Observables and their properties for a total system

An operator with the property exhibited in eqn (5.5) is said to be Hermitian if it satisfies this equation for all functions P defined in the function space in which the operator is defined. The mathematical requirement for Hermiticity of H expressed in eqn (5.5) places a corresponding physical requirement on the system—that there be a zero flux in the vector current through the surface S bounding the system To illustrate this and other properties of the total system we shall assume, without loss of generality, a form for H corresponding to a single particle moving under the influence of a scalar potential F(r) 

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y) 

For a bound system, is square integrable and, hence, and its derivatives vanish on all elements dS = dSn of the surface when the surface is removed to infinity. Thus the right-hand side of eqn (5.10) rushes for the total system with boundaries at infinity and H is Hermitian.Mowever, if the integration is limited to a subsystem 1 bounded by a surface which , 

Depending on the nature of the state function, some observables yield sharp values— F is an eigenfunction of the observable—while others yield 

0he equation of motion for the average value of an observable A can be obtained directly from Heisenberg s equation for A t) (Messiah 1958, p. 319). The result is 

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