It may be a stupid question, but why particularly for probability density expression $k~|\Psi|^2 = k~\Psi^{*}\Psi$, it's assumed that $k=1$?
As it is now, then in a complex plane probability density is just a rectangular area for a complex vector. But why it has to be rectangular specifically? Why can't be $k=\pi$, so that re-defined probability density $\pi |\Psi|^2$ would mean a bounding circle area of complex vector:
Or any other complex plane area scaling value $k$? What would be implications of that to quantum mechanics?
Best Answer
It's a normalization convention for $\Psi$ - indeed, the only sensible one. If the probability density is $k|\Psi|^2$, just absorb a $\sqrt{k}$ factor into $\Psi$. This density should not be interpreted as an area. Indeed, the real reason we square has nothing to do with $2$-dimensional geometry.