Before we get into a formalism, we’ll explore the foundational conceptual ideas of the wave function, probability, and normalization. We will later describe in detail the mathematical formalism of the Quantum Mechanics, which in short involves $\ket\psi$ vectors in a Hilbert Space $\mathcal H$ that are normalizable to $\braket{\psi|\psi}=1$.
How can we interpret the wave function?
How do we calculate statistics?
Is normalization a valid constraint to impose?
We’ve discussed the wave function and extracting position and functions of position information, but what about momentum?
Can we know both/either position and momentum to complete certainty?
Consider the eigenvalue problem for the position operator
$$ \hat x\psi(x)=x\psi(x)=\lambda\psi(x) $$