In this problem we represent the spin eigenfunctions and operators as vectors and matrices.

a. The spin eigenfunctions are often represented as the column vectors

Show that α and β are orthogonal using this representation.

b. If the spin angular momentum operators are represented by the matrices

show that the commutation rule [ŝ_x ,ŝ_y] = ihŝ_z holds.

c. Show that

d. Show that α and β are eigenfunctions of sÌ‚_z and sÌ‚₂. What are the eigenvalues?

e. Show that α and β are not eigenfunctions of ŝx and ŝy.

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