Page 131 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

122 Harmonic oscillator
1=2
"
y
hn9jxjniˆ (hn9j^ a jni‡hn9j^ ajni)
2mù
1=2
" p  p 
ˆ ( n ‡ 1ä n9,n‡1 ‡ nä n9,nÿ1 )
2mù
so that
r 
(n ‡ 1)"
hn ‡ 1jxjniˆ (4:45a)
2mù
r 
n"
hn ÿ 1jxjniˆ (4:45b)
2mù
hn9jxjniˆ 0 for n9 6ˆ n ‡ 1, n ÿ 1 (4:45c)

If we replace n by n ÿ 1 in equation (4.45a), we obtain
r 
n"
hnjxjn ÿ 1iˆ
2mù
From equation (4.45b) we see that
hn ÿ 1jxjniˆ hnjxjn ÿ 1i
Likewise, we can show that

hn ‡ 1jxjniˆ hnjxjn ‡ 1i
In general, then, we have

hn9jxjniˆ hnjxjn9i
To ®nd the matrix element hn9j^ pjni, we use equations (4.43b) and (4.44) to
give

1=2
m"ù
y
hn9j^ pjniˆ i hn9j^ a ÿ ^ ajni
2
1=2
m"ù p  p 
ˆ i ( n ‡ 1ä n9,n‡1 ÿ nä n9,nÿ1 )
2
so that
r 
(n ‡ 1)m"ù
hn ‡ 1j^ pjniˆ i (4:46a)
2
r 
nm"ù
hn ÿ 1j^ pjniˆÿi (4:46b)
2
hn9j^ pjniˆ 0 for n9 6ˆ n ‡ 1, n ÿ 1 (4:46c)
We can easily show that