In quantum mechanics, the collapse of the wave function is described by the Born rule, which gives the probabilities for different measurement outcomes based on the initial quantum state. According to the Born rule, the probabilities are calculated using the wave function of the system before the measurement and are related to the squared magnitudes of the possible measurement outcomes.
When two particles are entangled, their quantum states are mathematically described by a joint or composite wave function that encompasses both particles. This joint wave function represents the entangled state of the system. When a measurement is made on one of the entangled particles, the wave function of the system collapses instantaneously, and the other particle's wave function also collapses. The outcome of the measurement on the first particle then becomes correlated with the collapsed state of the second particle.
The concept of the collapse of the wave function is a fundamental aspect of quantum mechanics and has been extensively tested through experiments. These experiments have shown that the correlations between entangled particles are stronger than what can be explained by classical theories, confirming the existence of entanglement and the collapse of the wave function upon measurement.
However, it's important to note that the collapse of the wave function is inherently a probabilistic process. While the outcome of a measurement is determined by the collapsed state of the wave function, the exact time of collapse cannot be inferred from either particle. The collapse is a non-local process, meaning that the effects of a measurement on one particle can be observed instantaneously in the other entangled particle, regardless of the spatial separation between them. This phenomenon has been experimentally verified in tests of Bell's inequalities and is a characteristic feature of entanglement.
It's worth mentioning that there are ongoing debates and research exploring alternative interpretations of quantum mechanics that propose different explanations for the collapse of the wave function. However, the standard understanding based on the Copenhagen interpretation, which incorporates the collapse of the wave function upon measurement, has been highly successful in explaining and predicting the results of experiments in quantum physics.