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ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2016/paper-1/1/b/solution

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Past exam of the theoretical physics II course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution by

Codex  0 2026-10-20

Put

x = r cos θ

e^{ik x}

solves the two-dimensional Helmholtz equation,

(\nabla^{2} + k^{2}) e^{ik x} = 0

. Expand its angular dependence as a Fourier series,

e^{ik r c o s θ} = \sum_{m \in Z} F_{m} (r) e^{im θ} .

Substitution into the polar-coordinate Laplacian,

\nabla^{2} = \partial_{r}^{2} + \frac{1}{r} \partial_{r} + \frac{1}{r ^{2}} \partial_{θ}^{2},

gives

r^{2} F_{m}^{''} + r F_{m}^{'} + (k^{2} r^{2} - m^{2}) F_{m} = 0.

With

ρ = k r

, this is the Bessel differential equation. Regularity of the original plane wave at the origin excludes the singular second-kind solution. Thus

F_{m} (r)

is a constant multiple of the regular first-kind function, and

e^{ik r c o s θ} = m \in Z \sum a_{m} J_{m} (k r) e^{im θ} .

For integer negative orders,

J_{- m} = (- 1)^{m} J_{m}

, so this indexing still supplies the regular radial solution. The radial equation fixes the form of each term; the small-radius plane-wave expansion fixes its coefficient.

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