Quantum average value theorem

One particular strength of perturbation theory is its intuitive simplicity. That may not be apparent when you ve just emerged from two pages of calculus, but the first- and second-order corrections to the energy are very useful conceptually. As we ve seen, the first-order correction is essentially the quantum average value theorem applied to the perturbation Hamiltonian H. In many cases. 

The derivation of the principle of statioiiary action for an atom in a molecule (eqn (8.143)) yields Schrodinger s equation of motion for the total system, identifies the observables of quantum mechanics with the variations of the state function, defines their average values, and gives their equations of motion. We have demonstrated in Chapter 6 how one can use the atomic statement of the principle of stationary action given in eqn (8.148) to derive the theorems of subsystem quantum mechanics and thereby obtain the mechanics of an atom in a molecule. The statement of the atomic action... 

According to the correspondence principle as stated by N. Bohr (1928), the average behavior of a well-defined wave packet should agree with the classical-mechanical laws of motion for the particle that it represents. Thus, the expectation values of dynamical variables such as position, velocity, momentum, kinetic energy, potential energy, and force as calculated in quantum mechanics should obey the same relationships that the dynamical variables obey in classical theory. This feature of wave mechanics is illustrated by the derivation of two relationships known as Ehrenfest s theorems. 

Alternatively, but with a highly phenomenological importance, the connection between the eigen-values and the averaged measured values of Eqs. (3.1) and (3.2), respectively, the variational theorem of the quantum mechanics can be established the state functions/for which the variation of the average measured value is zero, S(a) = 0, are the eigen-functions satisfying the Eq. (3.1). The reciprocal assertion is also valid. 

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