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9.1 Variation method                       233

                          Except for the restrictions stated above, the function ö, called the trial
                        function, is completely arbitrary. If ö is identical with the ground-state eigen-
                        function ø 0 , then of course the quantity E equals E 0 .If ö is one of the
                        excited-state eigenfunctions, then E is equal to the corresponding excited-state
                        energy and is obviously greater than E 0 . However, no matter what trial function
                        ö is selected, the quantity E is never less than E 0 .
                          To prove the variation theorem, we assume that the eigenfunctions ø n form
                        a complete, orthonormal set and expand the trial function ö in terms of that set
                                                         X
                                                     ö ˆ     a n ø n                       (9:3)
                                                           n
                        where, according to equation (3.28)
                                                      a n ˆhø n jöi                        (9:4)
                        Since the trial function ö is normalized, we have
                                          *                  +

                                           X          X           X X

                                 höjöiˆ        a k ø k     a n ø n  ˆ    a a n hø k jø n i
                                                                           k

                                             k        n            k   n
                                          X X              X
                                                                   2
                                       ˆ         a a n ä kn ˆ  ja n j ˆ 1
                                                  k
                                           k  n              n
                        We next substitute equation (9.3) into the integral for E in (9.2) and subtract
                        the ground-state energy E 0 , giving
                                                      X X
                                         ^                             ^
                            E ÿ E 0 ˆhöjH ÿ E 0 jöiˆ         a a n hø k jH ÿ E 0 jø n i
                                                               k
                                                       k   n
                                      X X                             X
                                                                             2
                                    ˆ        a a n (E n ÿ E 0 )hø k jø n iˆ  ja n j (E n ÿ E 0 )  (9:5)
                                              k
                                       k   n                           n
                        where equation (9.1) has been used. Since E n is greater than or equal to E 0 and
                           2
                        ja n j is always positive or zero, we have E ÿ E 0 > 0 and the theorem is
                        proved.
                          In the event that ö is not normalized, then ö in equation (9.2) is replaced by
                        Aö, where A is the normalization constant, and this equation becomes
                                                        2   ^
                                                E  jAj höjHjöi > E 0
                        The normalization relation is
                                                             2
                                               hAöjAöiˆjAj höjöiˆ 1
                        giving
                                                           ^
                                                       höjHjöi
                                                  E             > E 0                      (9:6)
                                                        höjöi
                          In practice, the trial function ö is chosen with a number of parameters ë 1 ,
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