Physical Chemistry , 1st ed.

The influence of the spherical harmonics part of the complete wavefunction is

seen in the second term on the left. Although this is a differential equation in

terms ofronly, the quantum number is present. This suggests that the solu-

tion to this differential equation depends on the quantum number just like

the quantum number mdepends on in the spherical harmonics.

The solutions to the differential equation 11.60 were known, just as the

spherical harmonics were known. One part of the solution ofRis an expo-

nential function with a negative exponent, similar to the solution for the har-

monic oscillator. The exponential that works in this case is er/na,where nis a

positive integer and ais the collection of constants given by

a^4 



e

0

2

^2

where all the constants in the definition ofahave their usual meanings. We

will see more of this expression later. These exponentials are multiplying a

polynomial, again in a situation similar to the harmonic oscillator wavefunc-

tions, which composes the rest of the solution to the differential equation

11.60. The polynomial is one of a set of polynomials that have varying num-

bers of terms and are called the associated Laguerre polynomials.A positive in-

teger index labels each associated Laguerre polynomial, and it is usually indi-

cated by the letter n. This nhas the same value as the nin the exponential part

of the solution for R. Also, for each nthere may be several Laguerre polyno-

mials, each one having a different value of(the quantum number from 3-D

rotational motion). But not just any value of: the associated Laguerre poly-

nomials restrict the possible values ofto any integer such that

n

Therefore, the integer nrestricts the possible integer values of(with 0 being

the minimum value of). Since nis a positive integer, there is a simple series

ofvalues for each n:

n possible ’s

1 0

2 0,1

3 0,1,2

4 0,1,2,3

5 0,1,2,3,4

..

..

..

Since the possible values ofmare restricted by the specific value of,nulti-

mately restricts the values ofmalso. However, we again see that the restric-

tion arises due to the inherent restrictions on the allowed mathematical solu-

tions of the Schrödinger equation.

The complete wavefunction for the hydrogen atom is a combination of

the spherical harmonic,Ym(1/ 2 )eim,m, and the exponential-

associated Laguerre polynomial combination, which is denoted Rn,:

(r,,) Rn, Ym

1

2 

eim,m Rn, (11.61)

with the following restrictions:

n1,2,3,...

n (11.62)

m

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