Physical Chemistry , 1st ed.

1. Single-valued (that is, a wavefunction must have only one possible F(x)
   value for each and every value ofx.)


2. Continuous


3. Differentiable (that is, there must be no mathematical reasons why the
   derivative ofcannot exist.)*
   Among other things, this last restriction prohibits functions that approach
   either positive or negative infinity, except maybe for individual points in the
   function. Another way to state this is that the function is bounded. For what-
   ever variable(s) exist in the wavefunctions, these limitations must be satisfied
   for the entire variable range. In some cases, the range of the variable may be
   to . In other cases, the variables may be limited to a certain range.
   Functions that meet all these criteria are considered acceptable wavefunc-
   tions. Those that do not may not provide any physically meaningful conclu-
   sions. Figure 10.1 shows some examples of acceptable and unacceptable
   wavefunctions.
   The final part of this first postulate is that all possible information about
   the various observable properties of a system must be derived from the
   wavefunction. This seems an unusual statement at first. Later in the chapter
   we can fully develop this idea. But the point should be made immediately
   upon introduction of the wavefunction: All information must be deter-
   mined solely from the function that is now defined as the wavefunction of
   the particle. This fact gives the wavefunction a central role in quantum me-
   chanics.

Example 10.1

Which of the following expressions are acceptable wavefunctions, and which

are not? For those that are not, state why.

a.f(x) x^2 1, where xcan have any value

b.f(x) x,x 0

c.



1

2 

sin^2

x

,

2

x

2

d.

4 

1

x

,0 x 10

e.

4 

1

x

,0 x 3

Solution

a.Not acceptable, because as xapproaches positive or negative infinity, the

function also approaches infinity. It is not bounded.

b.Not acceptable, because the function is not single-valued.

c.Acceptable, because it meets all criteria for acceptable wavefunctions.

d.Not acceptable, because the function approaches infinity for x4, which

is part of the range.

e.Acceptable, because the function meets all criteria for acceptable wave-

functions within that stated range of the variable x. (Compare this to the con-

clusion reached in part d.)

10.2 The Wavefunction 275

*Many ’s must also be square-integrable; that is, the integral of^2 must also exist.

However, this is not an absolute requirement.



To 

(a)



(b)



(c)



(d)
Figure 10.1 (a) An acceptable wavefunction is
continuous, single-valued, bounded, and inte-
grable. (b) This function is not single-valued and
is not an acceptable wavefunction. (c) This func-
tion is not continuous and is not an acceptable
wavefunction. (d) This function is not bounded
and is not an acceptable wavefunction.