Page 87 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
P. 87
78 General principles of quantum theory
^^ ^ ^ ^ ^
ABø i BAø i B(á i ø i ) 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 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 c 1 ø i1 c 2 ø i2
^
We need to determine the coef®cients c 1 , c 2 such that Bø i â i ø i , that is
^
^
c 1 Bø i1 c 2 Bø i2 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 0
c 1 B 21 c 2 (B 22 ÿ â i ) 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
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 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