can calculate the probability to find the electron at some position, $x$, at some time, $t$. I hope this simple description helps you understand how electrons interact.

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I have found that the weirdness of these sorts of problems go away when one considers that actually we are not doing physics but inference. Physics without inference is merely art (ie. the design of experiments, writing of papers, etc). The actual science itself is entirely reduced to methods of inference - how well can we model the experimental data?

Quantum mechanics is an inference method developed in the last century to deal with certain problems which could not be dealt with sufficiently in the context of classical probability. But the technique is in no way confined to only the microscopic domain, as we are beginning to learn. Indeed, the methods of quantum are beginning to be applied to cognitive science and ecology.

So the answer to your question is that you've posed the wrong question. Asking about the nature of matter is unanswerable and uninformative, and ultimately a question of philosophy. The problem of physics is to develop methods that allow us to predict the outcome of experiments, to the best of our ability. QM is the best method we have for this now - but it is merely a method of inference and doesn't actually tell us much about the objective universe. In fact, an 'objective universe' is itself merely an inference! Perhaps this is the most important lesson of QM.

While this will no doubt be unsatisfying for you, at least initially, with time it may grow on you. Also, you have a lot of QM to learn before it really settles in.

Finally, we are less interested in the wavelength of an object than the ratio of wavelength to 'radius'. For a gamma ray, this is still pretty high, so QM is important. For a basketball, this ratio is tiny, and makes QM irrelevant on the large scale (interference is 'happening' on the micro level in the basketball, but all the many interferences and micro-collisions average out over a huge number of particles to give the steady state of the macroscopic ball)

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