Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 2 / f

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Solution

Let

J (x) = \int_{x}^{b} κ (s) d s = η - I (x)

. The WKB approximation in the barrier can be rewritten as

E_{II} = \frac{D _{R} e ^{- J} + G _{R} e ^{J}}{κ}, D_{R} = G e^{η}, G_{R} = D e^{- η} .

The right Airy turning-point connection formula gives

E_{III} = \frac{2 D _{R} c o s ( Ψ - π /4 ) - G _{R} s i n ( Ψ - π /4 )}{k _{3}},

so

A_{t} = D_{R} + i G_{R} /2

and

A_{back} = D_{R} - i G_{R} /2

. Impose no incident wave from the right,

A_{back} = 0

. Then

D_{R} = \frac{i G _{R}}{2}, G = \frac{i D}{2} e^{- 2 η}, A_{t} = i D e^{- η}, A_{in} = D (1 + \frac{1}{4} e^{- 2 η}) .

Consequently, within the leading Airy turning-point connection formula calculation,

t = \frac{A _{t}}{A _{in}} = \frac{i e ^{- η}}{1 + e ^{- 2 η} /4}, r = \frac{A _{r}}{A _{in}} = \frac{1 - e ^{- 2 η} /4}{1 + e ^{- 2 η} /4} .

For an opaque barrier,

η ≫ 1

,

t \sim i e^{- η}, T = ∣ t ∣^{2} ≃ e^{- 2 η} .

The transmission coefficient is a ratio of Poynting fluxes; the

k^{- 1/2}

normalization already includes their local wavenumber factors. The displayed denominator is useful for consistent matching and Wronskian flux conservation, but does not claim accuracy for subleading terms beyond the WKB approximation, or for merging wave turning points.

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