I am having some problem understanding Born's rule. I am getting a little bit confused. Here it goes;
Let $f(x,t)$ be a solution of Schrodinger equation. Then Born's rule says that the square modulus of $f(x,t)$ gives the probability of locating a particle at position $x$ at time $t$.
Since quantum mechanics also says that a quantum state in superposition collapses to one of its eigenvalues upon observation, then I understood $x$ as an eigenvalue corresponding to the position operator and that Dirac-delta function is its eigenstate.
But then I do encounter another form of Born's rule which states that given that $g(x,t)$ is a linear combination of quantum states superimposed on each other, then the probability of $g(x,t)$ collapsing into one of its eigenfunctions, say $h(x,t)$ is the square modulus of the dot product of $g(x,t)$ and $h(x,t)$. Yes, I do understand how this dot product yields the coeficient of $h(x,t)$ but I don't understand well the connection between this latter rule and the former.
For instance, is it correct to say that the solution to Schrodinger equation, $y(x,t)$ is a linear combination of Gaussian bell functions that tends to Dirac-delta functions as we increase the certainty of the positions? If yes, is the square modulus of $y(x,t)$ equals to the square modulus of the dot product of $y(x,t)$ at the Dirac-delta function with eigenvalue of $x$ at time $t$?