Next: 6.3 Square barrier
Up: 6. Scattering from potential
Previous: 6.1 Non-normalisable wave functions
Consider a potential step
Figure 6.1:
The step potential discussed in the text
|
|
Let me define
| k0 |
= |
, |
(6.4) |
| k1 |
= |
. |
(6.5) |
I assume a beam of particles comes in from the left,
(x) = A0eik0x, x < 0. |
(6.6) |
At the potential step the particles either get reflected back
to region I, or are transmitted to region II. There can thus only
be a wave moving to the right in region II, but in region
I we have both the incoming and a reflected wave,
(x) |
= |
A0eik0x + B0e-ik0x, |
(6.7) |
(x) |
= |
A1eik1x. |
(6.8) |
We define a transmission and reflection coefficient as the ratio
of currents between reflected or transmitted wave and the incoming wave,
where we have cancelled a common factor
R =
T = . |
(6.9) |
Even though we have given up normalisability, we still have the two
continuity conditions. At x = 0 these imply, using
continuity of
and

,
| A0 + B0 |
= |
A1, |
(6.10) |
| ik0(A0 - B0) |
= |
ik1A1. |
(6.11) |
We thus find
| A1 |
= |
A0, |
(6.12) |
| B0 |
= |
A0, |
(6.13) |
and the reflection and transmission coefficients can thus be expressed as
| R |
= |

, |
(6.14) |
| T |
= |
. |
(6.15) |
Notice that R + T = 1!
Figure 6.2:
The transmission and reflection coefficients for a square
barrier.
|
|
In Fig. 6.2 we have plotted the behaviour of the
transmission and reflection of a beam of Hydrogen atoms impinging
on a barrier of height 2 meV.
Next: 6.3 Square barrier
Up: 6. Scattering from potential
Previous: 6.1 Non-normalisable wave functions