The collapse of the wave function is a fundamental concept in quantum mechanics that occurs upon measurement of a quantum system. According to the Copenhagen interpretation, which is one of the interpretations of quantum mechanics, the wave function represents the probabilistic description of a quantum system until it is measured.
When a measurement is performed on a quantum system, the wave function "collapses" or "reduces" to a particular eigenstate of the measured observable. The result of the measurement corresponds to one of the possible eigenvalues of the observable, and the system is said to be in the corresponding eigenstate after the measurement.
Mathematically, the collapse of the wave function is described by the projection postulate or the measurement postulate in quantum mechanics. According to this postulate, if a measurement of an observable is made on a quantum system, the wave function of the system collapses to an eigenstate of that observable associated with the measured eigenvalue. The probability of obtaining a specific eigenvalue is given by the squared magnitude of the corresponding coefficient in the wave function.
The collapse of the wave function is often represented by the wave function "collapsing" into a spike or a peak at a specific value corresponding to the measurement outcome. However, it is important to note that the collapse itself is a mathematical formalism used to describe the change in the quantum state after measurement, and the underlying physical process is still a topic of debate and interpretation in quantum mechanics. Various interpretations offer different explanations and philosophical views on the nature of the collapse, such as the Copenhagen interpretation, the many-worlds interpretation, and the decoherence theory, among others.