[chaosified]

Why does the wavefunction squared give the probability density? What is the motivation or justification for this?

PS: I have taken Introductory QM, and gone through Griffith's book. Could not come to a conclusive answer.

[Enthalpy]

Hi chaosified,

Is there any mathematical justification? Up to now I consider it an a priori of QM.

But for the founders of QM this was a natural choice, as it was known from optics for instance. The "first-power" of the E and H (and A and B and V and whatever) add linearly, with phase and where applicable orientation, which permits them to interfere. The squared module of E gives a power per area unit transported by an electromagnetic propagating wave, and the number of photons per second (of energy defined by the frequency) is a power too, so the probability of detecting a photon goes with the squared module of the amplitude.

Because the previous paragraph is easily misinterpreted: many books claim the electric field E is the wavefunction of the photon, but this fails at symmetry considerations, at entanglement and more. The clean wavefunction is just a complex scalar (the usual psi) that depends on the polarisation plus the usual things.

[Irlanur]

as with all this fundamental stuff it is an educated guess:

Hint: calculate the expectation value of the charge density of a classical system, which is given by a sum of dirac-delta functions.