Probability and probability amplitude The equation:
$$P = |A|^2$$
appears in many books and lectures, where $P$ is a "probability" and $A$ is an "amplitude" or "probability amplitude". What led physicists to believe that the square of modulus connects them in this way? 
How can we show that this relation really holds? In a wonderful QM lecture by J. J. Binney at Oxford University (minute 12:00) he states that this equation holds but doesn't explain why or where it comes from.
 A: If you can accept Schrödinger's equation, I can give you a motivation of Born's rule. The wave function psi(x) completely specifies the system's state (let's talk about an electron). Therefore, the probability (here: to find the electron at x) must be some functional of the wave function. Schrödinger's equation describes the temporal dynamics of the wave function. Considering this, you have the requirement that the functional must such that the dynamics does not change the total probability. Eventually, the expression should be easy. Then, you are left with the wave functions absolute square (not just the square!).
@nervxxx: This is not really true. In Maxwell's electrodynamics you can pretty easily derive a continuity equation (with source term) and identify it with the energy. Then you get an expression ~ E² + B² for the energy density. Note: you get E², not |E|². The electric field is always a real number. It is convenient to use complex numbers (including a wave's phase) in the calculation and take the real part later. But you have to take the real part before taking the square because the complex/real trick only works for linear operations. So you should write (Re E)².
A: This relationship is called the Born rule and it's one of the postulates of quantum mechanics. There have been various attempts to derive it, but none of them having been terribly convincing so at the moment we have to assume it is true. Fortunately experiment supports this assumption.
The rule was originally suggested by Max Born (hence the name) in 1926 right at the beginnings of quantum mechanics. He won the Nobel prize for the suggestion!
A: This relationship comes from the scattering problem where there are also such probabilistic things as a particle flux and a scattering cross section. The Schroedinger equation contains a conservation law like a continuity equation where the square of modulus appears. For a flux of particles you obtain a scattered flux (an average, deterministic value); for one particle of the flow you obtain a probability to scatter this or that way.
