Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 311 / 4 / b / ii

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- Solution ii

Solution

In the two constant regions define

ω_{\pm} = k_{z}^{2} + a_{\pm}^{2} ∣ k ∣^{2} .

The normalized positive-frequency modes are

u_{\pm} (η) = \frac{e ^{- i ω_{\pm} η}}{2 ω _{\pm}},

up to the common spatial Fourier normalization. They satisfy the unit Wronskian condition

i (u_{\pm}^{*} u_{\pm}^{'} - u_{\pm} u_{\pm}^{*'}) = 1,

which is the mode form of the Klein-Gordon inner product. Because the mode equation contains no delta function at

η = 0

, both

Φ

and

Φ^{'}

are continuous there.

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