Next: 6. Scattering from potential
Up: 5. Infinite well
Previous: 5.1 Zero of energy
Figure 5.1:
The change in the wave function in region III, for the lowest
state, as we increase the depth of the potential well. We have used
a = 10-10 m, and
k0a = 2, 3, 4, 5, 6, 7, 8, 9 and 10.
|
|
As stated before the continuity arguments for the derivative of the
wave function do not apply for an infinite jump in the potential energy.
This is easy to understand as we look at the behaviour of a low
energy solution in one of the two outside regions (I or III). In this
case the wave function can be approximated as
e kr, k = , |
(5.1) |
which decreases to zero faster and faster as V0 becomes
larger and larger. In the end the wave function can no longer
penetrate the region of infinite potential energy. Continuity of
the wave function now implies that
(a) =
(- a) = 0.
Defining
= sqrt E, |
(5.2) |
we find that there are two types of solutions that satisfy the
boundary condition:
(x) = cos( x), (x) = sin( x). |
|
|
(5.3) |
Here
= . |
(5.4) |
We thus have a series of eigen states
(x),
l = 1, ... ,
.
The energies are
El = . |
(5.5) |
Figure 5.2:
A few wave functions of the infinite square well.
|
|
These wave functions are very good to illustrate the idea of normalisation.
Let me look
at the normalisation of the ground state (the lowest state), which is
for
- a < x < a, and 0 elsewhere.
We need to require
| (x)|2dx = 1, |
(5.7) |
where we need to consider the absolute value since A1 can be complex.
We only have to integrate from - a to a, since the rest of the integral is
zero, and we have
| (x)|2dx |
| |
= |
| A|2 cos2
dx |
|
| |
= |
| A|2 cos2 y dy |
|
| |
= |
| A|2  (1 + cos 2y )dy |
|
| |
= |
| A|2 . |
(5.8) |
Here we have changed variables from x to
y =
.
We thus conclude that the choice
A =  |
(5.9) |
leads to a normalised wave function.
Next: 6. Scattering from potential
Up: 5. Infinite well
Previous: 5.1 Zero of energy