Expected values of derivatives

In order to show that the expected-value and derivative operations commute, we begin with the definition of the derivative in terms of a limit 16 

The two random variables on the right-hand side are completely determined by the two-time PDF, /u,u (V, V x, t, t ). Thus, the expected value of the time derivative can be defined by 

The expected value appearing in the final expression of (2.82) can be rewritten as 

15 As discussed by Pope (2000), a time average is a random variable, and thus a poor starting point for deriving a mathematically well defined closure theory  

16 The demonstration is done here for t. However, the same steps can be repeated for each of the components of x. 

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Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

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