10.9 Average Values and Other Properties
There are other common observables in addition to energy. One could oper-
ate on the wavefunction with the position operator,xˆ, which is simply multi-
plication by the coordinate x, but multiplying the sine functions of the parti-
cle-in-a-box by the coordinate xdoes not yield an eigenvalue equation.The’s
of equation 10.11 are not eigenfunctions of the position operator.
This should not be cause for concern. The postulates of quantum mechan-
ics do not require that acceptable ’s be eigenfunctions of the position oper-
ator. (They require a special relationship with the Hamiltonian operator, but
not any other.) This does not imply that we cannot extract anyinformation
about the position from the wavefunction, only that we cannot determine
eigenvalue observables for position. The same is true for other operators, like
momentum.
The next postulate of quantum mechanics that we will deal with concerns
observables like this. It is postulated that although specificvalues of some ob-
servables may not be forthcoming from all wavefunctions,averagevalues of
these observables might be determined. In quantum mechanics, the average
valueorexpectation value A of an observable Awhose operator is Aˆis given
by the expression
A
ab*Aˆd (10.13)
Equation 10.13, which is another postulate of quantum mechanics, assumes
that the wavefunction is normalized. If it is not, the definition of an average
value expands slightly to
ab*Aˆd
A
ab*dAn average value is just what it says: if one were to take repeated measurements
of the same quantity and average them together, what would that average value
be? Quantum mechanically, if one were able to take an infinite number of mea-
surements, the average value would be the average of all of those infinite mea-
surements.
What is the difference between an average value as determined by equation
10.13 and the single eigenvalue of an observable determined from an eigen-
value equation? For some observables, there is no difference. If you know that
a particle-in-a-box is in a certain state, it has a certain wavefunction. According
to the Schrödinger equation, you know its exact energy. The average value of
that energy is the same as its instantaneous energy, because while in the state
described by that wavefunction, the energy does not change. However, some
observables cannot be determined from all wavefunctions using an eigenvalue
equation. The wavefunctions for the particle-in-a-box, for example, are not
eigenfunctions of the position or momentum operators. We cannot determine
instantaneous, exact values for these observables. But we candetermine aver-
age values for them. (Recognize that although the uncertainty principle denies
us the opportunity to know the specificvalues of the position and momentum
of any particle simultaneously, there is no restriction on knowing averageval-
ues of the position or the momentum.)
10.9 Average Values and Other Properties 293