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Prove that U is unitary 2. Consider an ensemble of electrons with an isotropic distribution of spin values. (a) What is the probability of finding an electron, chosen at random, with the following spin components? [6] i. S = +h/2, -ħ/2 ii. Sy=+h/2, -h/2 iii. S₂ = +h/2, -ħ/2 (b) In the coupled basis of angular momentum eigenstates, the total angular momentum eigen- states can be expanded as [16] lj,m) = [ Σ m1+m2=m (i1, j2; m₁, m₂|j, m) |j1, j2; m1, m₂). Write down all the allowed expansions of total angular momentum eigenstates for one such electron with l 2 and for the smallest value of j. Use the table of Clebsch-Gordan coefficients to simplify the associated coefficients of your expansion. = and first excited states for the