Physical Chemistry , 1st ed.

tions in terms that accepted the uncertainty principle, and the Born interpre-

tationis generally considered to be the correct way of thinking of. Because

of the uncertainty principle, Born suggested that we not think ofas indi-

cating a specificpath of an electron. It is very difficult to establish absolutely

that a particular electron is in a particular place at a particular time. Rather,

over a long period of time, the electron has a certain probability of being in a

certain region. The probability can be determined from the wavefunction .

Specifically, Born stated that the probability Pof an electron being in a certain

region between points aand bin space is

P

a

b

*d (10.7)

where * is the complex conjugateof(where every iin the wavefunction is

replaced with i),dis the infinitesimal of integration covering the dimen-

sional space of interest [dxfor one dimension, (dx dy) for two dimensions,

(dx dy dz) for three dimensions, and (r^2 sin dr dd) for spherical polar co-

ordinates], and the integral is evaluated over the interval of interest (between

points aand b, in this case). Note that * and are simply being multiplied

together (which is sometimes written as ^2 ). The operation of multiplication

is assumed the way the integrand (the part inside the integral sign) is written.

The Born interpretation also requires that a probability be evaluated over a

definite region,not a specific point, in space. Thus, we should think ofas an

indicator of the probabilitythat the electron will be in a certain region of space.

The Born interpretation affects the entire meaning of quantum mechanics.

Instead ofgiving the exact location of an electron, it will provide only the

probability of the location of an electron. For those who were content with un-

derstanding that they could calculate exactlywhere matter was in terms of

Newton’s laws, this interpretation was a problem since it denied them the abil-

ity to state exactlyhow matter was behaving. All they could do was state the

probabilitythat matter was behaving that way. Ultimately, the Born interpreta-

tion was accepted as the proper way to consider wavefunctions.

Example 10.6

Using the Born interpretation, for an electron having a one-dimensional

wavefunction  2 sin xin the range x0 to 1, what are the follow-

ing probabilities?

a.The probability that the electron is in the first half of the range, from

x0 to 0.5

b.The probability that the electron is in the middle half of the range, from

x0.25 to 0.75

Solution

For both parts, one needs to solve the following integral:

P

a

b

( 2 sin x)*( 2 sin x) dx

but between different initial and final limits. Since the wavefunction is a real

function, the complex conjugate does not change the function, and the inte-

gral becomes

P 2 

a

b

sin^2 xdx

where the constant 2 has been taken outside the integral sign. This integral

282 CHAPTER 10 Introduction to Quantum Mechanics

Figure 10.3 Max Born (1882–1970). His in-

terpretation of the wavefunction as a probability

rather than an actuality changed the common

understanding of quantum mechanics.

AIP Emilio Sergre Visual Archives (Gift of Jost Lemmerich)