quantum mechanics - Mathematical probabilistic interepretation of probability amplitude

As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question:

I'm looking at the possibility of using probability amplitude functions to represent probability distributions on surfaces. From my perspective, a probability amplitude function is a function $\psi:\Sigma\rightarrow\mathbb{C}$ satisfying $\int_\Sigma |\psi|^2=1$ for some domain $\Sigma$ (e.g. a surface or part of $\mathbb{R}^n$)-- obviously these are some of the main objects manipulated in quantum physics! In other words, $\psi$ is a complex function such that $|\psi|^2$ is a probability density function on $\Sigma$.

From this purely probabilistic standpoint, is it possible to understand why multiple $\psi$'s can represent the same probability density $|\psi|^2$? What is the most generic physical interpretation?

That is, if I write down any function $\gamma:\Sigma\rightarrow\mathbb{C}$ with $|\gamma(x)|=1\ \forall x\in\Sigma$, then $|\psi\gamma|^2=|\psi|^2|\gamma|^2=|\psi|^2$, and thus $\psi$ and $\psi\gamma$ represent the same probability distribution on $\Sigma$. So why is this redundancy useful mathematically?

Answer

Different wave functions with the same $|\psi(x)|^2$ represent different physical states (unless they are proportional). Different states means that one gets different measurable results on at least one kind of measurements.

The same $|\psi(x)|^2$ gives the same probability density for position measurements (only), but generally not for measurements of other observables such as momentum. For the momentum probability density, the absolute squares of the Fourier transform counts, and this is usually different if only the $|\psi(x)|^2$ are the same.

The mathematical content of the wave function is the following (from which the above follows): The inner product of $\psi$ with $A\psi$ gives the expectation value of the operator $A$ for a system in state $\psi$. For example, if you take $A$ to be multiplication by the characteristic function of a region in $R^3$ you get the probability for being in that region. The position operator is simply multiplication by $x$, while the momentum operator is a multiple of differentiation.

For going deeper, try my online book http://lanl.arxiv.org/abs/0810.1019, written for mathematicians without any background knowledge in physics.