Expectation Value
The probabilistic average or expected value of many physical measurements performed on an ensemble of identically prepared quantum states.
Governing formula
⟨A⟩ = ∫ ψ* Â ψ dx / ∫ ψ* ψ dx
SI unit
Same as measured quantity
In depth
Because individual quantum measurements yield random eigenvalues according to probability distributions, the expectation value ⟨A⟩ provides the statistical mean value. Ehrenfest's theorem shows expectation values obey classical physics laws.
Examples in the real world
Calculating the average radial distance ⟨r⟩ = 1.5 a₀ of an electron in a 1s hydrogen atom orbital.