Page 86 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 86

3.5 Simultaneous eigenfunctions                 77
                                           …
                                       X
                        f (q 1 , q 2 , ...) ˆ  ø (q9 1 , q9 2 , ...)f (q9 1 , q9 2 , ...)w(q9 1 , q9 2 , ...)dq9 1 ,dq9 2 , ...
                                              i
                                        i
                                       3 ø i (q 1 , q 2 , ...)
                        Interchanging the order of summation and integration gives
                                                … "                              #
                                                  X
                                 f (q 1 , q 2 , ...) ˆ  ø (q9 1 , q9 2 , ...)ø i (q 1 , q 2 , ...)
                                                        i
                                                    i
                                                3 f (q9 1 , q9 2 , ...)w(q9 1 , q9 2 , ...)dq 1 dq 2 ...
                        so that the completeness relation takes the form
                                      X

                         w(q9 1 , q9 2 , ...)  ø (q9 1 , q9 2 , ...)ø i (q 1 , q 2 , ...) ˆ ä(q 1 ÿ q9 1 )ä(q 2 ÿ q9 2 ) ...
                                           i
                                       i
                                                                                          (3:32)



                                            3.5 Simultaneous eigenfunctions
                        Suppose the members of a complete set of functions ø i are simultaneously
                                                               ^
                                                                     ^
                        eigenfunctions of two hermitian operators A and B with eigenvalues á i and â i ,
                        respectively
                                                       ^
                                                      Aø i ˆ á i ø i
                                                      ^
                                                      Bø i ˆ â i ø i
                                                                                              ^
                                                                      ^
                        If we operate on the ®rst eigenvalue equation with B and on the second with A,
                        we obtain
                                                ^ ^       ^
                                                BAø i ˆ á i Bø i ˆ á i â i ø i
                                                 ^^       ^
                                                ABø i ˆ â i Aø i ˆ á i â i ø i
                        from which it follows that
                                               ^^
                                                    ^ ^
                                              (AB ÿ BA)ø i ˆ [A, B]ø i ˆ 0
                                                                                      ^ ^
                        Thus, the functions ø i are eigenfunctions of the commutator [A, B] with
                        eigenvalues equal to zero. An operator that gives zero when applied to any
                                                                               ^
                                                                                     ^
                        member of a complete set of functions is itself zero, so that A and B commute.
                                                                ^
                                                                       ^
                        We have just shown that if the operators A and B have a complete set of
                                                       ^
                                                             ^
                        simultaneous eigenfunctions, then A and B commute.
                          We now prove the converse, namely, that eigenfunctions of commuting
                        operators can always be constructed to be simultaneous eigenfunctions.
                                     ^
                                                                          ^
                                                          ^ ^
                                                                                 ^
                        Suppose that Aø i ˆ á i ø i and that [A, B] ˆ 0. Since A and B commute, we
                        have
   81   82   83   84   85   86   87   88   89   90   91