Observables A and B are compatible experiment For any initial state, subsequent measurements of A and B produce certain outcomes which are consistent with the first measurements. math Common eigenstates of A and B form a complete basis. * ( = > [ A , B ] = 0 Dirac's theorem * with a simple example. Consider two Hermitian operators A and B with the matrix representation 10 0 0 0 0 A=A= 0 0i B = B =0 -1/2 -i/2 (1) 0 10 1/2 -1/2 in a three-dimensional Hilbert space; these matrices are defined with respect to basis vectors: | 1 ), | 2), | 3). (a) Show that the two operators commute by matrix multiplication. Since they commute, Dirac's theorem tells us it should be possible to find common eigenstates which form a complete basis, i.e., they span the Hilbert space. We label each common eigenstate according to the eigenvalues: Ala, b) = ala, b); Bla, b) = bla, b). (2) (b) Determine three orthogonal | a, b)'s, specifying the values of a and b within each ket, and express each | a, b ) as a column vector in the basis of | 1 ), | 2), 1 3).1 (c) Suppose a quantum state is initialized as | 2), and then a simultaneous measurement of A and B is made. What is the probability to obtain the measurement outcome (A, B) = (-1,0), according to Born's rule? (d) Suppose (A, B) = (-1,0) has been measured and | 2 ) has collapsed. Immediately after this measurement, a second simulta neous measurement of (A, B) is made. Specify the possible outcomes of (A, B) that are measurable with a nonzero probability, and specify each nonzero probability, according to Born's rule. (e) In one or two sentences, relate your findings in (c-d) to experiment in the top figure