BioPHYSICAL chemistry

These equations are the same as for the one-dimensional case, so we
can write the solutions (Figure 10.9) directly as:

(10.47)

(10.48)

Unlike the one-dimensional problem, two states now can have equal
energies – that is, they are degenerate – if Lxand Lyare equal. For example,
the nx=2, ny=1 and nx=1, ny=2 states are degenerate when Lx=Ly.
This degeneracy in energy is reflective of the symmetry.

Tunneling

For this problem, there was zero probability of finding the particle outside
the box because we had said that the walls were infinitely high. What
happens when we relax this condition and merely say that the walls are
high but not infinitely high? From a classical viewpoint, a ball has a certain
amount of energy. If the ball is in a valley between two hills it will simply
roll back and forth between the hills. The ball will have its energy all in
the form of kinetic energy at the bottom of the hills and be moving at its
maximum speed. As it moves up the hill the ball will slow down until it
stops and begins back down the hill. Because the ball does not have enough
energy to go over the hill it will always be found between the two hills.
In quantum mechanics, this problem corresponds to having two wells
that are separated by a wall that is finite (Figure 10.10). This is formally

EE E

h

m

n

L

n

xy L

x

x

y

y

=+=

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ +

⎛

⎝

⎜

⎜

2 2 ⎞

(^8) ⎠⎠

⎟

⎟

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

2

ψ
π
() ()()xy X xY y sin
L

nx

x L

x

x

, ==

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎛

⎝

⎜^2

⎜⎜

⎞

⎠

⎟

⎟

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

2

L

ny

y L

y

y

sin

π

Figure 10.9The solutions for the two-dimensional particle in a box.

CHAPTER 10 PARTICLE IN A BOX AND TUNNELING 209

nx  ny  1 nx  ny  2