Physical Chemistry , 1st ed.

Every term in equation 11.10 has the exponential ex

(^2) /2
in it, so it can be al-
gebraically divided out. Its residual influence on equation 11.10, in the form of
the ’s and x’s in the second derivative expression, is obvious. Further, the
terms in f(x),f(x), and f(x) can be grouped together and simplified so that
the substituted Schrödinger equation becomes [omitting the (x) part of the
power series f]
f 2 xf^2

m
2

E

f^0 (11.11)

This equation has terms arising from the power series f, its first derivative f,

and its second derivative f. The terms in ^2 x^2 fhave canceled. Since we are

assuming that fis a power series, we can actually write out, term by term, what

the derivatives are. Rewriting the original power series first, the derivatives are

f(x) 



n 0

cnxn (from 11.7 above)

f



n 1

ncnxn^1

f



n 2

n(n1)cnxn^2

The constants cnare unaffected by the derivation, since they are constants. The

starting value of the index nchanges with each derivative. In the first derivative,

since the first term of the original function fis constant, we lose the n 0

term. Now the n1 term is a constant, since the power ofxfor the n 1

term is now 0, that is,x^1 ^1 x^0 1. In the second derivative, the n1 term,

a constant in the fexpansion, itself becomes zero for the second derivative,

and so the summation starts at n2. You should satisfy yourself that this is

indeed the case, and that the above three expressions with the given summa-

tion boundaries are correct (of course, the infinity boundary does not change).

Since the first term in the summation for fbecomes 0 in f, the first deriv-

ative fdoes not change if we add a 0 as a first term and then start the sum-

mation at n0. Understand that this does not change f, since the first term,

the n0 term, is simply zero. But this does allow us to start the summation

at n0 instead ofn1 (the importance of this will be seen shortly).

Therefore, we can write fas

f



n 0

ncnxn^1 (11.12)

Again, this does not change the power series itself; it only changes the initial

value of the index n. The same tactic canbe taken with f, but mathematically

this will not lead anywhere. Rather, by doing a two-step redefinition of the in-

dex,we can achieve much more. Since the index nis simply a counting num-

ber used to label the terms in the power series, we can shift the index by sim-

ply redefining, say, an index ias i n2. Since this means that ni2,

the expression for the second derivative fcan be rewritten by substituting for

every n:

f



i 2  2

(i2)(i 2 1)ci+2xi^2 ^2

which simply becomes

f



i 0

(i2)(i1)ci+2xi

320 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom