In quantum mechanics, the expectation value refers to the predicted average value of a measurable quantity, such as position, momentum, energy, or other physical properties, based on the wave function describing a system.
When a physical quantity is measured in a quantum system, the outcome is probabilistic. The expectation value represents the average result of multiple measurements of that quantity performed on identical systems prepared in the same state. It is calculated by taking the weighted average of all possible measurement outcomes, with the probability of each outcome determined by the square of the wave function (or the probability amplitude) associated with the corresponding eigenstate of the observable being measured. The concept of expectation value is fundamental in quantum mechanics, providing a way to predict the average outcome of measurements in quantum systems described by wave functions.
