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

On the unbounded line the momentum density of states in a reference length $L$ is $Ldk/(2\pi )$. Count both signs of $k$, each with zero-point energy $|k|/2$ and the same frequency regulator:The extensive divergent term agrees exactly with c. Subtract it before taking the cutoff away:The force tending to change the interval length isIts 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.