(Solution) - a Consider two kets and Suppose -(2025 Original AI-Free Solution)

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a. Consider two kets | ? > and | ? >. Suppose < ??| ?>, < ??| ?), and < ??| ?>, < ??|?), are all known, where | ??>, | ??), form a complete set of base kets. Find the matrix representation of the operator | ? > < ? | in that basis.
b. We now consider a spin ½ system and let | ? > and | ? > be | sz = h/2 | and s = h/2), respectively. Write down explicitly the square matrix that corresponds to Ia)(/I in the usual (s. diagonal) basis.