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The entity "light" behaves as a wave & particle.

The wave is actually probability wave . That is, to every point in a light wave we can attach a numerical probability that a photon can be detected in any small volume centred on that point. -Resnick, Halliday, Walker - Principles of Physics.

  • Now, the light wave transports energy & momentum. Doesn't also an electron wave carry energy & momentum? If so, how?

  • A light wave propagates the energy in its field components. But how does an electron wave do this if it does so?

  • Unlike light, electron-waves are not made by electric or magnetic fields. But then the question arises: if light wave carries away energy, why doesn't a matter wave do the same?

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"The wave is actually probability in the sense that it assigns probability to the space coordinates of detecting photon at a certain time."

No, the emergence of the classical EM wave from the quantum wavefunction of the photon is not trivial, because a classical EM wave is made up of many photons. In particular, it is not the case that the classical EM wave is the wavefunction of a photon. (Even more particular, it is difficult to even speak of the wavefunction of a photon, since photons usually arise in a quantum field theoretic ("second-quantized") description where the notion of wavefunction does not exist (but is, if you insist on something comparable, replaced by a wavefunctional))

Also, do not speak of "the electron wave". While an electron - like all quantum objects - carries wave-like properties and can be described by a wavefunction (which is not a function as you might imagine it if we incorporate its spin in the sense that it does not take values in the real or complex numbers), it isn't a wave in any classical sense, and also still carries particle-like properties. It is a quantum object, neither fully wave nor fully particle at any time.

Nevertheless, all quantum objects (and hence all things you might decsribe as "matter waves") of course carry energy - the energy that is in their rest mass and the energy that is in their momentum, though the energy of any given quantum state may not be well-defined, but "smeared out" over a range of energies.

To ask how physical objects carry the properties they do is, deep down, not sensible. How does a classical particle carry momentum? By having mass and velocity! But how does it carry mass and velocity? By...um...moving and stuff. How does it move? Um... You get the idea. "Why/How" is a question that can be asked infinitely many times, but only answered finitely many before you hit a point where the only answer is because it seems that way.

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  • $\begingroup$ Some remark - not exactly objection. You are aware, aren't you? That there are additional views about the quantum objects. You should have told the user that this is the standard view. Adepts of other interpretations may jump on what you said with arguments that never end. $\endgroup$ – Sofia Mar 8 '15 at 2:12
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    $\begingroup$ @ACuriousMind Re the "wave-function of the photon" thing - I can restrict the the full space of states to a single photon and have a "wave-function of a photon" which is meaningful in certain contexts. And its Schrodinger equation will be isomorphic to the Maxwell's equations of the classical wave. $\endgroup$ – Slaviks Mar 9 '15 at 15:46
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    $\begingroup$ @Slaviks: That why I said usually - I know you can get a wavefunction for a photon, it's just not the usual way to think about photons, since they usually can be created and destryoed as we like. $\endgroup$ – ACuriousMind Mar 9 '15 at 16:16
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    $\begingroup$ @Sofia Telling you "what is the wave-function" is strictly not part of physics. "Strictly" in the same sense as "deep down" and "not part" in the sense of "not sensible" of ACuriousMind's answer. Mind you, I am not advocating "shut-up-and-calculate" interpretation, I am asserting that physics is what is interpretation-invariant. $\endgroup$ – Slaviks Mar 9 '15 at 17:29
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    $\begingroup$ @user36790: One may indeed express the wavefunction in terms of the fields as $\psi(x) = \left(\begin{matrix}\vec E \\ \mathrm{i}c\vec B\end{matrix}\right)$, see e.g. this paper, but this requires already a relativistic approach and is subtle. Asking whether the wavefunction "exists" or not is, to me, a meaningless question - it is a working tool to predict nature, and we don't usually ask ourselves whether "the Hamiltonian" or "momentum" or other things "really exist" as long as they are useful in predictions. $\endgroup$ – ACuriousMind Mar 16 '15 at 10:59
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"The wave is actually probability in the sense that it assigns probability to the space coordinates of detecting photon at a certain time. Now, the wave transports energy & momentum."

As you say that you quote words from a book, then, you have to know that there are many books and many authors, each one with his/her opinion. There are four basic opinions about the wave-function:

  • Standard Quantum Mechanics - we don't know what is the wave-function, and we can only calculate results of experiments.
  • Bohmian Mechanics - the wave-function is a field, not one that is known to us from the classical physics (e.g. not e.m., not gravitational), and this field guides particles like electrons, photons, which trajectories to follow. E.g. in interference experiments, these particles are guided to go with bigger probability to some places than to others.
  • G.R.W. (Ghirardi-Rimini-Weber) - the wave-function is a wave with no structure of particles in it, but when meeting classical apparatuses, it produces an effect of discrete responses.
  • MWI (many-worlds interpretation) which I am not going to discuss.

(There are also other interpretations but the most elaborated are the ones I quoted above.)

Now, to different details of your question, (my answers are in agreement to the most widespread views, not biased by one interpretation or another):

We don't know that the wave is probability. Probability is a mathematical thing that we write on the paper. In nature exists matter, under different forms: waves that carry by themselves their energy and momentum, waves whose energy and momentum are transmitted with the help of the particles of some medium (as in the case of water waves or the sound), definite objects big or small, and eventually other forms not yet known to us.

Now, if the waves were waves-of-probability as you are told, then how could they carry energy and momentum?

"Unlike light, electron-wave is not made by electric or magnetic fields."

Who says that a wave has to be electromagnetic? The gravitation waves aren't electromagnetic. The sound is not electromagnetic.

But, unfortunately, we don't know what is the wave of which you talk, i.e. the wave-function. It's more than a century that we stay with this problem. The electron-wave yes carries energy and momentum, however, we are unable to decide which one of the different interpretation is correct, and we also admit that there may exist another variant.

A. Peres, said once that it may be that we have to reconsider our concepts about space, time, and others.

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