Solution

For a wave incident from the left, define the reflection and transmission amplitudes by

ψ (x) \sim {e^{ik x} + r e^{- ik x}, t e^{ik x}, x \to - \infty, x \to + \infty, E = \frac{ℏ ^{2} k ^{2}}{2 m} .

(391)

Reflection invariance gives the same amplitudes for incidence from the right. If the incoming amplitudes from the left and right are ordered first, and the outgoing amplitudes toward the left and right second, the One-dimensional S-matrix is

S = (r t t r) .

(392)

For

V (x) = V_{0} δ (x)

, the wavefunction is continuous and its derivative has the jump

ψ^{'} (0^{+}) - ψ^{'} (0^{-}) = \frac{2 m V _{0}}{ℏ ^{2}} ψ (0) .

(393)

The scattering ansatz gives

1 + r = t, 2 ik r = \frac{2 m V _{0}}{ℏ ^{2}} t .

(394)

With

γ = \frac{m V _{0}}{ℏ ^{2} k},

(395)

we obtain the scattering by a delta potential amplitudes

t = \frac{1}{1 + iγ}, r = \frac{- iγ}{1 + iγ} .

(396)

They satisfy

∣ r ∣^{2} + ∣ t ∣^{2} = 1, r^{*} t + t^{*} r = 0.

(397)

Therefore

S^{†} S = (∣ r ∣^{2} + ∣ t ∣^{2} t^{*} r + r^{*} t r^{*} t + t^{*} r ∣ r ∣^{2} + ∣ t ∣^{2}) = I,

(398)

so the scattering matrix is unitary.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 35D / a / Solution

 Ancestors (11)