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 Incoming links: Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7 / c

Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7 / b / Solution 2026-10-25

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The Hamiltonian density is

H = [\dot{ϕ}^{2} + (\partial_{x} ϕ)^{2} + m^{2} ϕ^{2}] /2

. In the vacuum only

⟨ a_{n} a_{m}^{†} ⟩ = δ_{nm}

contributes. Hence

⟨ \dot{ϕ}^{2} ⟩ = \sum_{n} \frac{ω _{n}}{L} sin^{2} (k_{n} x), ⟨(\partial_{x} ϕ)^{2} ⟩ = \sum_{n} \frac{k _{n}^{2}}{ω _{n} L} cos^{2} (k_{n} x), ⟨ ϕ^{2} ⟩ = \sum_{n} \frac{s i n ^{2} ( k _{n} x )}{ω _{n} L} .

(1)

Substitution, using

ω_{n}^{2} = k_{n}^{2} + m^{2}

, yields

⟨ H (x)⟩ = n = 1 \sum \infty [\frac{ω _{n}}{2 L} - \frac{m ^{2}}{2 ω _{n} L} cos (2 k_{n} x)] .

(2)

This Dirichlet scalar vacuum energy density illustrates why a massive field's local unregularized energy density need not be uniform. The cosine integrates to zero over the interval, so

E_{0}^{BC} (L) = \int_{0}^{L} d x ⟨ H (x)⟩ = \frac{1}{2} n = 1 \sum \infty ω_{n} .

(3)

These divergent expressions are understood with a common regulator before exchanging sums, integrals or derivatives. Normal ordering the Hamiltonian operator separately in each geometry would erase the vacuum terms one is trying to compare; a common regulated subtraction is used instead in c and d.

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Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7 / d / Solution 2026-10-25

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

L

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} .

(1)

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} .

(2)

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 .

(3)

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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