Page 328 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Series solutions of differential equations 319

series expansion of u is valid for many differential equations in which p(x), q(x), and/
2
or r(x) are functions other than polynomials, but such differential equations do not
occur in quantum-mechanical applications.
The Frobenius procedure consists of the following steps.
1. Equations (G.2), (G.3), and (G.4) are substituted into the differential equation (G.1)
to obtain a series of the form
1
X k‡sÿ2 k‡sÿ1 k‡s
a k [(k ‡ s)(k ‡ s ÿ 1) p(x)x ‡ (k ‡ s)q(x)x ‡ r(x)x ] ˆ 0
kˆ0
2. The terms are arranged in order of ascending powers of x to obtain
1
X k‡sÿ2
c k x ˆ 0 (G:5)
kˆá
where the coef®cients c k are combinations of the constant s, the coef®cients a k , and
the coef®cients in the polynomials p(x), q(x), and r(x). The lower limit á of the
summation is selected such that the coef®cients c k for k , á are identically zero,
but c á is not.
3. Since the right-hand side of equation (G.5) is zero, the left-hand side must also
equal zero for all values of x in an interval that includes x ˆ 0. The only way to
meet this condition is to set each of the coef®cients c k equal to zero, i.e., c k ˆ 0 for
k ˆ á, á ‡ 1, .. .
4. The coef®cient c á of the lowest power of x in equation (G.5) always has the form
c á ˆ a 0 f (s), where f (s) is quadratic in s because the differential equation is
second-order. The expression c á ˆ a 0 f (s) ˆ 0 is called the indicial equation and
has two roots, s 1 and s 2, assuming that a 0 6ˆ 0.
5. For each of the two values of s, the remaining expressions c k ˆ 0 for k ˆ á ‡ 1,
á ‡ 2, .. . determine successively a 1 , a 2 , .. . in terms of a 0 . Each value of s yields
a different set of values for a k ; one set is denoted here as a k , the other as a9 k .
6. The two mathematical solutions of the differential equation are u 1 and u 2
2
u 1 ˆ a 0 x [1 ‡ (a 1 =a 0 )x ‡ (a 2 =a 0 )x ‡ ]
s 1
2
u 2 ˆ a9 0 x [1 ‡ (a9 1 =a9 0 )x ‡ (a9 2 =a9 0 )x ‡ ]
s 2
where a 0 and a9 0 are arbitrary constants. Physical solutions are obtained by applying
boundary and normalization conditions to u 1 and u 2 .
7. For some differential equations, the two roots s 1 and s 2 of the indicial equation
differ by an integer. Under this circumstance, there are two possible outcomes: (a)
steps 1 to 6 lead to two independent solutions, or (b) for the larger root s 1 , steps 1
to 6 give a solution u 1 , but for the root s 2 the recursion relation gives in®nite values
for the coef®cients a k beyond some speci®c value of k and therefore these steps fail
to provide a second solution. For some other differential equations, the two roots of
2 See for example E. T. Whittaker and G. N. Watson (1927) A Course of Modern Analysis, 4th edition
(Cambridge University Press, Cambridge), pp. 194±8; see also the reference in footnote 1 of this
Appendix.