I read in my physics book (NCERT class 11, page no. 374) that overlapping waves algebraically add to give a resultant wave and it is also mentioned that this superposition principle implies that the overlapping waves do not alter the travel of each other.

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Now I have also read about the uncertainty in the behaviour of electrons (on being a particle or a wave).

Isn't the same principle applicable for electrons ? If not, then please give the reason.

If they were waves and suppose two electrons were shooted towards each other, they should not affect the travel of each other but if they were a particle they should fly away being deviated.

  • $\begingroup$ I suggest adding the page of where it is in the book. $\endgroup$
    – Buraian
    Apr 16 at 10:00
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    $\begingroup$ @Buraian done .. $\endgroup$
    – Ankit
    Apr 16 at 10:13
  • $\begingroup$ To quote George Orwell in Animal Farm: not all waves are equal. $\endgroup$ Apr 16 at 10:19
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    $\begingroup$ Isn't this very common question just an artifact of our inadequate discussing the subject in textbooks? It seems to me it all depends on the measurement; some quantities like momentum yield wave-like properties, but the particle number measurement yields integers. There is no controversy, IMO. $\endgroup$
    – dominecf
    Apr 16 at 10:43
  • $\begingroup$ Here is something on how light can be something like both a particle and a a wave. Electrons are like like in this way. How can a red light photon be different from a blue light photon?. Here is a question about the wave function, which gives rise to the wavelike and particle like properties of the electron. Does the collapse of the wave function happen immediately everywhere? $\endgroup$
    – mmesser314
    Apr 16 at 13:36

We can say with certainty that an electron exhibits both particle-like and wave-like behaviours.

Experiments suggest that an electron is a point-like fundamental particle (ie one with no internal structure) with a radius of less than 10−22 metres. However, electrons can also be diffracted as if they were waves, the wavelength being dependent upon the electron's energy.

Mathematically, waves can be superimposed to create a resultant wave, and where the waves are travelling their direction of motion is unchanged by the superposition. However, you must take care when applying the mathematical ideal to physical cases. To take an extreme example to make the point, you cannot superimpose a wave of light and a wave on the surface of water, and where light and water waves do interact the direction of the light is not unchanged!

In the case you raise, electrons definitely do interact with each other, so clearly their behaviour cannot be modelled as if they were waves purely. However, since the electrons can be diffracted, their behaviour cannot be modelled by treating them as particles purely.


The wave function of a particles is described by the Schrödinger equation, which is compatible with the principle of superposition. However, the wave function has no physical meaning - it is its squared the probability, and the square of the wave function does not agree with the principle of superposition. Check Eisberg's book, for example, or Morrison's.

  • $\begingroup$ "the wave function has no physical meaning" aside from the fact that it embodies all the theoretical physical knowledge that we have. $\endgroup$
    – my2cts
    Apr 16 at 18:11
  • $\begingroup$ Yes, but you can't measure it. $\endgroup$ Apr 16 at 20:51
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    $\begingroup$ The role of wavefunction is much discussed in philosophy of physics. You can measure it in the sense that you can obtain, by measurement, information about what the wavefunction is in any given case. But of course it is not so clearly a physical thing as an electron or an electromagnetic field or something like that. $\endgroup$ Apr 17 at 10:05

Isn't the same principle [super-position] applicable to the electron?

Yes is is.

Diracs equation gives us the quantum field that describes the electron-positron field. Despite its name, it is not a quantised field. A better name for it would be a relativistic classical field. It is akin to the electromagnetic field. And hence we have super-positions. The quanta of the electron-positron field is found only after quantising the field.

In fact, Geoffrey Chew, a quantum theorist, objected to quantum field theory on positivistic grounds since the so-called quantum field is unobservable. However, the quanta is observable. He advocated instead the bootstrap based on the observable scattering matrix.

How to describe the ontology of quanta is an open question and is usually tackled in the philosophy or foundations of quantum mechanics. Personally, I find Anaximander's notion of the apeiron, the unlimited or indefinite, useful to describe the underlying quantum reality given that after Bell & EPR experiments that value indefiniteness characterises quantum ontology as opposed to classical ontology where value definiteness is taken for granted. It's worth noting that the apeiron is akin to Bohms notion of a beable. It was described by Aristotle in his Physics as:

Of the unlimited [the apeiron], there is no starting point, since, if there were, it would have a limit. Further, it is incapable of coming to be and passing away ... that is why, as we say, there is no starting point of the unlimited, but it is rather the unlimited that seems to be the starting point of the other things, and to encompass everything and steer everything (as it is said by those who do not posit other causes beyond the unlimited, such as Understanding or Love) and is divine. For it is immortal and indestructible, as Anaximander says, as do most of the physicists.

We can then suggestively coin the term apeiriton as a minimal element of definite reality that arises from the apeiron. This would be what we think of as quanta in general terms.


It is only in the simplest cases (called "linear") that overlapping waves simply add without disturbing one another. In more general cases waves add in more complicated ways and then they can affect one another.

With electrons, the wave associated with each individual electron only adds in a simple way with itself. This is observed for example when a wave impinges on a pair of slits and then diffracts and the two parts then go on to interfere. The two interfering parts simply add and nothing more complicated happens. But the wave associated with one electron combines with the wave associated with some other electron in much more complex ways, and now the charge has to be considered, and the fact that electrons repel one another.


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