The wave function is an abstract mathematical concept and cannot be "measured" directly. So what is it then?
The modern theory of quantum mechanics is formulated using rigorous mathematical formalisms. In this modern formulation, the wave function, commonly denoted as , 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 discrete degrees of freedom and continuous variables , the wave function can be written as . The wave function is an abstract mathematical construct and cannot be "measured" directly. The squared modulus of the wave function, , is interpreted as the probability density. In other words, defines a probability distribution and therefore, for every , it satisfies
The mathematical formalism of quantum mechanics defines an inner product on the space of all wave functions. For any two wave functions and , the inner product 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 and is interpreted as the probability of the wave function collapsing to the wave function :