Physical Chemistry , 1st ed.

This system is illustrated in Figure 10.8, and actually describes a large num-
ber of physically real systems. For example, a very fine metal point can be
brought very close—within several angstroms, but still not in physical con-
tact—to a clean surface. The gap between the two pieces of matter represents
a finite potential energy barrier whose height is higher in energy than the en-
ergy of the electrons on either side.
The acceptable wavefunctions of an electron on one side of the system must
be determined by application of the postulates of quantum mechanics. In par-
ticular, the Schrödinger equation must be satisfied by any wavefunction that a
particle can have. Inside the regions in Figure 10.8 where the potential energy
is zero, the wavefunctions are similar to the particle-in-a-box. But for the re-
gion where the potential energy has a nonzero, noninfinite value,



2



m

2





x

2

    2 VˆE

must be solved. Assuming that the potential energy Vis some constant indepen-
dent ofxbut larger than E, this expression can be algebraically rearranged into

2 m(V

^2

E)







x

2

    2 

This second-order differential equation has a known analytic solution. The
general wavefunctions that satisfy the above equation are

AekxBekx (10.14)

where

k

2 m(V

^2

E)

(^) 
1/2
Note the similarity of the wavefunction in equation 10.14 and the exponential
form of the wavefunctions for the particle-in-a-box (shown in the first foot-
note in Section 10.8). In this case, however, the exponentials have real expo-
nents, not imaginary exponents.
Without additional information about the system, we cannot say much
about the exact form (in terms ofAand B) of the wavefunctions in this region.
For example,over the entire space must be continuous, and that places some
restrictions on the values ofAand Bin terms of the length of the zero-poten-
tial region and the wavefunctions in that region. But there is one thing we can
note immediately: the wavefunction is not zero in the region of Figure 10.8
10.10 Tunneling 297
x  0 x  a
x-axis
V  
V  K ()
V  0 V  0
Energy
Figure 10.8 A potential energy diagram where tunneling can occur. Many real systems mimic
this sort of potential energy scheme. Tunneling is an observable phenomenon that is not pre-
dicted by classical mechanics.