Physical Chemistry , 1st ed.

Example 10.12

Determine x , the average value of the position of an electron having the

lowest energy level (n1) in a particle-in-a-box.

Solution

By definition, the average value of the position x for the lowest energy level is

x 

a

0



2

a

sin^

a

x

(^) * x (^) 

2

a

sin^

a

x

(^) dx
where all of the functions inside the integral sign are being multiplied to-
gether, the limits of the system are 0 to a, and dis dx. Because the function
is real, the complex conjugate doesn’t change anything and the expression be-
comes (because multiplication is commutative)
x 

2

a



a

0

x sin^2

a

x

dx

This integral also has a known solution (see Appendix 1). On solving, this ex-

pression becomes

2

a

(^) 
x
4
2

4
x
a
sin

2

a

x



8

a

2

    2 cos

2

a

x

(^) a 0
When this is evaluated, the average value for the position is
x 
2
a
Thus, this particle having the given wavefunction has an average position
in the middle of the box.
The above example illustrates two things. First, average values canbe deter-
mined for observables that cannot be determined using an eigenvalue equation
(which a postulate of quantum mechanics requires of its observables); and sec-
ond, average values should make sense. It would be expected that for a parti-
cle bouncing back and forth in a box, its average position be the middle of the
box. It should spend as much time on one side as on the other, so its average
position would be right in the middle. This is what equation 10.13 provides,
at least in this case: an intuitively reasonable value. There are many examples
in quantum mechanics where a reasonable average is produced, albeit from a
different argument than classical mechanics. This simply reinforces the applic-
ability of quantum mechanics. The average value of the position of the parti-
cle in a box is a/2 for anyvalue of the quantum number n. Evaluate, as an ex-
ercise, the average value of the position observable for
 3 (x) 

2

a

sin^

3

a

x

where the subscript on indicates that this wavefunction has the quantum

number n3. The solution for the integral used for the average value shows

that the quantum number n, whatever it is, has no effect in the determination

of x. (These conclusions only apply to stationary states of the particle-in-a-

box. If the wavefunctions are not stationary states, x and other average values

would not necessarily be intuitively consistent with classical mechanics.)

294 CHAPTER 10 Introduction to Quantum Mechanics