Physical Chemistry , 1st ed.

This is not a problem (as inspection of the cosine/sine form of the wavefunc-
tion shows). They must be continuous and differentiable. Again, exponential
functions of this sort are mathematically well behaved.
They must also be single-valued, and this presents a potential problem.
Because the particle is traveling in a circle, it retraces its path after 360° or 2
radians. When it does so, the ‘’single-valued’’ condition of acceptable wave-
functions requires that the value of the wavefunction be the same when the
particle makes a complete circle. (This is also sometimes called a circular
boundary condition.) Mathematically, this is written as

() ( + 2)

We can use the form of the wavefunction in equation 11.36 and simplify in
steps:

AeimAeim(^2 )

eimeimeim2

1 e^2 im

where Aand eimhave been canceled out sequentially in each step, and in
the last step the symbols in the exponent have been rearranged. This last
equation is the key. It is probably better followed if we use Euler’s theorem
(eicos isin ) and write the imaginary exponential in terms of sine
and cosine:

e^2 imcos 2misin 2m 1

In order for this equation to be satisfied, the sine term must be exactly zero
(because the number 1 has no imaginary part to it) and the cosine term must
be exactly 1. This will occur only when 2mis equal to any multiple of 2(in-
cluding 0 and negative values):

2 m0, 2 , 4 , 6 ,...

This means that the number mmust have only whole number values:

m0,1,2,3,...

Thus, the constant min the exponential cannot be any arbitrary constant, but
it must be an integerin order to have a properly behaved wavefunction.
Therefore the wavefunctions are not just arbitrary exponential functions, but
a set of exponential functions where the exponents must have certain specified
values. The number mis a quantum number.
In order to normalize the wavefunction, we need to determine d and the
limits of the integral. Since the only thing changing is , the infinitesimal for
integration is simply d. The value ofgoes from 0 to 2before it starts to
repeat the space it is covering, so the limits of integration are 0 to 2. The nor-
malization of the wavefunction proceeds as follows.

N^2 

2 

0

(eim)*eimd 1

For the first time in these model systems, the complex conjugate changes
something in the wavefunction: it affects iin the exponent of the function. The
first exponent becomes negative:

N^2 

2 

0

eimeimd 1

11.6 Two-Dimensional Rotations 335