In quantum mechanics, the average or mean value of an observable quantity is represented by the expectation value. The expectation value is the average value you would expect to measure if you performed repeated measurements of the same observable on identical quantum systems prepared in the same state.
Mathematically, the expectation value of an observable is obtained by taking the inner product of the quantum state with the corresponding operator representing the observable. For a quantum state represented by the wavefunction ψ(x), the expectation value of an observable A is given by the integral:
⟨A⟩ = ∫ ψ*(x) A ψ(x) dx
Here, ψ*(x) represents the complex conjugate of the wavefunction, A is the operator corresponding to the observable, and dx represents the infinitesimal element of position or other relevant variable.
The expectation value provides information about the average outcome of measuring the observable on an ensemble of identically prepared quantum systems. However, it's important to note that the result of a single measurement may not necessarily match the expectation value, as quantum mechanics incorporates inherent uncertainty due to the wave-like nature of particles and the process of measurement itself.