13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Casimir effect | 3/6 | https://en.wikipedia.org/wiki/Casimir_effect | reference | science, encyclopedia | 2026-05-05T10:54:44.616020+00:00 | kb-cron |
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{\displaystyle F(p)=-\left.{\frac {\delta \langle E(s)\rangle }{\delta s}}\right\vert _{p}\,.}
This value is finite in many practical calculations. Attraction between the plates can be easily understood by focusing on the one-dimensional situation. Suppose that a moveable conductive plate is positioned at a short distance a from one of two widely separated plates (distance l apart). With a ≪ l, the states within the slot of width a are highly constrained so that the energy E of any one mode is widely separated from that of the next. This is not the case in the large region l where there is a large number of states (about l/a) with energy evenly spaced between E and the next mode in the narrow slot, or in other words, all slightly larger than E. Now on shortening a by an amount da (which is negative), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to −da/a, whereas all the l/a states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to −da/l (note the different denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the l/a modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make a slightly smaller, the plates drawing each other closer, across the thin slot.
== Derivation of Casimir effect assuming zeta-regularization == In the original calculation done by Casimir, he considered the space between a pair of conducting metal plates at distance a apart. In this case, the standing waves are particularly easy to calculate, because the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the plates lie parallel to the xy-plane, the standing waves are
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{\displaystyle \psi _{n}(x,y,z;t)=e^{-i\omega _{n}t}e^{ik_{x}x+ik_{y}y}\sin(k_{n}z)\,,}
where ψ stands for the electric component of the electromagnetic field, and, for brevity, the polarization and the magnetic components are ignored here. Here, kx and ky are the wavenumbers in directions parallel to the plates, and
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{\displaystyle k_{n}={\frac {n\pi }{a}}}
is the wavenumber perpendicular to the plates. Here, n is an integer, resulting from the requirement that ψ vanish on the metal plates. The frequency of this wave is
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{\displaystyle \omega _{n}=c{\sqrt {{k_{x}}^{2}+{k_{y}}^{2}+{\frac {n^{2}\pi ^{2}}{a^{2}}}}}\,,}
where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes. Since the area of the plates is large, we may sum by integrating over two of the dimensions in k-space. The assumption of periodic boundary conditions yields,
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{\displaystyle \langle E\rangle ={\frac {\hbar }{2}}\cdot 2\int {\frac {A\,dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\,,}
where A is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is
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{\displaystyle {\frac {\langle E(s)\rangle }{A}}=\hbar \int {\frac {dk_{x}\,dk_{y}}{(2\pi )^{2}}}\sum _{n=1}^{\infty }\omega _{n}\left|\omega _{n}\right|^{-s}\,.}
In the end, the limit s → 0 is to be taken. Here s is just a complex number, not to be confused with the shape discussed previously. This integral sum is finite for s real and larger than 3. The sum has a pole at s = 3, but may be analytically continued to s = 0, where the expression is finite. The above expression simplifies to:
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{\displaystyle {\frac {\langle E(s)\rangle }{A}}={\frac {\hbar c^{1-s}}{4\pi ^{2}}}\sum _{n}\int _{0}^{\infty }2\pi q\,dq\left|q^{2}+{\frac {\pi ^{2}n^{2}}{a^{2}}}\right|^{\frac {1-s}{2}}\,,}
where polar coordinates q2 = kx2 + ky2 were introduced to turn the double integral into a single integral. The q in front is the Jacobian, and the 2π comes from the angular integration. The integral converges if Re(s) > 3, resulting in
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{\displaystyle {\frac {\langle E(s)\rangle }{A}}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}}}{\frac {1}{3-s}}\sum _{n}\left|n\right|^{3-s}=-{\frac {\hbar c^{1-s}\pi ^{2-s}}{2a^{3-s}(3-s)}}\sum _{n}{\frac {1}{\left|n\right|^{s-3}}}\,.}
The sum diverges at s in the neighborhood of zero, but if the damping of large-frequency excitations corresponding to analytic continuation of the Riemann zeta function to s = 0 is assumed to make sense physically in some way, then one has