Physical Chemistry , 1st ed.

The two exponential functions cancel each other out, leaving only the infini-

tesimal. The normalization is completed:

N^2 

2 

0

d 1

N^2  02  1

N^2 (2)  1

N^2 

2

1





N

1

2 



where again only the positive square root is used. The complete wavefunction

for two-dimensional rotational motion, then, is:

m

1

2 

eim m0,1,2,3,... (11.37)

The normalization constant is the same for all wavefunctions and does not de-

pend on the quantum number m. Figure 11.9 shows plots of the first few ’s.

The magnitudes of the ’s are reminiscent of circular standing waves, and

these are also suggestive of de Broglie’s picture of electrons in a circular orbital.

It is only suggestive, and this analogy is not meant to hint that this is a true

description of electron motion.

Now the energy eigenvalues of the system can be evaluated. It is given by

the Schrödinger equation, of course:





2 I

2









2

 2 E

By inserting the general form of the wavefunction given in equation 11.37, we get





2



I

2









2

 (^2) 

1

2 

eimE

1

2 

eim

The second derivative of the exponential is easily evaluated as m^2 eim.

(The constant 1/ 2 is not affected by the derivative.) Substituting and rear-

ranging the constants to keep the terms in the original wavefunction grouped

together:



m

2

2

I

^2



1

2 

eimE

1

2 

eim

This shows that the eigenvalue is m^2 ^2 /2I. Since the eigenvalue of the

Schrödinger equation corresponds to the energy observable, the conclusion

is that

E

m

2

2

I

^2

 (11.38)

where m0,1,2, etc. A certain specified mass at a fixed distance rhas a

certain moment of inertia I. Planck’s constant is a constant, so the only vari-

able in the expression for energy is an integer m. Therefore,the total energy of

a rotating particle is quantizedand depends on the quantum number m.The

following example shows how these quantities come together to yield units of

energy.

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