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
Ramsauer-Townsend effect
energy of electrons scattering from noble gases can be tuned to values with no scattering: no classical explanation
Scanning tunnelling microscope (STM) =
T = [ 1+ (V^2sinh^2(2la)) / (4E(V-E) ]^(-1)
≈ 16E(V-E) / V^2 exp[-2Ll]
For large barrier opacity ,
2la = Ll»_space; 1
Alpha decay
is emission of He nuclei (2n2p) from atomic nuclei
In alpha decay the nucleus is bound by
strong nuclear force vs coulomb repulsion of protons
strong wins at small radii, Coulomb at large.
alpha is very tightly bound:
can spontaneously form by nuclear disintegration, but potential barrier to emission
Gamow modelled barrier as a series of barriers
T = Πi Ti(Vi,ai)
