Quantum distributions operator identities

Postulate 4 If is not an eigenfunction of the quantum mechanical operator, then a series of measurements on identical systems of particles will yield a distribution of results, such that Equation (3.32) will describe the average (or expectation ) value of the observable (assuming the wave function is normalized). 

In several cases, the polarizability distribution can be found by chemical intuition. For instance, in the case of naphthalene, which is made up of two identical fragments, the polarizability can be decomposed into two equivalent parts. Also, group or atom contributions can be deduced from a variety of schemes such as Stone s approach [74], the theory of atoms in molecules [75], the localization of molecular orbitals into chemical functions [76], atom/ bond additivity [77], the use of the acceleration gauge for the electric dipole operator [78], quantum mechanically determined induction energies [79], or calculated molecular quadrupole polarizabilities and their derivatives with respect to molecular deformations [80]. Several of these models consider charge... 

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

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