= Solution
On the unbounded line the momentum density of states in a reference length $L$ is $L\,dk/(2\pi)$. Count both signs of $k$, each with <zero-point energy> $|k|/2$ and the same frequency regulator:
$$
\boxed{E_0(L,\Lambda)=\frac L2\int_{-\infty}^\infty\frac{dk}{2\pi}|k|e^{-|k|/(\pi\Lambda)}
=\frac L{2\pi}\int_0^\infty dk\,ke^{-k/(\pi\Lambda)}
=\frac{\pi L\Lambda^2}{2}.}
$$
The extensive divergent term agrees exactly with <7/c>. Subtract it before taking the cutoff away:
$$
E_{\rm Cas}(L)=\lim_{\Lambda\to\infty}[E_0^{\rm BC}(L,\Lambda)-E_0(L,\Lambda)]
=-\frac\pi{24L}.
$$
The force tending to change the interval length is
$$
\boxed{F_L=-\frac{dE_{\rm Cas}}{dL}=-\frac\pi{24L^2},\qquad
F_L=-\frac{\pi\hbar c}{24L^2}\ \text{with units restored}.}
$$
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 \b[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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