BioPHYSICAL chemistry

expressed by saying that the potential energy

at x=0 is finite with a specified value that is

greater than the energy of the ball. Since the

walls at x=−Land x=+Lare still infinite,

we know that the wavefunction must have a

value of zero at these positions. However, we

can no longer specify the value of the wave-

function at x=0. Rather, this value must be

solved based upon the value of the potential

at this position. We know that the probability

must be very small at this position, but in

quantum mechanics the wavefunction has a

non-zero value. The ball cannot be found at

x=0 for any period of time since that would

violate classical mechanics and the corres-

pondence principle. However, because it has

a non-zero value, it has a finite probability

of being at x=0 for a fleeting moment that

allows the ball to move into the second well.

Thus, in quantum mechanics, it is possible

for a particle to enter a “classically forbidden

region” provided the visit is only temporary.

To determine the conditions that permit

tunneling of an object, consider a particle with

an energy Ein a large box (Figure 10.11).

Within most of the box the potential is zero

except for a small region that has a barrier

with a small width aand a potential V 0. A

wavefunction outside the barrier is a traveling wave since the potential

is zero in this region. A wavefunction incident to the barrier can then

be described as a simple sine function. In the barrier region, the energy

Eof the particle is less than the potential V 0 , and hence the particle is

classically forbidden to pass through the barrier. To determine the wave-

function, Schrödinger’s equation is written for that region:

(10.49)

Since the energy Eand potential V 0 are constants, the solution of this

equation can be written as an exponential function:

ψ(x) ∝e−κx 0 ≤x≤a (10.50)

κ

()

=

2 0 −

2

mV E

Z

(EV x) () ()

mx

−=− 0 xxa≤≤

2

ψψ 2 2 0

Z d

d

2

210 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Energy

Reaction coordinate

A  B A  B

Classical

Energy

Reaction coordinate

A  B

ψ^2

A  B

Quantum

Figure 10.10
Tunneling across
an energy barrier
permits an electron
to have a probability
to be located in
both wells.