Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 36B / d

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- Solution d

Solution

Since

tan δ_{0} = γ / (K_{0} - k)

,

σ_{0} (k) = \frac{4 π}{k ^{2}} \frac{γ ^{2}}{( K _{0} - k ) ^{2} + γ ^{2}} .

The exact nearby stationary point satisfies

K_{0} - k = γ^{2} / K_{0} + O (γ^{4} / K_{0}^{3})

, so to leading order in

γ / K_{0}

the local maximum is at

k = K_{0}

. This is a scattering resonance with

E_{R} = \frac{ℏ ^{2} K _{0}^{2}}{2 m}, Γ = \frac{2 ℏ ^{2} K _{0} γ}{m},

where

Γ

is the full energy width of the Breit-Wigner peak.

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