Page 88 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.5 Simultaneous eigenfunctions 79
(1)
(1)
ø (1) ˆ c ø i1 ‡ c ø i2
i
2
1
ø (2) (2) (2)
2
1
i ˆ c ø i1 ‡ c ø i2
which satisfy the relations
(1)
^
Bø (1) ˆ â ø (1)
i i i
(2)
^
Bø (2) ˆ â ø (2)
i
i
i
and are, therefore, simultaneous eigenfunctions of the commuting operators A ^
^
and B.
This analysis can be extended to three or more operators. If three operators
^
^ ^
A, B, and C have a complete set of simultaneous eigenfunctions, then the
^
^
^
^
^
argument above shows that A and B commute, B and C commute, and A and C ^
^
^
commute. Furthermore, the converse is also true. If A commutes with both B
^
^
^
and C, and B commutes with C, then the three operators possess simultaneous
eigenfunctions. To show this, suppose that the three operators commute with
^
^
one another. We know that since A and B commute, they possess simultaneous
eigenfunctions ø i such that
^
Aø i ˆ á i ø i
^
Bø i ˆ â i ø i
^
We next operate on each of these expressions with C, giving
^ ^
^ ^
^
^
CAø i ˆ A(Cø i ) ˆ C(á i ø i ) ˆ á i (Cø i )
^ ^
^^
^
^
CBø i ˆ B(Cø i ) ˆ C(â i ø i ) ˆ â i (Cø i )
^ ^ ^
Thus, the function Cø i is an eigenfunction of both A and B with eigenvalues á i
and â i , respectively. If á i and â i are non-degenerate, then there is only one
^
eigenfunction ø i corresponding to them and the function Cø i is proportional
to ø i
^
Cø i ˆ ã i ø i
^ ^
^
and, consequently, A, B, and C possess simultaneous eigenfunctions. For
degenerate eigenvalues á i and/or â i , simultaneous eigenfunctions may be
constructed using a procedure parallel to the one described above for the
doubly degenerate two-operator case.
^
^
^
^
^
We note here that if A commutes with B and B commutes with C, but A does
^
^
^
^
not commute with C, then A and B possess simultaneous eigenfunctions, B and
^
^
^
C possess simultaneous eigenfunctions, but A and C do not. The set of
^
^
^
^
simultaneous eigenfunctions of A and B will differ from the set for B and C.
An example of this situation is discussed in Chapter 5.