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In quantum mechanics, I understand that the wave function represents the state of a particle and that the square of the wave function tells us the probability of a particle being found at a particular place in space and time.

When I think of the quantum double slit experiment, however, it seems that the wave function needs to be more than just a mathematical construct, because it physically diffracts and interferes with itself.

So do we have any idea what the wave physically is? What is the mechanism by which the diffraction and interference occur?

I don't know anything about quantum field theory and so perhaps this is all taken care of by some theory like that. Or is this all still a complete mystery?

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    $\begingroup$ By way of clarifying what kind of answer you're looking for, what would you consider a good answer to "What is light, physically"? $\endgroup$
    – WillO
    Apr 4 at 12:45
  • $\begingroup$ I suppose that I would talk about an electromagnetic wave and of electric and magnetic fields and the mechanisms by which they interact. $\endgroup$ Apr 4 at 12:48
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    $\begingroup$ Related/possible duplicate: physics.stackexchange.com/q/17107/50583 and its linked questions $\endgroup$
    – ACuriousMind
    Apr 4 at 12:52
  • $\begingroup$ Okay, so light is a wave in the electromagnetic field. The electron wave function is a wave in the electron field. Are you looking for more than that? If so, what is the "light" analogue of what you're looking for? $\endgroup$
    – WillO
    Apr 4 at 12:56
  • $\begingroup$ Saying that it's a wave in some other kind of field is somewhat satisfactory for me. I understand that more that saying it's just a mathematical representation. Is it correct to say that we can't directly measure this field in the same way we can measure an EM field though? We just infer it? $\endgroup$ Apr 4 at 13:05
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All still a complete mystery. Broadly speaking, “ontic” interpretations of quantum mechanics hold that the wave function has objective physical existence, while “epistemic” interpretations hold that it only represents an observer’s knowledge of the state of a system. So physicists don’t even agree about that.

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  • $\begingroup$ Fascinating. And those who interpret the wave function as representative, do they offer any explanation of how a mere representation can interact with other representations?? $\endgroup$ Apr 4 at 12:46
  • $\begingroup$ Nope, not that I’ve ever seen. $\endgroup$
    – tparker
    Apr 4 at 12:56
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    $\begingroup$ Related: physics.stackexchange.com/q/290522/149907 $\endgroup$
    – Urb
    Apr 4 at 13:28
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Depends on what interpretation of QM you take.

However, just because a think undergoes interference does not mean that it is "physical".

You can think of the interference of the wave-function as the creation of some "hot spots" where the electron is more likely to be detected, and some "cold-spots" where it is less likely to be detected. The net result is an interference pattern.

However, it does not mean that the electron "interfered" with itself, the electron is not a wave, its probability distribution may be modelled as a wave which interferes with itself.

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  • $\begingroup$ Are there other examples of things that can interact / interfere with other things in the world (like slits) but not themselves be physical? $\endgroup$ Apr 4 at 12:50
  • $\begingroup$ Electromagnetic waves (it really depends on what you mean by "physical" of course). $\endgroup$ Apr 4 at 12:53
  • $\begingroup$ But I never see anyone argue that an electromagnetic wave is just a "mathematical representation" or anything like that (as people do with a wave function). $\endgroup$ Apr 4 at 12:59
  • $\begingroup$ Well an electromagnetic wave is really just a perturbation in the electromagnetic field. Are electromagentic fields physical? I don't know, but what I can say is that they are a mathematical representation of "something". edit: for example, would you say that a force is a physical thing, or a mathematical representation of an interaction? $\endgroup$ Apr 4 at 13:01
  • $\begingroup$ I guess the important point to be made is why do you think that a wave-function "physically" interferes with itself? Would you say that if you take two random sine waves with different phases, the fact that they form an interference pattern makes them physical? $\endgroup$ Apr 4 at 13:08
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See how single electrons, build up an interference pattern in the double slit experiment.

