Physical Chemistry , 1st ed.

Mathematically, the function fhas not changed.What has changed is the index,
which has shifted by 2. It is the same second derivative function determined
originally.
Of course, it doesn’t matter what letter is used to designate the index. If that
is the case, why not use n? The second derivative fbecomes

f



n 0

(n2)(n1)cn+2xn (11.13)

which is the useful form of the second derivative.
The reason all this manipulation has taken place is so that when the sum-
mations are substituted into the Schrödinger equation, all terms can be grouped
under the same summation sign (and this cannot be done unless the summa-
tion index starts at the same number and means the same thing in all expres-
sions). Now the summations for f,f, and fcan be substituted into equation
11.11. The resulting equation is


n 0

(n2)(n1)cn+2xn 2 x



n 0

ncnxn^1 

2



m

2

E





n 0

cnxn 0

Because all of the summations in the above equation start at zero, go to infin-
ity, and use the same index, it can be rewritten as a single summation. This is
the reason for getting the indices to be the same for all summations. The equa-
tion becomes


n 0 

(n2)(n1)cn+2xn 2 xncnxn^1 

2



m

2

E

cnxn 0

This equation can be simplified by recognizing that the x’s in the second term
can be combined so that the power on xbecomes n, and further recognizing
that all three terms now have xraised to the power ofn. Making the combi-
nation and factoring xnout of all terms yields


n 0 

(n2)(n1)cn 2  2 ncn

2



m

2

E

cnxn 0 (11.14)

Now we need to determine the values of the constants cn. Recall that this
equation was determined by substituting a trial wavefunction into the
Schrödinger equation, so that if the harmonic oscillator system has wavefunc-
tions that are eigenfunctions of the Schrödinger equation, those wavefunctions
would be of the form given in equation 11.8 [that is,ex

(^2) /2
f(x)]. By
identifying the constants, we complete our determination of the wavefunctions
of a harmonic oscillator.
Equation 11.14 is an infinite sum that equals exactly zero. This is a some-
what remarkable conclusion: if one were to add up all infinite terms in the
sum, the total would be exactly zero. The only way to guarantee this for all val-
ues ofxis ifeverycoefficient multiplying xnin equation 11.14 were exactly
zero:
(n2)(n1)cn 2  2 ncn

2



m

2

E

cn0for any n

This does not mean that every coefficient cnis exactly zero [that would imply
that our power series f(x) is exactly zero]. It means that the entire expression
above must be zero. This requirement allows us to rewrite the above equation
to get a relationship between one coefficient cnand the coefficient two places
away,cn 2.

11.3 The Quantum-Mechanical Harmonic Oscillator 321