Page 104 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.8 Parity operator 95
^
^ ^
^ 2
Ð ø(q) ˆ Ð(Ðø(q)) ˆ Ðø(ÿq) ˆ ø(q)
Further, we see that
^ n
Ð ø(q) ˆ ø(q), n even
ˆ ø(ÿq), n odd
or
^ n
Ð ˆ 1, n even
^
ˆ Ð, n odd
^
The operator Ð is linear and hermitian. In the one-dimensional case, the
^
hermiticity of Ð is demonstrated as follows

1 ÿ1
^
höjÐjøiˆ ö (x)ø(ÿx)dx ˆÿ ö (ÿx9)ø(x9)dx9
ÿ1 1

1
^ ^
ˆ ø(x9)Ðö (x9)dx9 ˆhÐöjøi
ÿ1
where x in the second integral is replaced by x9 ˆÿx to obtain the third
integral. By applying the same procedure to each coordinate, we can show that
^
Ð is hermitian with respect to multi-dimensional functions.
^
The eigenvalues ë of the parity operator Ð are given by
^
Ðø ë (q) ˆ ëø ë (q) (3:65)
^
where ø ë (q) are the corresponding eigenfunctions. If we apply Ð to both sides
of equation (3.65), we obtain
^
^ 2
2
Ð ø ë (q) ˆ ëÐø ë (q) ˆ ë ø ë (q)
^ 2
2
Since Ð ˆ 1, we see that ë ˆ 1 and that the eigenvalues ë, which must be
^
real because Ð is hermitian, are equal to either ‡1or ÿ1. To ®nd the
eigenfunctions ø ë (q), we note that equation (3.65) now becomes
ø ë (ÿq) ˆÐ ø ë (q)
^
For ë ˆ 1, the eigenfunctions of Ð are even functions of q, while for ë ˆÿ1,
they are odd functions of q. An even function of q is said to be of even parity,
^
while odd parity refers to an odd function of q. Thus, the eigenfunctions of Ð
are any well-behaved functions that are either of even or odd parity in their
cartesian variables.
^
We show next that the parity operator Ð commutes with the Hamiltonian
^
operator H if the potential energy V(q) is an even function of q. The kinetic
energy term in the Hamiltonian operator is given by
!
" 2 " 2 @ 2 @ 2
2
ÿ = ˆÿ ‡ ‡
2m 2m @q 2 @q 2
1 2