# Why is the coefficient-squared the probability, and not just the coefficient itself?

Context: I have decided not to accept the postulates of quantum mechanics blindly as gospel. There must be a way someone arrived at those postulates, and I want to know the basic reasoning behind them, if not a derivation.

Problem: According to page 30 of this paper, the reason for the quadratic speedup in Grover's algorithm is the idea that probability in QM is the complex-conjugate squared of the probability amplitude. That seems a little too convenient for me. Who decided that the elements of the vector $$|\psi\rangle$$ are the probability amplitudes and not the probability itself? Can we not describe quantum mechanics wherein the latter is the case?

• Apr 2 at 14:55
• Apr 2 at 15:11
• There are actually a lot of proofs of this, but you should tell us which kind of axioms you are willing to accept. Have you read about Gleason’s theorem, for example?
– Plop
Apr 2 at 23:39

Strictly speaking, you could have a theory with L2-normed states and operations, but with a measurement that followed some other norm. You can write a quantum state vector simulator and make it sample measurements any way you want, and nothing comes down and smashes your computer for doing so. There's nothing mathematically impossible about it. However, in experiment, reality does seem to follow the L2-norm. If you measure how much light makes it through a polarizing filter with an angle $$\theta$$ between the filter and the polarization of the light, the amount that gets through is $$\cos^2 \theta$$ instead of $$\cos \theta$$ or $$\cos^3 \theta$$ or $$e^{\cos \theta}$$