What is Wave function and Born’s rule?

The state of a particle moving along $x$ is described by a complex wave function $\psi(x,t)$ such that

is the probability of finding the particle between $x$ and $x + \dd x$ at the time $t$ (Born’s rule); hence the normalisation $\int_{-\infty}^{\infty}|\psi|^2\dd x = 1$, and the expectation values $\langle x\rangle = \int x|\psi|^2\dd x$, $\Delta x^2 = \langle x^2\rangle - \langle x\rangle^2$. Wave functions obey the superposition principle: if $\psi_1$ and $\psi_2$ are possible states, so is $\alpha\psi_1 + \beta\psi_2$ (normalised), and the probabilities then contain the interference term $2\operatorname{Re}(\alpha^*\beta\,\psi_1^*\psi_2)$. A global phase factor $\eu^{\iu\theta}$ changes nothing; a relative phase between two superposed components changes the interference.