Page 332 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Series solutions of differential equations 323

Since the parameter ë is equal to a positive integer n, the energy E of the harmonic
oscillator in equation (G.7) is
1
E n ˆ (n ‡ )"ù, n ˆ 0, 1, 2, .. .
2
in agreement with equation (4.30). Setting ë in equation (G.10) equal to the integer n
gives
u0 ÿ 2îu9 ‡ 2nu ˆ 0 (G:17)
A comparison of equation (G.17) with (D.10) shows that the solutions u(î) are the
Hermite polynomials, whose properties are discussed in Appendix D. Thus, the
functions ö n (î) for the harmonic oscillator are
2
ö n (î) ˆ a n H n (î)e ÿî =2
where a n are the constants which normalize ö n (î). Application of equation (D.14)
yields the ®nal result
2
n
ð
ö n (î) ˆ (2 n!) ÿ1=2 ÿ1=4 H n (î)e ÿî =2
which agrees with equation (4.40).
Orbital angular momentum
We wish to solve the differential equation
^ 2 2 (G:18)
L ø(è, j) ˆ ë" ø(è, j)
^ 2
where L is given by equation (5.32) as
" #

1 @ @ 1 @ 2
^ 2 2 sin è ‡ (G:19)
L ˆÿ"
sin è @è @è sin è @j 2
2
We write the function ø(è, j) as the product of two functions, one depending only on
the angle è, the other only on j
ø(è, j) ˆ È(è)Ö(j) (G:20)
When equations (G.19) and (G.20) are substituted into (G.18), we obtain after a little
rearrangement

2
sin è d sin è dÈ 2 1 d Ö
È dè dè ‡ ë sin è ˆÿ Ö dj 2 (G:21)
The left-hand side of equation (G.21) depends only on the variable è, while the right-
hand side depends only on j. Following the same argument used in the solution of
equation (2.28), each side of equation (G.21) must be equal to a constant, which we
2
write as m . Thus, equation (G.21) separates into two differential equations

sin è d sin è dÈ 2 2
È dè dè ‡ ë sin è ˆ m (G:22)
and
2
d Ö ˆÿm Ö
2
dj 2 (G:23)
The solution of equation (G.23) is
Ö ˆ Ae imj (G:24)
where A is an arbitrary constant. In order for Ö to be single-valued, we require that
Ö(j) ˆ Ö(j ‡ 2ð)
or