Answers for Assigned Problems

4.27ab. (a) A = 1/5
             (b) <S_x> = 0
                   <S_y> = -(12/25) hbar
                   <S_z> = -(7/50) hbar

4.29. (a) l = ± hbar/2
              Eigenvectors given in class
         (b) + hbar/2 with probability |a-ib|² / 2
              - hbar/2 with probability |a+ib|² / 2
         (c) hbar² / 4 with probability one.

4.31.

S_x =

(0 1 0)
(1 0 1) hbar/sqrt(2)
(0 1 0)

(0 -1  0)
(1  0 -1) i hbar/sqrt(2)
(0  1  0)

(1  0  0)
(0  0  0) hbar
(0  0 -1)

4.32a. P₊(t) = (1/2)[1 + sin a  cos(gB₀t)]

4.38 (a) Proof.
        (b) The degeneracy of the nth level is (n+1)(n+2)/2 .

B. [p_i, L²] = i hbar S_jm { e_ijm [2 p_mL_j + i hbar S_n e_jmnp_n] } ¹ 0

Last revised 2005/04/18