Page 132 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.4 Matrix elements 123

hn9j^ pjniˆÿhnj^ pjn9i
2
The matrix element hn9jx jni is
" "
2 y 2 y 2 y y 2
hn9jx jniˆ hn9j(^ a ‡ ^ a) jniˆ hn9j(^ a ) ‡ ^ a ^ a ‡ ^ a^ a ‡ ^ a jni
2mù 2mù
From equation (4.34) we have

p p 
y 2  y
(^ a ) jniˆ n ‡ 1^ a jn ‡ 1iˆ (n ‡ 1)(n ‡ 2)jn ‡ 2i
p 
y y
^ a ^ ajniˆ n^ a jn ÿ 1iˆ njni
(4:47)
p 
y
^ a^ a jniˆ n ‡ 1^ ajn ‡ 1iˆ (n ‡ 1)jni
p p 
2 
^ a jniˆ n^ ajn ÿ 1iˆ n(n ÿ 1)jn ÿ 2i
so that
" p 
2
hn9jx jniˆ [ (n ‡ 1)(n ‡ 2)ä n9,n‡2 ‡ (2n ‡ 1)ä n9n
2mù
p 
‡ n(n ÿ 1)ä n9,nÿ2 ]

We conclude that
" p 
2 2
hn ‡ 2jx jniˆhnjx jn ‡ 2iˆ (n ‡ 1)(n ‡ 2) (4:48a)
2mù
"
2 1
hnjx jniˆ (n ‡ ) (4:48b)
mù 2
" p 
2 2
hn ÿ 2jx jniˆhnjx jn ÿ 2iˆ n(n ÿ 1) (4:48c)
2mù
2
hn9jx jniˆ 0, n9 6ˆ n ‡ 2, n, n ÿ 2 (4:48d)
2
The matrix element hn9j^ p jni is obtained from equations (4.43b) and (4.47)

m"ù m"ù
2 y 2 y 2 y
hn9j^ p jniˆÿ hn9j(^ a ÿ ^ a) jniˆÿ hn9j(^ a ) ÿ ^ a ^ a
2 2
y 2
ÿ ^ a^ a ‡ ^ a jni

m"ù p 
ˆÿ [ (n ‡ 1)(n ‡ 2)ä n9,n‡2 ÿ (2n ‡ 1)ä n9n
2
p 
‡ n(n ÿ 1)ä n9,nÿ2 ]
so that