Physical Chemistry , 1st ed.

for their quantized values depends on the system.Average values,rather than

quantized values, may be all that can be determined. (3) All of these model sys-

tems have approximate analogs in reality, so that the conclusions obtained

from the analysis of these systems can be applied approximately to known

chemical systems (much in the same way ideal gas laws are applied to the be-

havior of real gases). (4) Classical mechanics was unable to rationalize these

observations of atomic and molecular systems. It is this last point that makes

quantum mechanics worth understanding in order to understand chemistry.

11.9 The Hydrogen Atom: A Central Force Problem

It is a very short jump from the 3-D rigid rotor to the hydrogen atom. Hydrogen

is nothing more than a nucleus (a single proton) and an electron “in orbit”

about the nucleus. For a two-particle system with the motion occurring rela-

tively (that is, the electron is usually considered as moving around the nu-

cleus), the reduced mass must be used in any expression where mass would

appear. Instead of simple electronic motion, it is more properly thought of as

motion of two particles about a common center of mass. (The reduced mass

is very similar to the mass of the electron, but the difference is measurable.)

The final part of the quantum-mechanical description of the hydrogen

atom system deals with the third spherical polar coordinate r. In the 3-D rigid

rotor, we assumed a constant r. In earlier treatments of atoms (specifically, the

Bohr theory of the hydrogen atom), electrons were naively assumed to have

fixed orbits about nuclei. Classical mechanics provided the background for

such an assumption. Consider a rock tied to the end of a rope, swirled above

your head. Since you grip the rope tightly, of course the rock spins at a con-

stant radius! It would be contrary to experience to think that the radius of the

rope changes as the weight spins. Other circular motion reinforces this rea-

soning: merry-go-rounds, Ferris wheels, automobile tires, spinning tops—

almost every circular motion in our experience occurs at some fixed distance

from an axis.

Consider the atomic scale, however. The uncertainty principle suggests that

specifying a certain position of an electron is incompatible with other observ-

ables that we use to describe the state of the electron, like momentum and en-

ergy. Maybe we can’tfix the electron to a certain radius.

A proper quantum-mechanical treatment of H makes no presumptions about

the distance of an electron from a nucleus.Thus, the description of the hydro-

gen atom is the same as for the 3-D rigid rotor except it includes variation of

r, which ranges from 0 to . This is illustrated in Figure 11.16. The hydrogen

atom is defined like a 3-D rigid rotor, only now the radius can vary also.

Wavefunctions describing the motion of an electron in a hydrogen atom must

therefore satisfy the 3-D spherical polar Schrödinger equation

 2





2



r

1

2 



r

r^2 





r



r^2 s

1

in









sin 









r^2 s

1

in^2 









2

 (^2) Vˆ
E (11.56)
where the form of the Hamiltonian operator reflects the fact that all three
spherical polar coordinates,r,, and , can vary. Note the relationship between
equations 11.56 and 11.46, where the spherical polar coordinate ris not chang-
ing. Also note that we are using the reduced mass in the Schrödinger equa-
tion, not the mass of the electron.
352 CHAPTER 11 Quantum Mechanics: Model Systems and the Hydrogen Atom
e

p^
r
Figure 11.16 The hydrogen atom, as defined
in quantum mechanics. This system is defined
similarly to the 3-D rigid rotor (Figure 11.11) ex-
cept now rcan vary.