Page 327 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix G

Series solutions of differential equations

General procedure
The application of the time-independent Schrodinger equation to a system of chemical
È
interest requires the solution of a linear second-order homogeneous differential equa-
tion of the general form
2
d u(x) du(x)
p(x) ‡ q(x) ‡ r(x)u(x) ˆ 0 (G:1)
dx 2 dx
where p(x), q(x), and r(x) are polynomials in x and where p(x) does not vanish in
some interval which contains the point x ˆ 0. Equation (G.1) is linear because each
term contains u or a derivative of u to the ®rst power only. The order of the highest
derivative determines that equation (G.1) is second-order.Ina homogeneous differ-
ential equation, every term contains u or one of its derivatives.
The Frobenius or series solution method for solving equation (G.1) assumes that the
solution may be expressed as a power series in x
1
X
s
u ˆ a k x k‡s ˆ a 0 x ‡ a 1 x s‡1 ‡ (G:2)
kˆ0
where a k (k ˆ 0, 1, 2, ...) and s are constants to be determined. The constant s is
chosen such that a 0 is not equal to zero. The ®rst and second derivatives of u are then
given by
1
du X k‡sÿ1 sÿ1 s
ˆ u9 ˆ a k (k ‡ s)x ˆ a 0 sx ‡ a 1 (s ‡ 1)x ‡ (G:3)
dx
kˆ0
2
1
d u X k‡sÿ2 sÿ2 sÿ1
ˆ u0 ˆ a k (k ‡ s)(k ‡ s ÿ 1)x ˆ a 0 s(s ÿ 1)x ‡ a 1 (s ‡ 1)sx ‡
dx 2
kˆ0
(G:4)
A second-order differential equation has two solutions of the form of equation (G.2),
each with a different set of values for the constant s and the coef®cients a k .
Not all differential equations of the general form (G.1) possess solutions which can
1
be expressed as a power series (equation (G.2)). However, the differential equations
encountered in quantum mechanics can be treated in this manner. Moreover, the power
1 For a thorough treatment see F. B. Hildebrand (1949) Advanced Calculus for Engineers (Prentice-Hall,
New York).

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