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78                   General principles of quantum theory
                                                 ^^      ^ ^     ^           ^
                                                 ABø i ˆ BAø i ˆ B(á i ø i ) ˆ á i Bø i
                                                                           ^
                                                   ^
                             Therefore, the function Bø i is an eigenfunction of A with eigenvalue á i .
                                                                                     ^
                               There are now two possibilities; the eigenvalue á i of A is either non-
                             degenerate or degenerate. If á i is non-degenerate, then it corresponds to only
                                                                                 ^
                             one independent eigenfunction ø i , so that the function Bø i is proportional to
                             ø i
                                                            ^
                                                           Bø i ˆ â i ø i
                                                                                                    ^
                             where â i is the proportionality constant and therefore the eigenvalue of B
                             corresponding to ø i . Thus, the function ø i is a simultaneous eigenfunction of
                                  ^
                                        ^
                             both A and B.
                               On the other hand, suppose the eigenvalue á i is degenerate. For simplicity,
                             we consider the case of a doubly degenerate eigenvalue á i ; the extension to n-
                             fold degeneracy is straightforward. The function ø i is then any linear combina-
                             tion of two linearly independent, orthonormal eigenfunctions ø i1 and ø i2 of A ^
                             corresponding to the eigenvalue á i

                                                        ø i ˆ c 1 ø i1 ‡ c 2 ø i2
                                                                              ^
                             We need to determine the coef®cients c 1 , c 2 such that Bø i ˆ â i ø i , that is
                                                           ^
                                                  ^
                                                c 1 Bø i1 ‡ c 2 Bø i2 ˆ â i (c 1 ø i1 ‡ c 2 ø i2 )
                             If we take the scalar product of this equation ®rst with ø i1 and then with ø i2 ,
                             we obtain
                                                     c 1 (B 11 ÿ â i ) ‡ c 2 B 12 ˆ 0

                                                     c 1 B 21 ‡ c 2 (B 22 ÿ â i ) ˆ 0
                             where we have introduced the simpli®ed notation
                                                                    ^
                                                         B jk  hø ij j Bø ik i
                             These simultaneous linear homogeneous equations determine c 1 and c 2 and
                             have a non-trivial solution if the determinant of the coef®cients of c 1 , c 2
                             vanishes


                                                      B 11 ÿ â i  B 12
                                                                           ˆ 0

                                                        B 21    B 22 ÿ â i
                             or
                                              2
                                             â ÿ (B 11 ‡ B 22 )â i ‡ B 11 B 22 ÿ B 12 B 21 ˆ 0
                                              i
                                                                               (2)
                             This quadratic equation has two roots â (1)  and â , which lead to two
                                                                     i
                                                                               i
                                                             (1)  (1)     (2)  (2)
                             corresponding sets of constants c , c 2  and c , c . Thus, there are two
                                                                          1
                                                             1
                                                                              2
                             distinct functions ø (1)  and ø (2)
                                               i
                                                       i
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