Page 124 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.3 Eigenfunctions 115

and
y
^ a jniˆ c9 n jn ‡ 1i (4:32b)
where c n and c9 n are proportionality constants, dependent on the value of n, and
to be determined by the requirement that jn ÿ 1i, jni, and jn ‡ 1i are normal-
ized. To evaluate the numerical constants c n and c9 n, we square both sides of
equations (4.32a) and (4.32b) and integrate with respect to î to obtain

1 1
2 2
j^ aö n j dî ˆjc n j ö nÿ1 ö nÿ1 dî (4:33a)
ÿ1 ÿ1
and

1 1
y 2 2
j^ a ö n j dî ˆjc9 n j ö n‡1 ö n‡1 dî (4:33b)
ÿ1 ÿ1
The integral on the left-hand side of equation (4.33a) may be evaluated as
follows

1 1
2 y ^
j^ aö n j dî ˆ (^ aö )(^ aö n )dî ˆhnj^ a ^ ajniˆhnjNjniˆ n
n
ÿ1 ÿ1
Similarly, the integral on the left-hand side of equation (4.33b) becomes

1 1
y 2 y y y ^
j^ a ö n j dî ˆ (^ a ö )(^ a ö n )dî ˆhnj^ a^ a jniˆhnjN ‡ 1jniˆ n ‡ 1
n
ÿ1 ÿ1
Since the eigenfunctions are normalized, we obtain
2
2
jc n j ˆ n, jc9 n j ˆ n ‡ 1
Without loss of generality, we may let c n and c9 n be real and positive, so that
equations (4.32a) and (4.32b) become
p 
^ ajniˆ njn ÿ 1i (4:34a)
p 
y
^ a jniˆ n ‡ 1jn ‡ 1i (4:34b)
If the normalized eigenvector jni is known, these relations may be used to
obtain the eigenvectors jn ÿ 1i and jn ‡ 1i, both of which will be normalized.

Excited-state eigenfunctions
We are now ready to obtain the set of simultaneous eigenfunctions for the
^
^
commuting operators N and H. The ground-state eigenfunction j0i has already
been determined and is given by equation (4.31). The series of eigenfunctions
j1i, j2i, ... are obtained from equations (4.34b) and (4.18b), which give

d
jn ‡ 1iˆ [2(n ‡ 1)] ÿ1=2 î ÿ jni (4:35)
dî
Thus, the eigenvector j1i is obtained from j0i