Section 6 Flashcards
If E > V(±∞)
we have a scattering state
Scattering formalism
Incoming plane wave with t-dependence hidden:
Ψ(x) = Aexp[ikx]
Complex so non-zero flux
J = (ℏk)/m |A|^2
Reflection coefficient
R = J(R)/J(I) = |B|^2/|A|^2
Transmission coefficient
T = J(T)/J(I) = k2/k1 |C|^2/|A|^2
Flux conservation
J(I) = k2|C|^2 = k1(|A|^2-|B|^2)
Scattering from allowed potential step
R = |B|^2 /|A|^2 =
(k1-k2)^2/(k1+k2)^2
Scattering from allowed potential step
T = k2/k1 |C|^2 /|A|^2 =
1 - R = 4k1k2/(k1+k2)^2
Scattering from forbidden potential step
R =|B|^2 /|A|^2 =
|ik1+k2|^2 / |ik1-k2|^2 = 1
Scattering from forbidden potential step
T = 1 - R =
0
In forbidden potential steps barrier penetration occurs, but
can’t pass through, i.e. no flux
skin depth
d = 1/2k2
Scattering from potential well
4klA =
[ (k+l)^2 exp[-2ila] - (k-l)^2 exp[2ila] ] F exp[2ika]
Resonant scattering from potential well
Limit 1: E «_space;|V| => k «_space;l
4klA ≈ l^2 (exp[-2ila] - exp[2ila]) F exp[2ika]
= -2iFl^2 sin(2la) exp[2ika]
T = |F|^2 / |A|^2 = 0
100% transmission
Resonant scattering from potential well
Limit 2: E»_space; |V| => k ≈ l
4k^2A ≈ (2k)^2 exp[-2ila] F exp[2ika]
=> T = | exp[2ila] exp[-2ika] |^2
= 1
Perfect transmission
Resonant transmission when
2la = nπ