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2. NANOMEMS PHYSICS: Quantum Wave-Particle Phenomena 63

Lifshitz formula [75] if the material between two plates has properties that
are intermediate between those of the plates.
The startling aspect of the Casimir force is that it is a manifestation of the
purely quantum-mechanical prediction of zero-point vacuum fluctuations
[74-77] (see Appendix A), i.e., of the fact that, even in circumstances in
which the average electromagnetic field is zero, its average energy shows
fluctuations with small but non-zero value, i.e., there is virtually infinite
energy in vacuum. Research efforts aimed at the practical exploitation of this
extremely large energy source, residing in free space, are under way [85-87].
Calculating the Casimir force entails circumventing the fact that the zero-

point vacuum energy, E = 1 = ¦ ω diverges, and many techniques to
Field n
2 n
accomplish this have been developed [74-77], [88], [89], but including these
in our presentation is well beyond the scope of this article. The essence of
many of these calculations, however, is to compute the physical energy as a
difference in energy corresponding to two different geometries, e.g., the
parallel plates at a distance “a” apart, and these at a distance “b,” where the
limit as b tends to infinity is taken. For flat surfaces, the infinite part of the
energy cancels when the energy difference of the two configurations is
taken. The calculated zero-temperature Casimir energy for the space
between two uncharged perfectly conducting parallel plates, Figure 2-10,

z z
A A

Figure 2-10. Casimir effect geometry.

is given by,

π 2 c = 1
U () z = − , (51)
Casimir 3
720 z
and, the corresponding Casimir force per unit area is given by,