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 probabilities for position measurements (only), but generally not for measurements of other observables such as momentum. For the momentum probabilities, 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.

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.

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

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 probabilities for position measurements (only), but generally not for measurements of other observables such as momentum. For the momentum probabilities, 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.