The eigenfunctions α and β of the Hermitian operator S^z form a complete, orthonormal set, and any one-electron spin function can be written as c1α + c2β. We saw in Section 7.10 that functions can be represented by column vectors and operators by square matrices. For the representation that uses a and b as the basis functions,
a. Write down the column vectors that correspond to the functions a, b, and c1α + c2β;
b. Use the results of Section 10.10 to show that the matrices that correspond to S^x, S^y, S^z, and S^2 are

c. Verify that the matrices in (b) obey SxSy - SySx = ihSz [Eq. (10.2)].
d. Find the eigenvalues and eigenvectors of the Sx matrix.

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