Section 6 Flashcards

1
Q

If E > V(±∞)

A

we have a scattering state

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2
Q

Scattering formalism

Incoming plane wave with t-dependence hidden:

A

Ψ(x) = Aexp[ikx]

Complex so non-zero flux

J = (ℏk)/m |A|^2

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3
Q

Reflection coefficient

A

R = J(R)/J(I) = |B|^2/|A|^2

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4
Q

Transmission coefficient

A

T = J(T)/J(I) = k2/k1 |C|^2/|A|^2

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5
Q

Flux conservation

A

J(I) = k2|C|^2 = k1(|A|^2-|B|^2)

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6
Q

Scattering from allowed potential step

R = |B|^2 /|A|^2 =

A

(k1-k2)^2/(k1+k2)^2

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7
Q

Scattering from allowed potential step

T = k2/k1 |C|^2 /|A|^2 =

A

1 - R = 4k1k2/(k1+k2)^2

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8
Q

Scattering from forbidden potential step

R =|B|^2 /|A|^2 =

A

|ik1+k2|^2 / |ik1-k2|^2 = 1

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9
Q

Scattering from forbidden potential step

T = 1 - R =

A

0

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10
Q

In forbidden potential steps barrier penetration occurs, but

A

can’t pass through, i.e. no flux

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11
Q

skin depth

A

d = 1/2k2

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12
Q

Scattering from potential well

4klA =

A

[ (k+l)^2 exp[-2ila] - (k-l)^2 exp[2ila] ] F exp[2ika]

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13
Q

Resonant scattering from potential well

Limit 1: E &laquo_space;|V| => k &laquo_space;l

A

4klA ≈ l^2 (exp[-2ila] - exp[2ila]) F exp[2ika]

= -2iFl^2 sin(2la) exp[2ika]

T = |F|^2 / |A|^2 = 0

100% transmission

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14
Q

Resonant scattering from potential well

Limit 2: E&raquo_space; |V| => k ≈ l

A

4k^2A ≈ (2k)^2 exp[-2ila] F exp[2ika]

=> T = | exp[2ila] exp[-2ika] |^2
= 1

Perfect transmission

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15
Q

Resonant transmission when

A

2la = nπ

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16
Q

Ramsauer-Townsend effect

A

energy of electrons scattering from noble gases can be tuned to values with no scattering: no classical explanation

17
Q

Scanning tunnelling microscope (STM) =

A

T = [ 1+ (V^2sinh^2(2la)) / (4E(V-E) ]^(-1)

≈ 16E(V-E) / V^2 exp[-2Ll]

18
Q

For large barrier opacity ,

A

2la = Ll&raquo_space; 1

19
Q

Alpha decay

A

is emission of He nuclei (2n2p) from atomic nuclei

20
Q

In alpha decay the nucleus is bound by

A

strong nuclear force vs coulomb repulsion of protons

strong wins at small radii, Coulomb at large.

21
Q

alpha is very tightly bound:

A

can spontaneously form by nuclear disintegration, but potential barrier to emission

22
Q

Gamow modelled barrier as a series of barriers

A

T = Πi Ti(Vi,ai)