Expectation Values and Ensemble Averaging

As we saw in Chapter 6, in a spin system consisting of N nuclei, each of spin Y2, there are 2N spin product functions, labeled pt, tf 2, etc. (or alternatively in the ket notation, l , 2 , etc.). Each of the 2N wave functions /f describing the state of all nuclear spins in the molecule can always be described as a linear combination 

It is well known from basic quantum theory that the value of an observable quantity A, represented by the operator A, can be found for any pure state i as the expectation value, CT.)W  

Note that the matrix element Amn is a constant that is independent of the particular state of the molecule, and the products of the ck describe the extent to which each matrix element contributes to state i. 

For a macroscopic sample, consisting of a large number of chemically identical molecules, the spin product functions are the same for all individual molecules, but the expectation value of the operator must be averaged over all molecules (spin systems) in the ensemble. With the notation that has been introduced, this average may be obtained by considering the probability of occurrence of each pure state, as expressed in terms of the basis functions  

Equation 11.5 defines the elements of a matrix p, called the density matrix, which completely specifies the state of the ensemble, because it includes all the variables ofEq. 11.4. 

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