Simple Nature - Light and Matter

m/Tunneling through a bar-
rier. (As discussed on p. 894, it is
actually not quite correct to graph
the wavefunction of an electron
as a real number unless it is a
standing wave, which isn’t the
case here.)

where, according to our assumptions,ais independent ofx. We need

to find a function whose second derivative is the same as the original

function except for a multiplicative constant. The only functions

with this property are sine waves and exponentials:

d^2

dx^2

[qsin(rx+s) ] =−qr^2 sin(rx+s)

d^2

dx^2

[

qerx+s

]

=qr^2 erx+s

The sine wave gives negative values of a,a = −r^2 , and the

exponential gives positive ones,a=r^2. The former applies to the

classically allowed region withU < E.

This leads us to a quantitative calculation of the tunneling ef-

fect discussed briefly in the preceding subsection. The wavefunction

evidently tails off exponentially in the classically forbidden region.

Suppose, as shown in figure m, a wave-particle traveling to the right

encounters a barrier that it is classically forbidden to enter. Al-

though the form of the Schr ̈odinger equation we’re using technically

does not apply to traveling waves (because it makes no reference

to time), it turns out that we can still use it to make a reasonable

calculation of the probability that the particle will make it through

the barrier. If we let the barrier’s width bew, then the ratio of the

wavefunction on the left side of the barrier to the wavefunction on

the right is

qerx+s

qer(x+w)+s

=e−rw.

Probabilities are proportional to the squares of wavefunctions, so

906 Chapter 13 Quantum Physics