Expectation Value of Operators and Method to Calculate Expectation Values

In quantum mechanics, the expectation value of an operator corresponds to the average measurement of that operator on a wave function.

Mathematically, 

The expectation value of an operator A on a quantum state described by a wavefunction ψ is given by:

⟨A⟩ = ∫ ψ* A ψ dx

Here, ψ* represents the complex conjugate of the wavefunction ψ, A represents the operator, and the integral is taken over the entire region of interest. 

The expectation value is a complex number that combines both the real and imaginary parts.

Physically, 

Physically, the expectation value represents the average value that would be obtained by performing a large number of measurements of the corresponding observable on a large ensemble of identically prepared quantum systems. This is similar to what we call as average in classical mechanics. 

Average and the individual values in quantum mechanics: 

The expectation value can also be interpreted as the most probable outcome of a single measurement. However, due to the probabilistic nature of quantum mechanics, individual measurements may yield different results, but the average over many measurements will converge to the expectation value.