Page 125 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

116 Harmonic oscillator

d 2 2
ð
e
j1iˆ 2 ÿ1=2 î ÿ (ð ÿ1=4 ÿî =2 ) ˆ 2 1=2 ÿ1=4 îe ÿî =2
dî
the eigenvector j2i from j1i

d
2
2
1 1=2 ÿ1=4 ÿî =2 ÿ1=2 ÿ1=4 2 ÿî =2
j2iˆ î ÿ (2 ð îe ) ˆ 2 ð (2î ÿ 1)e
2 dî
the eigenvector j3i from j2i
d
2
2
ð
j3iˆ 6 ÿ1=2 î ÿ (2 ÿ1=2 ÿ1=4 (2î ÿ 1)e ÿî =2 )
dî
2
3
ð
ˆ 3 ÿ1=2 ÿ1=4 (2î ÿ 3î)e ÿî =2
and so forth, inde®nitely. Each of the eigenfunctions obtained by this procedure
is normalized.
When equation (4.18a) is combined with (4.34a), we have

d
jn ÿ 1iˆ (2n) ÿ1=2 î ‡ jni (4:36)
dî
Just as equation (4.35) allows one to go `up the ladder' to obtain jn ‡ 1i from
jni, equation (4.36) allows one to go `down the ladder' to obtain jn ÿ 1i from
jni. This lowering procedure maintains the normalization of each of the
eigenvectors.
Another, but completely equivalent, way of determining the series of eigen-
functions may be obtained by ®rst noting that equation (4.34b) may be written
for the series n ˆ 0, 1, 2, ... as follows

y
j1iˆ ^ a j0i
y 2
j2iˆ 2 ÿ1=2 y ÿ1=2 (^ a ) j0i
^ a j1iˆ 2
y 3
j3iˆ 3 ÿ1=2 y ÿ1=2 (^ a ) j0i
^ a j2iˆ (3!)
.
. .
Obviously, the expression for jni is
y n
jniˆ (n!) ÿ1=2 (^ a ) j0i
Substitution of equation (4.18b) for ^ a and (4.31) for the ground-state eigen-
y
vector j0i gives
d n
2
n
ð
jniˆ (2 n!) ÿ1=2 ÿ1=4 î ÿ e ÿî =2 (4:37)
dî
This equation may be somewhat simpli®ed if we note that