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Thus the outer product acting on a ket is just another ket; in other words, I,B ) (al can be regarded as an operator. 33) are equal, we may as well omit the dots and let 1 ,8 ) (aly) stand for the operator I ,B ) (al acting on ly) or, equivalently, the number (aly) multiplying 1 ,8 ) . ) Notice that the operator I ,B ) (a I rotates I y) into the direction of I ,B ). 35) then which is left as an exercise. In a second important illustration of the associative axiom, we note that (( ,B I) · (XIa)) = (( ,B I X) · (Ia)) .

29). 2. 12). 3 . 3 1 ) The matrix representation of an observable A becomes particularly simple if the eigenkets of A themselves are used as the base kets. First, we have A = LLi a") (a" IA i a') (a' l . :a 'la') (a'l a' Spin ! 34) � It is here instructive to consider the special case of spin systems. The base kets used are I Sz ; ±) , which we denote, for brevity, as I ± ) . 3. 1 1 ), can be written as 1 = 1 +) (+ 1 + 1 - ) (- 1 . ( 1 . 34), we must be able to write Sz as Sz = (h/2) [( 1 +) (+ 1 ) - ( 1 -) ( - 1 )] .

3) are postulated to form a complete set. 23 em, for example, whereas x is an operator. The state ket for an arbitrary physical state can be expanded in terms of { /x ' ) }: /a) = i: dx' lx') (x' /a). 4) We now consider a highly idealized selective measurement of the position ob­ servable. Suppose we place a very tiny detector that clicks only when the particle is precisely at x ' and nowhere else. Immediately after the detector clicks, we can say that the state in question is represented by j x') .

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A practical treatise on Fourier's theorem and harmonic analysis for physicists and engineers by Albert Eagle


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