where
(hbar, ħ) is the reduced Planck constant,
c is the speed of light,
a is the distance between the two plates.
The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of
shows that the Casimir force per unit area
Fc /
Ais very small, and that furthermore, the force is inherently of quantum-mechanical origin.
More recent theory
Concept of zero-point energy module
using the Casimir Effect
A very complete analysis of the Casimir effect at short distances is based upon a detailed analysis of the van der Waals force by Lifshitz. Using this approach, complications of the bounding surfaces, such as the modifications to the Casimir force due to finite conductivity, can be calculated numerically using the tabulated complex dielectric functions of the bounding materials. In addition to these factors, complications arise due to surface roughness of the boundary and to geometry effects such as degree of parallelism of bounding plates. For boundaries at large separations, retardation effects give rise to a long-range interaction. For the case of two parallel plates composed of ideal metals in vacuum, the results reduce to Casimir’s.
Measurement
One of the first experimental tests was conducted by Marcus Sparnaay at Philips in Eindhoven, in 1958, in a delicate and difficult experiment with parallel plates, obtaining results not in contradiction with the Casimir theory, but with large experimental errors.
The Casimir effect was measured more accurately in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen and Anushree Roy of the University of California at Riverside. In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a sphere with a large radius. In 2001, a group at the University of Padua finally succeeded in measuring the Casimir force between parallel plates using microresonators.
Regularisation
In order to be able to perform calculations in the general case, it is convenient to introduce a regulator in the summations. This is an artificial device, used to make the sums finite so that they can be more easily manipulated, followed by the taking of a limit so as to remove the regulator.
The heat kernel or exponentially regulated sum is
where the limit
is taken in the end. The divergence of the sum is typically manifested as
for three-dimensional cavities. The infinite part of the sum is associated with the bulk constant
Cwhich
does not depend on the shape of the cavity. The interesting part of the sum is the finite part, which is shape-dependent. The Gaussian regulator
is better suited to numerical calculations because of its superior convergence properties, but is more difficult to use in theoretical calculations. Other, suitably smooth, regulators may be used as well. The zeta function regulator
is completely unsuited for numerical calculations, but is quite useful in theoretical calculations. In particular, divergences show up as poles in the complex
s plane, with the bulk divergence at
s=4. This sum may be analytically continued past this pole, to obtain a finite part at
s=0.
Not every cavity configuration necessarily leads to a finite part (the lack of a pole at
s=0) or shape-independent infinite parts. In this case, it should be understood that additional physics has to be taken into account. In particular, at extremely large frequencies (above the plasma frequency), metals become transparent to photons (such as x-rays), and dielectrics show a frequency-dependent cutoff as well. This frequency dependence acts as a natural regulator. There are a variety of bulk effects in solid state physics, mathematically very similar to the Casimir effect, where the cutoff frequency comes into explicit play to keep expressions finite. (These are discussed in greater detail in
Landau and Lifshitz, "Theory of Continuous Media".)
Generalities
experimental setup for the conversion of
vacuum energy into mechanical-energy.
The Casimir effect can also be computed using the mathematical mechanisms of functional integrals of quantum field theory, although such calculations are considerably more abstract, and thus difficult to comprehend. In addition, they can be carried out only for the simplest of geometries. However, the formalism of quantum field theory makes it clear that the vacuum expectation value summations are in a certain sense summations over so-called "virtual particles". More interesting is the understanding that the sums over the energies of standing waves should be formally understood as sums over the eigenvalues of a Hamiltonian. This allows atomic and molecular effects, such as the van der Waals force, to be understood as a variation on the theme of the Casimir effect. Thus one considers the Hamiltonian of a system as a function of the arrangement of objects, such as atoms, in configuration space. The change in the zero-point energy as a function of changes of the configuration can be understood to result in forces acting between the objects.In the chiral bag model of the nucleon, the Casimir energy plays an important role in showing the mass of the nucleon is independent of the bag radius. In addition, the spectral asymmetry is interpreted as a non-zero vacuum expectation value of the baryon number, cancelling the topological winding number of the pion field surrounding the nucleon.
Casimir effect and wormholes
Exotic matter with negative energy density is required to stabilize a wormhole. Morris, Thorne and Yurtsever pointed out that the quantum mechanics of the Casimir effect can be used to produce a locally mass-negative region of space-time, and suggested that negative effect could be used to stabilize a wormhole to allow faster than light travel. This concept has been used extensively in Science Fiction.
Analogies
A similar analysis can be used to explain Hawking radiation that causes the slow "evaporation" of black holes (although this is generally visualized as the escape of one particle from a virtual particle-antiparticle pair, the other particle having been captured by the black hole).
Repulsive forces
There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. In a seminal paper, Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise. This has sparked interest in applications of the Casimir effect toward the development of levitating devices. Other scientists have also suggested the use of gain media to achieve a similar levitation effect, though this is controversial because these materials seem to violate fundamental causality constraints and the requirement of thermodynamic equilibrium. An experimental demonstration of the Casimir-based levitation was recently demonstrated by the Capasso group at Harvard through experiments involving a gold-coated particle and silica thin film immersed in bromobenzene.
Applications
It has been suggested that the Casimir forces have application in nanotechnology, in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, and so-called Casimir oscillators.
Classical 'Critical' Casimir Effect
In 2008, physicists in Germany made the first direct measurements of the “critical Casimir effect”, a classical analogue of the quantum Casimir effect. This effect had been theoretically predicted in 1978 by Michael Fisher and Pierre-Gilles de Gennes but all observations had been indirect.
In this experiment, the critical Casimir effect arises in a mixed liquid that is close to its critical point. The liquid used was a solution of water and the oil 2,6-lutidine which has a critical point of 34°C at normal atmospheric pressure. As this liquid approaches its critical point, the oil and water starts separate into small regions whose size and shape are subject to statistical fluctuations and that exhibit random Brownian motion. To demonstrate the effect, a tiny coated Styrofoam ball is suspended in the liquid close to the wall of its coated glass container. The ball and the container coatings are the same and both have a preference for either oil or water. As the liquid nears its critical point, total internal reflection microscopy is used to detect displacements of the ball. From the sudden movements detected only towards the glass, the classical Casimir force was calculated to be approximately 600 fN (6 x 10−13 N). To tune the effect for repulsion, the coatings of the glass and the ball are changed so that one prefers oil and the other water.
While the German physicists say this reverse critical Casimir effect could be useful in nanoelectromechanical systems, its dependence upon a very specific temperature presently limits its usefulness.
dude i need that . nobody cooks for me 
you mean those particles pass the plates? what's the essence of those particles ?
qorma Sabzi is even better 
