**Casimir forces**(rus. сила Казимира

*otherwise*эффект Казимира) — force caused by the distortion of zero-point oscillations spectrum in vacuum.

### Description

From the macroscopic view, the Casimir force is negligible. However, for objects with nanometer size and, accordingly, extremely small mass, the Casimir force becomes quite noticeable and has to be considered when designing nanoelectromechanical systems.

To get a clearer idea of the physical sense of this force we should use the Schrodinger equation solution for a harmonic oscillator, which gives us the following energy eigenvalues :

where is the oscillator's own frequency so is the harmonic oscillator quantum equal to the difference in energy of levels with quantum numbers and . This solution shows that even if the quantum number is the energy of a harmonic oscillator is not zero but . The value was given the name of zero-point energy of a harmonic oscillator.

If we extend this logic to electromagnetic radiation quanta – photons (and use the secondary quantisation approach involving photon origination and annihilation operators) then, at some approximation, the occurrence of the Casimir force can be explained as follows: in the absence of any objects the entire space of physical vacuum is filled with an endless number of harmonics of zero-point oscillations of electromagnetic field (even in the absence of photons, as was shown above, the vacuum energy is not equal to zero) with an infinite set of wavelengths, respectively.

In the particular case of two uncharged conducting parallel plates, the Casimir force is the force of their mutual attraction to each other. The presence of two conducting plates restricts the space so that between the plates there is a standing wave with a wavelength of where is the harmonic number (1, 2, 3, etc.). At the same time, outside the plates the physical vacuum space is left undisturbed, and it is the space that puts pressure on the plates, tending to bring them closer to each other.

Within the framework of the original calculations made by Dutch scientists Hendrik Casimir and Dirk Polder in 1948 [1] this force was found out to have the following value per unit of area :

The first experiments to detect the Casimir force were made as early as in 1958 [2] but their accuracy was pretty low. A more precise measurement of the Casimir force was made by Steve Lamoreaux in 1997. [3]. In 2012 first NEMS taking this effect into account were introduced [4].

To get a clearer idea of the physical sense of this force we should use the Schrodinger equation solution for a harmonic oscillator, which gives us the following energy eigenvalues :

_{},

where is the oscillator's own frequency so is the harmonic oscillator quantum equal to the difference in energy of levels with quantum numbers and . This solution shows that even if the quantum number is the energy of a harmonic oscillator is not zero but . The value was given the name of zero-point energy of a harmonic oscillator.

If we extend this logic to electromagnetic radiation quanta – photons (and use the secondary quantisation approach involving photon origination and annihilation operators) then, at some approximation, the occurrence of the Casimir force can be explained as follows: in the absence of any objects the entire space of physical vacuum is filled with an endless number of harmonics of zero-point oscillations of electromagnetic field (even in the absence of photons, as was shown above, the vacuum energy is not equal to zero) with an infinite set of wavelengths, respectively.

In the particular case of two uncharged conducting parallel plates, the Casimir force is the force of their mutual attraction to each other. The presence of two conducting plates restricts the space so that between the plates there is a standing wave with a wavelength of where is the harmonic number (1, 2, 3, etc.). At the same time, outside the plates the physical vacuum space is left undisturbed, and it is the space that puts pressure on the plates, tending to bring them closer to each other.

Within the framework of the original calculations made by Dutch scientists Hendrik Casimir and Dirk Polder in 1948 [1] this force was found out to have the following value per unit of area :

_{}.

The first experiments to detect the Casimir force were made as early as in 1958 [2] but their accuracy was pretty low. A more precise measurement of the Casimir force was made by Steve Lamoreaux in 1997. [3]. In 2012 first NEMS taking this effect into account were introduced [4].

#### Author

- Lourie Sergey

#### Sources

- Casimir H. B.G., Polder D. The Influence of Retardation on the London — van der Waals Forces // Physical Review. 1948. V. 73, №4. P. 360–372.
- Sparnaay M. J. Measurement of attractive forces between flat plates // Physica. 1958. V. 24, №6–10. P. 751–764.
- Lamoreaux S.K. Demonstration of the Casimir Force in the 0.6 to 6 μm Range // Phys. Rev. Lett. 1997. V. 78, №1. P. 5–8.
- J. Zou, Z. Marcet, A. W. Rodriguez, M. T. H. Reid, A. P. McCauley, I. I. Kravchenko, T. Lu, Y. Bao, S. G. Johnson, H. B. Chan. Geometry-dependent Casimir forces on a silicon chip // arXiv:1207.6163v1