Page 257 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

248 Approximation methods

explicitly introduced, then the perturbed ground-state eigenfunction ø 0 to ®rst
order is
mù 1=4 ë
2
2
4
ø 0 ø (0) ‡ ø (1) ˆ 1 ÿ (4î ‡ 12î ÿ 9) e ÿî =2 (9:48)
0
0
ð" 16
As a second example, we suppose that the potential energy V for the
perturbed harmonic oscillator is
1=2
3
m ù 5
2
2 2
3
V ˆ kx ‡ cx ˆ mù x ‡ ë x 3 (9:49)
1
1
2 2 "
3
5
where c ˆ ë(m ù =") 1=2 is again a small quantity and ë is dimensionless. The
^
perturbation H9 for this example is
1=2
3
m ù 5
^
^
H9 ˆ H (1) ˆ ë x 3 (9:50)
"
3
The matrix elements hn9jx jni for the unperturbed harmonic oscillator are
given by equations (4.50). The ®rst-order correction term E (1) is obtained by
n
substituting equations (9.50) and (4.50e) into (9.24), giving the result
1=2
3
m ù 5
3
E (1) ˆ ë hnjx jniˆ 0 (9:51)
n "
Thus, the ®rst-order perturbation to the eigenvalue is zero. The second-order
term E (2) is evaluated using equations (9.34), (9.50), and (4.50), giving the
n
result
E n E (2)
n
^
^
^
^
jH (1) j 2 jH (1) j 2 jH (1) j 2 jH (1) j 2
n‡3,n
n‡1,n
nÿ1,n
nÿ3,n
ˆÿ ÿ ÿ ÿ
E (0) ÿ E (0) E (0) ÿ E (0) E (0) ÿ E (0) E (0) ÿ E (0)
nÿ3 n nÿ1 n n‡1 n n‡3 n
" #
3
3
3
3
ë m ù 5 hn ÿ 3jx jni 2 hn ÿ 1jx jni 2 hn ‡ 1jx jni 2 hn ‡ 3jx jni 2
2
3
ˆÿ ‡ ‡ ‡
" (ÿ3"ù) (ÿ"ù) "ù 3"ù
ˆÿ (30n ‡ 30n ‡ 11)ë "ù (9:52)
1
2
2
8
9.5 Degenerate perturbation theory
The perturbation method presented in Section 9.3 applies only to non-degen-
erate eigenvalues E (0) of the unperturbed system. When E (0) is degenerate, the
n
n
denominators vanish for those terms in equations (9.40) and (9.41) in which
(0) (0)
E k is equal to E , making the perturbations to E n and ø n indeterminate. In
n