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19.4 Show that the Pauli matrices S. = th (1 6 ). s, - 1(" " ). s - # (8 9 ). which are used as the operators corresponding to intrinsic spin of sh in non- relativistic quantum mechanics, satisfy S; = S; = S; = th'l, and have the same commutation properties as the components of orbital angular momentum. Deduce that any state [y) represented by the column vector (a, b)' is an eigenstate of S' with eigenvalue 3h-/4. 19.5 Find closed-form expressions for cos C and sin C, where C is the matrix c- (1 4) Demonstrate that the 'expected" relationships cos' C + sin C =I and sin 20 = 2 sin CcosC are valid. 19.6 Operators A and B anticommute. Evaluate (A + B)?" for a few values of n and hence propose an expression for for in the expansion (A + B) = = > Car 4212 12. 7=0 Prove your proposed formula for general values of n, using the method of induction. Show that cos(A + B) = > > dur A 2-2 12. 1=0 720 where the dar are constants whose values you should determine. By taking as A the matrix A = ( ] ( ), confirm that your answer is consistent with that obtained in exercise 19.5. 19.7 Expressed in terms of the annihilation and creation operators A and A discussed in the text, a system has an unperturbed Hamiltonian Ho = hoA'A. The system is disturbed by the addition of a perturbing Hamiltonian H1 = gho(A + A! ), where g is real. Show that the effect of the perturbation is to move the whole energy spectrum of the system down by gho. 19.8 For a system of N electrons in their ground state (0), the Hamiltonian is H = S- 2m Show that [p.. . x,] = -2ihp.., and hence that the expectation value of the double commutator [ [ x. H ], x], where x = >=, *m, is given by (0 | [ [ x, H] ,x] |0) = Nh m