Page 329 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

320 Appendix G

the indicial equation are the same (s 1 ˆ s 2 ) and therefore only one solution u 1 is
obtained. In those cases where steps 1 to 6 give only one solution u 1 , a second
3
solution u 2 may be obtained by a slightly more complex procedure. This second
solution has the form
1
X
u 2 ˆ cu 1 ln x ‡ c b k x k‡s 2
kˆ0
where c is an arbitrary constant and the coef®cients b k are related to the coef®cients
a k . However, a solution containing ln x is not well-behaved and the arbitrary
constant c is set equal to zero in quantum-mechanical applications.
8. The interval of convergence for each of the series solutions u 1 and u 2 may be
determined by applying the ratio test. For convergence, the condition

a k‡1
lim jxj , 1

k!1 a k
must be satis®ed. Thus, a series converges for values of x in the range
1 1
ÿ , x ,
R R
where R is de®ned by

a k‡1
R lim

k!1 a k
For R equal to zero, the corresponding series converges for ÿ1 , x , 1.If R
equals unity, the corresponding series converges for ÿ1 , x , 1.

Applications
È
In Chapters 4, 5, and 6 the Schrodinger equation is applied to three systems: the
harmonic oscillator, the orbital angular momentum, and the hydrogen atom, respec-
tively. The ladder operator technique is used in each case to solve the resulting
differential equation. We present here the solutions of these differential equations
using the Frobenius method.

Harmonic oscillator
The Schrodinger equation for the linear harmonic oscillator leads to the differential
È
equation (4.17)
2
d ö(î) 2 2E
ÿ ‡ î ö(î) ˆ ö(î) (G:6)
dî 2 "ù
If we de®ne ë by the relation
2E
2ë ‡ 1 ˆ (G:7)
"ù
and introduce this expression into equation (G.6), we obtain
2
ö0 ‡ (2ë ‡ 1 ÿ î )ö ˆ 0 (G:8)
3 See footnote 1 of this Appendix.