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Expedient of introduction fractional integro-differentiations in a quantum theory

Keywords:

Z. Z. Alisultanov, R. P. Mejlanov


In article attempt of introduction fractional integro-differentiations in a quantum theory becomes. Possibility of introduction of fractional derivatives on time in a quantum theory of an electromagnetic field and fractional derivatives on co-ordinate in a quantum theory of many particles is viewed. In operation it is shown, that it is possible to present an electromagnetic field featured by fractional Maxwell equations as plurality of the perpetual number “fractional” oscillators. At electromagnetic field quantization there is a necessity to recast an of Heisenberg equation of motion. To gain fractional Maxwell equations it is necessary to use fractional on time the of Heisenberg equation. Proceeding from it, the deduction becomes, that by viewing of similar systems it is necessary to use the fractional equation of Heisenberg, instead of a fractional Schrodinger equation. Since transition from Heisenberg representation to Schrodinger in case of fractional derivatives is related to major difficulties. Using a source function method attempt of introduction of fractional derivatives on coordinate in system of many particles is made. Interaction between particles can lead to change of dependence of a kinetic energy from an impulse. Being grounded on this fact it is possible to guess, that interaction can lead to fractionally power characteristic, so and to fractional derivatives on co-ordinate. Further, connection between function of interaction of quasiparticles and fractionally power characteristic is established. Such approach allows to understand most accurately physics of the process featured by the equations of the fractional order. The important problem of quantum statistics is examination of the multicomponent systems, in particular an electron – phonon system. In this connection it its shown, that is fractional – the differential approach to system of many particles allows to explain the observational effects on examination an electron – phonon interaction.
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