Physical Chemistry , 1st ed.

where pxand pyare the linear momenta in the xand ydirections. At this time,

we are ignoring the vector property of the momenta (except for its zdirection)

for the sake of simplicity.

In terms of the angular momentum, the kinetic energy of a particle having

mass mand revolving at a distance rabout a center is

K

2 m

Lz^2

r^2



L

2 I

z^2 (11.31)

where Ihas been defined as mr^2 and is called the moment of inertia.(You

should be aware that there are different expressions for the moment of inertia

of a physical object depending on the shape of the object. The expression I

mr^2 is the moment of inertia for a single mass moving in a circular path.)

Quantum mechanically, since operators for linear momenta are defined, an

operator for the angular momentum can also be defined:

Lˆzixˆ





y

yˆ





x

 (11.32)

By analogy, therefore, one can write the Schrödinger equation for this system

in terms of equations 11.31 and 11.32 as



L

2

ˆz

I

2

E (11.33)

As useful as the angular operator will be, it is still not in its best form, since

using it in the Hamiltonian will still lead to an expression in terms ofxand y.

Instead of using Cartesian coordinates to describe the circular motion, we will

use polar coordinates. In polar coordinates,the entire two-dimensional space

can be described using a radius from the center,r, and an angle measured

from some specified direction (typically the positive xaxis). Figure 11.8 shows

how the polar coordinates are defined. In polar coordinates, the angular mo-

mentum operator has a very simple form:

Lˆzi







 (11.34)

By using this form of the angular momentum, the Schrödinger equation for

two-dimensional rotation becomes





2 I

2









2

 2 E (11.35)

Equation 11.35 shows that even though we call this system “two-dimensional

motion,” in polar coordinates only one coordinate is changing: the angle .

Equation 11.35 is a simple second-order differential equation that has known

analytic solutions for , which is what we are trying to find. The possible ex-

pressions for are

Aeim (11.36)

where the values of the constants Aand mwill be determined shortly,is the

polar coordinate introduced above, and iis the square root of1. (Do not

confuse the constant mwith the mass of a particle.) The astute reader will rec-

ognize that this wavefunction can be written in terms of (cos misin m),

but the exponential expression above is the more useful form.

Although the wavefunction above satisfies the Schrödinger equation, proper

wavefunctions must also have other properties. First, they must be bounded.

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



(r, )

r

y

x

Figure 11.8 Two-dimensional polar coordi-
nates are defined as a distance from an origin,r,
and an angle with respect to some arbitrary
direction. Here,is the angle made with the
positive xaxis.