Physical Chemistry , 1st ed.

The postulate regarding eigenfunctions and eigenvalue equations gets more
specific: the only possiblevalues of the observables are those that are eigen-
values of the wavefunction when operated upon by the corresponding oper-
ator. No other values will be observed. Frequently, as we will see, this implies
that many observables on the atomic scale are quantized. In addition, not all
experimental quantities are determined by any given wavefunction. Rather, a
given wavefunction is an eigenfunction of some operators (and so we can de-
termine the values of those observables) but not an eigenfunction of other
operators.

Example 10.4

What is the value of the momentum observable if the wavefunction 

is ei^4 x?

Solution

According to the postulate stated above, the momentum is equal to the eigen-

value produced by the expression

i





x

ei^4 x

When this expression is evaluated, we get

i





x

ei^4 x(i)(i4)ei^4 x 4 ei^4 x

or, more succinctly,

i





x

ei^4 x 4 ei^4 x

where we have used i i1 to get the final expression. This is an eigen-

value equation with an eigenvalue of 4 . Therefore, the value of the

momentum from this wavefunction is  4 . Numerically, 4  equals

(4)(6.626 10 ^34 J s)/2

4.218 10 ^34 J s 4.218 10 ^34 kg m^2 /s.

In quantum mechanics, the eigenvalue equations that we will consider have
real numbers as values of eigenvalues. Although we have already seen eigen-
functions and operators with the imaginary root iin them, when solving for
the eigenvalue itself these imaginary parts must cancel out to yield a real
number for the eigenvalue.Hermitian operatorsare operators that always have
real (nonimaginary) numbers as eigenvalues (that is,Kin equation 10.2 will
always be a real number or a collection of constants that have real values). All
operators that yield quantum mechanical observables are Hermitian opera-
tors, since in order to be observed a quantity must be real. (Hermitian oper-
ators are named after Charles Hermite, a nineteenth-century French mathe-
matician.)

10.4 The Uncertainty Principle

Perhaps the most unusual part of quantum mechanics is the statement called
the uncertainty principle. Occasionally it is called Heisenberg’s uncertainty
principle or the Heisenberg principle, after the German scientist Werner

10.4 The Uncertainty Principle 279