Page 133 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

124 Harmonic oscillator

2 2 m"ù p 
hn ‡ 2j^ p jniˆhnj^ p jn ‡ 2iˆÿ (n ‡ 1)(n ‡ 2) (4:49a)
2
2
1
hnj^ p jniˆ m"ù(n ‡ ) (4:49b)
2

m"ù p 
2 2
hn ÿ 2j^ p jniˆhnj^ p jn ÿ 2iˆÿ n(n ÿ 1) (4:49c)
2
2
hn9j^ p jniˆ 0, n9 6ˆ n ‡ 2, n, n ÿ 2 (4:49d)
k
Following this same procedure using the operators (^ a ^ a) , we can ®nd the
y
k
k
matrix elements of x and of ^ p for any positive integral power k. In Chapters
4
3
9 and 10, we need the matrix elements of x and x . The matrix elements
3
hn9jx jni are as follows:
3=2
" p 
3 3
hn ‡ 3jx jniˆhnjx jn ‡ 3iˆ (n ‡ 1)(n ‡ 2)(n ‡ 3) (4:50a)
2mù
3=2
(n ‡ 1)"
3
3
hn ‡ 1jx jniˆhnjx jn ‡ 1iˆ 3 (4:50b)
2mù
3=2
n"
3
3
hn ÿ 1jx jniˆhnjx jn ÿ 1iˆ 3 (4:50c)
2mù
3=2
" p 
3 3
hn ÿ 3jx jniˆhnjx jn ÿ 3iˆ n(n ÿ 1)(n ÿ 2) (4:50d)
2mù
3
hn9jx jniˆ 0, n9 6ˆ n 1, n 3 (4:50e)
4
The matrix elements hn9jx jni are as follows
2
" p 
4 4
hn ‡ 4jx jniˆhnjx jn ‡ 4iˆ (n ‡ 1)(n ‡ 2)(n ‡ 3)(n ‡ 4)
2mù
(4:51a)
2
1 " p 
4 4
hn ‡ 2jx jniˆhnjx jn ‡ 2iˆ (2n ‡ 3) (n ‡ 1)(n ‡ 2) (4:51b)
2 mù
2
3 "
4 2 1
hnjx jniˆ n ‡ n ‡ (4:51c)
2 mù 2
2
1 " p 
4 4
hn ÿ 2jx jniˆhnjx jn ÿ 2iˆ (2n ÿ 1) n(n ÿ 1) (4:51d)
2 mù
2
" p 
4 4
hn ÿ 4jx jniˆhnjx jn ÿ 4iˆ n(n ÿ 1)(n ÿ 2)(n ÿ 3) (4:51e)
2mù
4
hn9jx jniˆ 0, n9 6ˆ n, n 2, n 4 (4:51f)