 Thus one obtains the expectation value of formula_25 up to linear order in the perturbation.The brackets formula_27 mean an equilibrium average with respect to the Hamiltonian formula_28 Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for formula_29.The above expression is true for any kind of operators.