On the nature of the wave function

The modern theory of quantum mechanics is formulated using rigorous mathematical formalisms. In this modern formulation, the wave function, commonly denoted as $\Psi$, is defined as a function of the degrees of freedom of a quantum system such as the positions or momenta of particles and their spin, which describes the state of the system. A complex-valued function, the wave function assigns a complex number to each element of its domain, i.e. every point in space or every possible spin value of each particle.

Given $n$ discrete degrees of freedom ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ and $m$ continuous variables ${\omega }_1,{\omega }_2,\dots ,{\omega }_m$, the wave function can be written as $\Psi (\mathbf{\alpha },\mathbf{\omega },t)$. The wave function is an abstract mathematical construct and cannot be "measured" directly. The squared modulus of the wave function, $|\Psi {|}^2={\Psi }^{\ast }\Psi$, is interpreted as the probability density. In other words, $|\Psi {|}^2$ defines a probability distribution and therefore, for every $t$, it satisfies

The mathematical formalism of quantum mechanics defines an inner product on the space of all wave functions. For any two wave functions $\Phi$ and $\Psi$, the inner product $(\Phi ,\Psi )$ is defined as

Upon measurement of an observable, the wave function "collapses" to a new wave function. The modulus squared of the inner product of two wave functions $\Phi$ and $\Psi$ is interpreted as the probability of the wave function $\Psi$ collapsing to the wave function $\Phi$: