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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.

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Is this a history question of a "now" question? – dmckee Jan 23 '13 at 14:59
hm a good motivation would be due to the diffraction pattern experiments? that light goes like $\vec{E} e^{ikx}$ and the intensity which we observe is $|\vec{E}|^2$, and then because electrons also show diffraction patterns so somehow we can analogously describe them as some wave $\sim A e^{ikx}$ with probability $P = |A|^2$. although I think historically it was the other way round, QM was born first before those experiments were done.. – nervxxx Jan 23 '13 at 16:36
@dmckee I am learning QM and the problem is that it is hard for me to just believe in this equation. It has no background and is completely made up not derived. Some even write it down like $P \propto |A|^2$ insead of $P = |A|^2$. I mean compared to Einsteins relativity where everything was derived this is ugly to me (personal opinion). Are there at least any similar cases where we take a square of something to get probability? If we re going to guess theories like this they are sooner or later going to fail (my oppinion). – 71GA Jan 23 '13 at 19:34
@71GA: every theory is based on some assumptions. Born's law is one of the assumptions that QM is based on. I suspect most of us would agree the GR is more beautiful than GM, but then GR is unusually elegant. Having said this we know GR fails at singularities, so GR must be an effective theory that is a subset of a bigger theory and who knows whether the bigger theory will be as elegant. You ask "we also have all the rights to do this?". We have the right to build any mathetical model we want - the question is whether it corresponds to reality. QM does! – John Rennie Jan 24 '13 at 11:04
Comment formatting is intentionally restricted, and using block equation is discouraged: the comments are emphatically not a discussion engine and should not be used as such. Using block equations in lieu of horizontal rules will be treated as abuse. – dmckee Jan 24 '13 at 15:18
up vote 2 down vote accepted

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.

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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!

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"[...], following a slightly earlier paper in which he famously omitted the absolute value squared signs (though he corrected this in a footnote added in proof)."( – Keep these mind Jan 23 '13 at 19:04
This postulate seems like a "dogma" to me. – 71GA Jan 23 '13 at 19:39
It works though, and unlike religious dogma it gets tested every day. – John Rennie Jan 23 '13 at 20:42

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.

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