In quantum mechanics, the expectation value of a wave function represents the average value that a physical observable will have when a measurement is performed on a system described by that wave function. It provides a probabilistic prediction for the outcome of a measurement.
Mathematically, the expectation value of an observable, represented by an operator A, is calculated by taking the inner product (or the overlap) of the wave function ψ with the operator A applied to ψ, and then integrating over all possible values of the system's variables. It is denoted as ⟨A⟩ or ⟨ψ|A|ψ⟩.
Physically, the expectation value represents the mean value of a measurement outcome that we would obtain if we performed the measurement on an ensemble of identically prepared systems, each described by the same wave function ψ. It gives us information about the most probable outcome of the measurement.
Importantly, the expectation value is not necessarily equal to any particular measurement result; rather, it represents the average value we expect to obtain over repeated measurements on identically prepared systems. Quantum mechanics is inherently probabilistic, so different measurements on identically prepared systems can yield different outcomes, but the expectation value provides a statistical prediction for the long-term behavior.
For example, consider the expectation value of position (x) for a particle described by a wave function ψ(x). The expectation value ⟨x⟩ gives us the average position we would expect to measure if we performed repeated measurements on identically prepared particles. Similarly, the expectation value of momentum (p) gives the average momentum we would expect to measure.
The concept of expectation values is fundamental in quantum mechanics and plays a central role in predicting the statistical behavior of physical systems at the quantum level. It allows us to make probabilistic predictions about the outcomes of measurements, providing a bridge between the mathematical formalism of quantum mechanics and observable quantities in the physical world.