Where do wave functions come from?

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Summary:

Variables to wave functions to 2nd quantization
In classical mechanics, we have either Newton’s laws or a Lagrangian in terms of coordinates and their derivatives (or momenta) and we can solve them for the behavior of the system in terms of these variables, which are what we observe (measure).

In QM, we quantize classical mechanics by making operators out of these quantities and by making some of them non-commutative. They then need to operate on something, so the wave function (or state vector) was invented. But what was that? Only with the Born rule did the square of the wave function come to represent the probability of the system’s being in a certain state, in which the state variables may take on eigenvalues given by the momentum and position operators operating on the state vector.

Then along comes QFT, wherein we quantize the state vectors (because we treat them as fields) by the same trick of forcing commutation relations onto them. The same question arises: What do they operate on? Well, we use the same Dirac notation, but it's not clear to me just what this new thing is.

And I am intrigued by the same trick being iterated and reiterated. Is there some interpretation of this I have missed?
 

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In some ways, your question reflects on why the Math works so well in these circumstances. There is no answer to that or what many of these QM concepts relate to our simple reality.

NOVA has an episode on this question:

 
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Summary:: Variables to wave functions to 2nd quantization

Then along comes QFT, wherein we quantize the state vectors (because we treat them as fields) by the same trick of forcing commutation relations onto them. The same question arises: What do they operate on? Well, we use the same Dirac notation, but it's not clear to me just what this new thing is.
Don't think of QFT as second quantization. Think of it as first quantization of the classical continuum field, such as the electromagnetic field. The Born rule gives the probability that the field has a particular shape ##\phi({\bf x})## at time ##t##. Does it help?
 
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