Physical Chemistry , 1st ed.

has a known solution. It is



a

b

sin^2 xdx

2

x



4

1

sin 2 xba

where we have substituted into the general form of the integral for the con-

stants in this particular example (you should verify this substitution yourself ).

a.Evaluating for the region x0 to 0.5:

P2[0.25  0 (0 0)]

P2(0.25)

P0.50

which as a percentage is 50%. This should, perhaps, be expected: in one-half

of the region of interest, the probability of the electron being there is one half,

or 50%.

b.For x0.25 to 0.75:

P (^2) 0.375 
4

1

(1) 0.125 

4

1

(^1) 
P2(0.409)
P0.818
which means that the probability of finding the electron in the middle half
of this region is 81.8%—much greater than half! This result is a consequence
of the wavefunction being a sine function. It also illustrates some of the more
unusual predictions of quantum mechanics.
The Born interpretation makes obvious the necessity of wavefunctions be-
ing bounded and single-valued. If a wavefunction is not bounded, it approaches
infinity. Then the integral over that space, the probability, is infinite.
Probabilities cannot be infinite. Since probability of existence represents a
physical observable, it must have a specific value; therefore,’s (and their
squares) must be single-valued.
Because the wavefunction in this last example does not depend on time, its
probability distribution also does not depend on time. This is the definition of
a stationary state: a state whose probability distribution, related to (x)^2 by
the Born interpretation, does not vary with time.

10.6 Normalization

The Born interpretation suggests that there should be another requirement for
acceptable wavefunctions. If the probability for a particle having wavefunction
were evaluated over the entire space in which the particle exists, then the
probability should be equal to 1, or 100%. In order for this to be the case,
wavefunctions are expected to be normalized.In mathematical terms, a wave-
function is normalized if and only if







*d 1 (10.8)

The limits and are conventionally used to represent “all space,” al-
though the entire space of a system may not actually extend to infinity in both
directions. The integral’s limits would be modified to represent the limits of

10.6 Normalization 283