 | Assume E>V0. Let
Then
are the most general solutions. |
f is continuous at x=0. Therefore A1+A1'=A2+A2' .
¶f/¶x is continuous at x=0. Therefore ik1 A1-ik1A1'=ik2A2-ik2A2' .
We have two equations and four unknowns. No unique solution exists. Let us limit ourselves to particles approaching from the left being transmitted or reflected at the barrier. Assume A2'=0
We can then solve for A1'/A1 and A2/A1 . We find
.
The ratios are real and positive, there is no phase shift upon reflection and transmission.
Probability of Reflection (R) and Transmission (T):
Our solutions in regions 1 and 2 are plane wave solutions. They are not square integrable, and a proper normalization is not possible. Plane wave solutions are useful in describing steady streams of particles. To evaluate the probability of reflection and transmission for such beams, we compare the reflected and transmitted flux with the incident flux.
Let P(x) be the probability density at x, let x0 be some point in region 1, and let x0’ be some point in region 2. We then have
incident flux µ v1Pi(x0)
reflected flux µ v1Pr(x0)
transmitted flux µ v2Pt(x0')
The reflection coefficient R is
,
and the transmission coefficient T is
.
We have
Classically the probability of reflection is zero.
In Quantum Mechanics R ® 0, T ® 1 if E>>V0 and k2 ® k1.
| Assume E<V0. Let |
Then
and
are the most general solutions.
We need B2=0 for the solution to remain finite at x ® ¥.
f is continuous at x=0. Therefore A1+A1'=B2'.
¶f/¶x is continuous at x=0. Therefore k1 A1-ik1A1'=-ir2B2'.
We find
.
The ratios are complex, a phase shift appears upon reflection.
We compute R by comparing the incident and the reflected flux,
Therefore T=0, there is no transmitted flux.
Note: As V0 ® ¥, r2 ® ¥. Then A1'/A1 ® -1, B2'/A1 ® 0, we have a phase shift of 1800.
If V0 = ¥ then A1'=-A1, B'2=0, f2(x)=0, f(x) is continuous at x=0 since f(0)=0, but ¶f/¶x is not continuous. We have:
Link:
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and
