The Postulates of Quantum Mechanics
The state of a quantum physical system is completely described by a state vector $|psirangle$ residing in a complex separable Hilbert space $mathcal{H}$. For spatial degrees of freedom, the spatial wavefunction $psi(x)$ represents the projection $langle x | psi rangle$.
The probability density of locating the particle in an infinitesimal interval $dx$ about point $x$ is given by:
P(x) dx = |psi(x,t)|^2 dx = psi^*(x,t) psi(x,t) dx