Page 164 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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Problems                              155
                                                B
                                                            m 5 3


                                                                  m 5 2

                                                                     m 5 1


                                                                      m 5 0


                                                                     m 5 21

                                                                  m 5 22


                                                           m 5 23


                        Figure 5.6 Possible orientations in a magnetic ®eld B of the orbital angular momentum
                        vector L for the case l ˆ 3




                                                       Problems
                                                       ^
                                                          ^
                                                             ^
                         5.1 Show that each of the operators L x , L y , L z is hermitian.
                         5.2 Evaluate the following commutators:
                                                                         ^
                                             ^
                                                            ^
                                ^
                            (a) [L x , x]  (b) [L x , ^ p x ]  (c) [L x , y]  (d) [L x , ^ p y ]
                                                                                         ^
                         5.3 Using the commutation relation (5.10b), ®nd the expectation value of L x for a
                            system in state jlmi.
                                                                            ^
                                                                     ^
                         5.4 Apply the uncertainty principle to the operators L x and L y to obtain an expres-
                                     ^
                                        ^
                            sion for ÄL x ÄL y. Evaluate the expression for a system in state jlmi.
                                                                          ^
                                                                ^
                                                ^ 2
                         5.5 Show that the operator J commutes with J x and with J y .
                                            ^
                                     ^
                         5.6 Show that J ‡ and J ÿ as de®ned by equations (5.18) are adjoints of each other.
                         5.7 Prove the relationships (5.19a)±(5.19g).
                         5.8 Show that the choice for c ÿ in equation (5.24) is consistent with c ‡ in equation
                            (5.22).
                                                                     ^
                                                               ^
                         5.9 Using the raising and lowering operators J ‡ and J ÿ, show that
                                                   ^
                                                               ^
                                               hjmjJ x jjmiˆhjmjJ y jjmiˆ 0
                        5.10 Show that
                                            ^ 2
                                                        ^ 2
                                                                            2
                                                                1
                                       hjmjJ jjmiˆhjmjJ jjmiˆ [j(j ‡ 1) ÿ m ]" 2
                                            x            y      2
                                                                                ^
                                                                                    ^
                                                                 ^
                                                                    ^
                        5.11 Show that jj, mi are eigenfunctions of [J x , J ‡ ] and of [J y , J ‡ ]. Find the
                            eigenvalues of each of these commutators.
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