Quantum Mechanics 183
Hence each energy eigenfunction can be expressed as a linear combination of the two
wave functions(^) n einxL (5.49)
(^) n einxL (5.50)
Inserting the first of these wave functions in the eigenvalue equation, Eq. (5.48), we
have
ˆp (^) npn (^) n
(^) n einxL (^) npn (^) n
so that pn (5.51)
Similarly the wave function nleads to the momentum eigenvalues
pn (5.52)
We conclude that (^) nand (^) nare indeed the momentum eigenfunctions for a parti-
cle in a box, and that Eq. (5.47) correctly states the corresponding momentum
eigenvalues.
5.9 FINITE POTENTIAL WELL
The wave function penetrates the walls, which lowers the energy levels
Potential energies are never infinite in the real world, and the box with infinitely hard
walls of the previous section has no physical counterpart. However, potential wells
with barriers of finite height certainly do exist. Let us see what the wave functions and
energy levels of a particle in such a well are.
Figure 5.7 shows a potential well with square corners that is Uhigh and Lwide
and contains a particle whose energy E is less than U. According to classical
mechanics, when the particle strikes the sides of the well, it bounces off without
entering regions I and III. In quantum mechanics, the particle also bounces back
and forth, but now it has a certain probability of penetrating into regions I and III
even though EU.
In regions I and III Schrödinger’s steady-state equation is
(EU) 0
2 m
2
d^2
dx^2
n
L
n
L
n
L
in
L
2
L
1
2 i
i
d
dx
i
2
L
1
2 i
2
L
1
Momentum 2 i
eigenfunctions for
trapped particle
I E II III
U
- x 0 L +x
EnergyFigure 5.7A square potential well
with finite barriers. The energy E
of the trapped particle is less than
the height Uof the barriers.bei48482_ch05.qxd 1/17/02 12:17 AM Page 183