k-+=

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lim la(k)

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a-+O

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lim la(k)

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(2.156)

36Notice that N n disappears as we go to the continuum because then w = 0 is a point of measure zero. 37 As long as the plates are brought from infinity in a adiabatic way. In 4 we will see what happens when cavity walls are moved in a nonadiabatic way, an effect that is often called the dynamic Casimir effect.

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THE CASIMIR FORCE

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Including this cutoff function, (2.155) becomes

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(2.157) Let us examine the integrals over v in (2.157). They have the general form (2.158) Changing the variable of integration from v to Vi simpler expression,

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== (2/3)( v + e)3/2, we obtain the

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(2.159)

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If we take the limit a ---T 0 after doing the integral in (2.159), the various derivatives of a(~) are given by

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d a d C4 = 0, ... a~O ~ a~O ." a~O ~ a~O ." (2.160) Now if we expand (2.157) into an Euler-Maclaurin series [433], we find that d a a C a 1 ----;u: = - 2c 2 l' dd C2 = -4.", l' d,lC3 1m .", 1m 1m

If the relevant physics is independent of the manner in which the reflectivity falls off with frequency, we are allowed to take the limit a ---T 0 in (2.161). Then all derivatives of a vanish at ~ = 0, except the third derivative, and we find that

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(2.162) This potential implies an attractive force per unit area, or pressure on the plates, given by 1 dU (2.163) L2 dl This is the Casimir force, or more precisely, pressure. For a plate separation of 1 cm, this pressure is only 1.3 x 10- 18 dyn/cm2. This is so weak that it is even less than half of the electrostatic pressure on the plates if they were charged with a single electron [Le., 47re2/ L 4 ~ 2.9 X 10- 18 dyn/cm 2 (where e is the charge of the electron)). But on the scale of nanomachinery (millimeter to nanometer range) the Casimir force is dominant [83, 309, 352, 353].

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FIATLUXI

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The first experiment on the Casimir force was done by Sparnaay in 1957, nine years after Casimir's paper was published. Unfortunately, the experimental error was far too large for this to be a decisive confirmation of Casimir's prediction. Since then, the experimental accuracy has improved immensely. But there are still very few experiments in comparison with the number of theoretical papers on this subject. For a list of the experimental papers on the Casimir force, see the recommended reading at the end of the chapter. Last but not least, we promised to show you the physical reason why the regularization trick works. To do so, we must calculate first the mode spectral density for both free space and for the parallel-plate case. The mode spectral density is the function p(w) of the frequency w that gives the number of modes having a frequency between wand w + dJ.,;. So, in terms of the mode spectral density, the zero-point energy (without including the cutoff) is given by

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00 dJ.,; p(w) -hw. 1

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(2.164)

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For the free-space case (i.e., when both L

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00 and l

00), (2.164) yields

00 dJ.,; PFS(W) -hw = 3" 1 1 V o 2

1 dk z 2-2 nw .

(2.165)

Changing the integration variables on the right-hand side of (2.165) from Cartesian to spherical, we find that

00 dkx 100 dky 100 dkz2-hw= 100 k 1

dkhw

17r/2 d(-cosB) 27r dcp

27r foo dJ.,;

w2hw.

(2.166)

Substituting (2.166) in (2.165), we see that we can take PFS(W) as given by38 PFS(W)

7r4V W . 2c3

(2.167)

For the parallel-plate case, (2.164) yields

dJ.,;

ppp(w)

2 = 7r2~ 10

dk x

00 n {K21. + ~ Kl + (n7r)2} ' T

(2.168)

38Sometimes in the literature (e.g . in [274] and [305]). PFS(W) is quoted without the factor 4 seen in (2.\66), The reason for the missing factor 4 in those references is that unlike here. they use periodic boundary conditions for kx and k y so that these components of the wave vector are discretized in units of L/(27r) instead of L/7r. as we have here.