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So I have just started studying Quantum mechanics and like a lot, and struggling with the concepts, so it’s really difficult (if possible) to relate to reality.

Nevertheless, some intuition will still help for approaching the subject, and they include:

What is really the wave function? I know when you square is the probability function, but by itself, does it mean anything physically?

And also does a wavefunction exist in reality if it is not normalised, is there anything else other than so the area is one (when the wavefuncion is normalised)

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  • $\begingroup$ QM has at least 20 different intuitive interpretations. $\endgroup$
    – safesphere
    Oct 27 '19 at 4:18
  • $\begingroup$ @safesphere Many of them unsustainable. $\endgroup$
    – my2cts
    Oct 27 '19 at 8:13
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I personally find it clearer to do the apparently more formal approach, which is studying kets in Hilbert spaces.

Say you have a particle. Whatever be the relevant physical information about it, let us call it the 'state' of that particle. Now imagine an all-powerful object which stores all this information. Denote it by $|\psi\rangle$, and call it a "ket" vector. Consider the collection of all such ket vectors. Convince yourself(naively) that such a collection must be a linear space(think experimental results which point to superposition principle etc). Realise that experiment suggests there is no idea of a classical trajectory for quantum system.

So we don't have numbers like $x,p$ which evolve in $t$. Instead, the only way to associate such a number(position, momentum, etc) would be to conduct a measurement on the system, and it is clear that this measurement itself doesn't represent any kind of time evolution of $x,p$, etc. So it captures no information about how is the system moving through time. How then, do these 'states'(i.e. the kets) evolve? After all, we do require a causal time evolution equation. This brings in the Schrodinger equation(which you can motivate by a bunch of ways; the clearest being 'deriving' it from the propagator, because the latter is more believable to postulate and easier to arrive at. Personal choice.)

Now, we have an equation which, because it is causal, is a first order in time differential equation. The kets were vectors in a hilbert space, and now we choose a basis to expand this ket out(just as in linear algebra, when you 'add vectors', you first choose a basis and then just add their components).

Now, consider the position of your particle, $x$. The corresponding state vectors(that capture information about the position of your particle) form a continuum of states, $|x\rangle$. As it turns out, they form some kind of a basis for your Hilbert space(that is a VERY imprecise statement, but sufficient for now). And you expand out your state ket in terms of this basis; $$|\psi\rangle =\int dx{ \psi(x)|x\rangle}$$, where there is an implicit integral over $x$ on the RHS(if it were instead a discrete index, then there would be a sum, as in $v=\Sigma v_i e_i$). And this defines $\psi(x)$-it is the merely the components of your state ket in position basis.

Now, there are other ways to motivate this, but I find them very handwavy. An example(along the lines of Resnick-Eisberg)is-you know from elementary physics that the intensity of a wave goes as it's amplitude squared. And this intensity is a measure of the energy carried by the wave. Consider a beam of light, and quantum physics tells you it is made up of photons. So, the intensity is essentially in a sense the number of photons, and so the amplitude squared is therefore a similar quantity. The greater the amplitude(squared), the greater the no. of photons. Now, you merely go a step back and work with the amplitude, instead of amplitude squared. But clearly, this is not very satisfying.

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  • $\begingroup$ This a convenient notation but an explanation of what a wave function is. $\endgroup$
    – my2cts
    Oct 27 '19 at 8:17

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