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Unformatted text preview: If we multiply the first equation by and the second by then we obtain (65) and, using the fact that is an eigenfunction of and , this becomes (66) so that if we subtract the first equation from the second, we obtain (67) For this to hold for general eigenfunctions, we must have , or . That is, for two physical quantities to be simultaneously observable, their operator representations must commute. Section 8.8 of Merzbacher [ 2 ] contains some useful rules for evaluating commutators. They are summarized below. (68) (69) (70) (71) (72) (73) (74) If and are two operators which commute with their commutator, then (75) (76) We also have the identity (useful for coupled-cluster theory) (77) Finally, if then the uncertainties in A and B, defined as , obey the relation 1 (78) This is the famous Heisenberg uncertainty principle. It is easy to derive the well-known relation (79) from this generalized rule....
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- Fall '10