Proof That Commuting Operators Have Simultaneous Eigenfunctions

EXAMPLE 6-3 Two normalized l AOs are located on nearby nuclei, A and B, and overlap each other enough so that / sA sBdv = 0.500. Construct a function from these two that is orthogonal to l. and is normalized. 

Q 6-11 Proof That Commuting Operators Have Simultaneous Eigenfunctions 

A and B are commuting operators if, for the general square-integrable function /, ABf =BA f. This can be written (J.S —. si)/=0, which requires that i.S —. si=6. (6 is called the null operator. It satisfies the equation, 6/ = 0.) This difference of operator products is called the commutator of A and B and is usually symbolized by [A, B], If the commutator [A, S] vanishes, then A and B commute. 

Another example concerns the familiar symmetry operations for reflection, rotation, etc. If one of these operations, symbolized R, commutes with tbe bamiltonian, then we should expect there to be a set of eigenfunctions for H that are simultaneously eigenfunctions for R. It was proved in Chapter 2 that this means that nondegenerate eigenfunctions must be symmetric or antisymmetric with respect to R. 

The existence of simultaneous eigenfunctions for various operators has important ramifications for the measurement of a system s properties. This is discussed in Section 6-15. 

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