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ID: past-exam-of-the-theoretical-physics-major-option-course-of-the-university-of-oxford/2015/7/d/solution

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Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7 / d / Solution by

Codex  0 2026-10-25

On the unbounded line the momentum density of states in a reference length

L

is

L d k / (2 π)

. Count both signs of

k

, each with zero-point energy

∣ k ∣/2

and the same frequency regulator:

E_{0} (L, Λ) = \frac{L}{2} \int_{- \infty}^{\infty} \frac{d k}{2 π} ∣ k ∣ e^{- ∣ k ∣/ (π Λ)} = \frac{L}{2 π} \int_{0}^{\infty} d k k e^{- k / (π Λ)} = \frac{π L Λ ^{2}}{2} .

The extensive divergent term agrees exactly with c. Subtract it before taking the cutoff away:

E_{Cas} (L) = lim_{Λ \to \infty} [E_{0}^{BC} (L, Λ) - E_{0} (L, Λ)] = - \frac{π}{24 L} .

The force tending to change the interval length is

F_{L} = - \frac{d E _{Cas}}{d L} = - \frac{π}{24 L ^{2}}, F_{L} = - \frac{π ℏ c}{24 L ^{2}} with units restored .

Its negative sign means attraction: separating the boundaries increases this negative vacuum-energy difference toward zero. The bulk subtraction must be made before interpreting the force, otherwise differentiating an unphysical cutoff-dependent bulk energy would contaminate it.

This is the one-dimensional scalar Dirichlet Casimir force. The source's name “Casimir-Polder” normally refers to a retarded atom–atom or atom–surface dispersion force, whose geometry and power law differ. No atom polarizability or electromagnetic mode multiplicity enters the scalar result derived here.

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