Also here

dblsl

The accumulation of the random looking points is the interference pattern of the wavefunction solution for scattering "electron through two slits", i.e. the $Ψ^*Ψ$. The wavefunction itself $Ψ$ is not measurable, it is a mathematical function.

When the parabola of a trajectory is calculated for shooting on a target, the parabola is a mathematical function, it describes the path of the projectile, but the parabola is not physically measurable by itself. It is a mathematical tool. the same is true for $Ψ$.

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In QM we speak of the "physical content" of the wavefunction, not its "physical-ness". The solution Schrodinger's equation gives a specific set of wavefunctions that are eigenfunctions of the Hamiltonian.

The physical content of these wavefunction can be understood to mean that in some cases expectation values of observables obtained from these wavefunctions will return definite values, such as the energy levels of stationary states, whereas in other cases, there will only be a choice of numbers with greater or lesser probability, such as the position of an electron in an atom.

The physical content of the wavefunction can be extracted through integrals involving the wavefunction and its complex conjugate, which is also a path to constructing the interference pattern you ask about.

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QFT and most of the interpretation issues in QM are only marginally relevant to answer your question.

The key point is the role of the wavefunction in the theory and how the theoretical descriptions are connected to the real world.

The second point is quite simple but not trivial. Theoretical concepts are how we describe and understand the world, and there is no "deeper" alternative. We can never say what a phenomenon is. We can just build concepts to model as faithfully as possible what happens around us. To be precise and quantitative math plays an important role, but it is an auxiliary role. The bare fact that we have a mathematical description of some phenomenon does not imply that such a description is totally disjoint from what can be measured. Instead, using mathematics is the way to allow quantitative measurements. When we have a mathematical model which uniquely maps measurable phenomena into mathematical objects, we can use the latter as proxies of the former.

Things get different if there is not a unique mapping. In such cases, we say that a mathematical quantity of a physical theory is not measurable. And consequently, we assign to it a reduced physical meaning.

The simplest example I can imagine is the case of energy. Our theories always introduce energy in the form of differences in energy. Therefore, variations of energy are measurable while absolute energies are not.

Slightly more complicated, but closer to wavefunctions, the case of the electromagnetic (em) potentials in classical electrodynamics: em fields are directly measurable, while there is a one-to-many relation with the em potentials (gauge invariance of the theory). Once again, the missing one-to-one correspondence between phenomena and concepts of the theory makes the classical scalar and vector potential non-measurable.

The case of the wavefunction in QM is similar. It is a concept required to describe what we measure, but there is no unique correspondence with the physical situation.

Within the wave formulation of QM, the non-uniqueness has two origins. On the one hand, all the observable consequences of the theory do not use the wavefunction directly but only quantities defined within an arbitrary phase-factor, analogously to the previously cited examples from classical physics. On the other hand, the time dependence of the observables may be entirely assigned to the wavefunction or the operators or even shared between operators and wavefunctions (the cases of the Schrödinger, Heisenberg, and Interaction pictures). The possibility of such choices further increases the non-uniqueness of the wavefunction.

In conclusion, the question about the physical meaning of formalism is always a matter of establishing a unique correspondence between concepts and measurements. However, when a theoretical concept fails to have a unique link with measurements, we consider it not directly physical but an auxiliary concept.

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The jury is still out on thos one. There are around twenty or so interpretations of quantum mechanics and no real consensus about what a quantum wave represents. Although it is often said that the Copenhagen interpretation by Bohr is the position of most physicosts - this to me seems a cop out - simce they take a positivistic and operational view and refuse to say what the quantum wave is.

It's worth contrasting this with what we would ordinarily take to be a wave. Consider a violin string. Plucking it gives the fundamental harmonic tones and overtones. These build up a Hilbert space of harmonics. But whilst we see a wave on the string and would most naturally describe it in this way, the Hilbert space of harmonics seems like a concept that we are using to model the situation, rather than the thing - the wave on the string - under consideration.

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