Physical Chemistry , 1st ed.

of electrons to pass a very, very small gap between a sharp tip and a surface.

Since the amount of tunneling varies exponentially with distance, even very

small distance changes can yield very large differences in the amount of elec-

tron tunneling (measured as a current, since current is the flow of electrons).

The extreme sensitivity of the STM allows one to make pictures of smooth sur-

faces on an atomic scale. Figure 10.11 shows an image measured by an STM.

Tunneling is a real, detectable phenomenon. It is not predicted by classical

mechanics (and would be forbidden by it) but it arises naturally out of quan-

tum mechanics. Its existence is the first real-life example given here of the

strange and wonderful world of quantum theory.

10.11 The Three-Dimensional Particle-in-a-Box

The one-dimensional particle-in-a-box can be expanded to two and three di-

mensions very easily. Because the treatments are similar, we consider just the

three-dimensional system here (and we trust that the student will be able to

simplify the following treatment for a two-dimensional system; see exercise

10.53 at the end of this chapter). A general system, showing a box having its ori-

gin at (0, 0, 0) and having dimensions a b c, is illustrated in Figure 10.12.

Once again we define the system with V0 inside the box and Voutside

the box. The Schrödinger equation for a particle in a three-dimensional box is



2 m

^2

(^) 




x

2

2  (^) 



y

2

2  (^) 



z

2

(^2) E (10.15)
The three-dimensional operator ^2 /x^2 ^2 /y^2 ^2 /z^2 is very common and
is given the symbol ^2 , called “del-squared” and referred to as the Laplacian
operator:
^2




x

2

2  (^) 



y

2

2  (^) 



z

2

    2 (10.16)

The 3-D Schrödinger equation is usually written as



2 m

^2

^2 E

In the rest of this text, we use the symbol ^2 to represent the Laplacian for a

particular system that operates on in the Schrödinger equation.

We determine the acceptable wavefunctions for this system by trying an-

other assumption. Let us assume that the complete three-dimensional (x,y,

z), which must be a function ofx,y, and z, can be written as a product of three

functions, each of which can be written in terms ofonlyone variable. That is:

(x,y,z) X(x) Y(y) Z(z) (10.17)

where X(x) is a function only ofx(that is, independent ofyand z),Y(y) is a

function solely ofy, and Z(z) is a function solely ofz. Wavefunctions that can

be written this way are said to be separable.Why make this particular assump-

tion? Because then in the evaluation of the del-squared part of the Schrödinger

equation, each second derivative will act on only one of the separate functions

and the others will cancel, making an ultimate solution of the Schrödinger

equation that much simpler.

We also simplify the notation by dropping the parenthetical variables on the

three functions. The Schrödinger equation in equation 10.15 becomes



2 m

^2

(^) 




x

2

2  (^) 



y

2

2  (^) 



z

2

(^2) XYZE XYZ
10.11 The Three-Dimensional Particle-in-a-Box 299
Figure 10.11 An STM image of a ring of Fe
atoms on a copper surface.
(0, 0, c)
(0, 0, 0)
(a, 0, 0)
(0, b, 0)
V  0
V  
z
y
x
Figure 10.12 The three-dimensional particle-
in-a-box. An understanding of its wavefunctions
is based on the wavefunctions of the 1-D particle-
in-a-box, and it illustrates the concept of separa-
tion of variables. Generally,abc, although
when abcthe wavefunctions may have some
special characteristics.
Courtesy IBM