Page 256 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.4 Perturbed harmonic oscillator 247
E (2)
n
^
^
^
^
jH (1) j 2 jH (1) j 2 jH (1) j 2 jH (1) j 2
nÿ4,n
n‡4,n
nÿ2,n
n‡2,n
ˆÿ (0) ÿ (0) ÿ (0) ÿ (0)
E nÿ4 ÿ E (0) E nÿ2 ÿ E (0) E n‡2 ÿ E (0) E n‡4 ÿ E (0)
n
n
n
n
" #
2
4
4
4
4
4
ë m ù 6 hn ÿ 4jx jni 2 hn ÿ 2jx jni 2 hn ‡ 2jx jni 2 hn ‡ 4jx jni 2
ˆÿ ‡ ‡ ‡
" 2 (ÿ4"ù) (ÿ2"ù) 2"ù 4"ù
2
2
3
1
ˆÿ (34n ‡ 51n ‡ 59n ‡ 21)ë "ù (9:45)
8
The perturbed energy E n to second order is, then
E n E (0) ‡ E (1) ‡ E (2)
n
n
n
3
2
1
2
2
3
ˆ (n ‡ )"ù ‡ (n ‡ n ‡ )ë"ù ÿ (34n ‡ 51n ‡ 59n ‡ 21)ë "ù
1
1
2 2 2 8
(9:46)
(2)
In the expression (9.45) for the second-order correction E , the summation
n
on the right-hand side includes all states k other than the state n, but only for
the states (n ÿ 4), (n ÿ 2), (n ‡ 2), and (n ‡ 4) are the contributions to the
(2) (2)
summation non-vanishing. For the two lowest values of n, giving E 0 and E ,
1
only the two terms k ˆ (n ‡ 2) and k ˆ (n ‡ 4) should be included in the
summation. However, the terms for the meaningless values k ˆ (n ÿ 2) and
k ˆ (n ÿ 4) vanish identically, so that their inclusion in equation (9.45) is
(2) (2)
valid. A similar argument applies to E 2 and E , wherein the term for the
3
meaningless value k ˆ (n ÿ 4) is identically zero. Thus, equation (9.46)
applies to all values of n and the perturbed ground-state energy E 0 , for
example, is
21 2
1
E 0 ( ‡ ë ÿ ë )"ù
3
2 4 8
The evaluation of the ®rst- and second-order corrections to the eigenfunc-
tions is straightforward, but tedious. Consequently, we evaluate here only the
®rst-order correction ø (1) for the ground state. According to equations (9.33),
0
(9.43), and (4.51), this correction term is given by
^
^
20
40
ø (1) ˆÿ H (1) ø (0) ÿ H (1) ø (0)
0 (0) (0) 2 (0) (0) 4
E ÿ E E ÿ E
2 0 4 0

4
2
4
ëm ù 3 h2jx j0i (0) h4jx j0i (0)
ˆÿ ø 2 ‡ ø 4
" 2"ù 4"ù
ë p 
(0)
ˆÿ p [6ø (0) ‡ 3ø ] (9:47)
4

2
4 2
If the unperturbed eigenfunctions ø (0) and ø (0) as given by equation (4.41) are
2
4