Application of Double Quantization in Landau Diamagnetism

For the calculation of the diamagnetic permeability is considered a set of noninteracting electrons in a finite, sufficiently large amount of magnetic. The Hamiltonian of the considered structure with one-electron functions, involving the operators of birth and destruction is reduced to conjunction operators, the first of which is simply the kinetic energy of the electrons, and the other two are considered as small perturbations. Using the procedure of perturbation theory is calculated the energy of the magnetic field in the first and second order. It is shown that the amendment of the first order equal to zero and the second order is computed by using the introduction of the Fermi momentum in the case of temperatures close to zero. The result for the energy is represented in the form of a number from quadratic in the vector potential terms. Further tying together the result is a representation of energy using the current density of electrons able to find the connection between the components of the current density and the corresponding components of the vector potential. A similar relationship obtained by using Fourier transform of Maxwell’s equations. When comparing the obtained ratios obtained an expression for diamagnetic permeability, which is accurate to dimensional multipliers, associated with the choice of system of units, coincides with the classical result obtained by a different method.

References

[1] Batyev E.G. Pauli paramagnetism and Landau diamagnetism. Phys. Usp., 2009, vol. 52, pp. 1245-1246. DOI: 10.3367/UFNe.0179.200912i.1333

[2] Einsenstein J.P., MacDonald A.H. Bose - Einstein condensation of excitons in bilayer electron systems. Nature, 2004, vol. 432, pp. 691-694.

[3] Landau L.D. Diamagnetismus der Metalle. Z. Phys. 1930, bd. 64, s. 629.

[4] Bandyopadhyay M., Dattagupta S. Landay - Drude diamagnetism: fluctuation, dissipation and decoherence. Journal of Physics: Condensed Matter, 2006, vol. 18, no. 44, pp. 10029-10043.

[5] Aliev I.N., Kopylov I.S. Use of Dirac monopoles formalism in some magnetism problems. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2015, no. 6, pp. 25-39 (in Russ.). DOI: 10.18698/1812-3368-2015-6-25-39

[6] Aliev I.N., Melikyants D.G. On potentials in Londons’ electrodynamics. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2016, no. 2, pp. 42-50 (in Russ.). DOI: 10.18698/1812-3368-2016-2-42-50

[7] Aliev I.N., Tolmachev V.V. Optiko-mekhanicheskaya analogiya i uravnenie Shredingera [Optical-mechanical analogy and the Schrodinger equation]. Moscow, MGTU im. N.E. Baumana Publ., 1998. 80 p.

[8] Aliev I.N., Kopylov I.S. Applying the Lagrange multipliers method to the calculation of DC magnetic field. Dinamika slozhnykh sistem [Dynamics of Complex Systems], 2015, no. 4, pp. 3-10 (in Russ.).

[9] Landau L.D., Lifshitz Е.М. Quantum mechanics (Vol. 3. Course of theoretical physics). Pergamon Press, 1965.

[10] Landau L.D., Lifshitz E.M. Mechanics (Vol. 1. Course of theoretical physics). Pergamon Press, 1969.

[11] Llano M., Tolmachev V.V. Multiple phases in a new statistical boson-fermion model of superconductivity. Physica A. 2003, vol. 317, pp. 546-564.

[12] Landau L.D. Sobranie trudov. V 2 t. Т. 2 [Collected papers in 2 vol. Vol. 1.]. Ed. by Е.М. Lifshitz. Moscow, Nauka Publ., 1969. 510 p.

[13] Batyev E.G. Properties of an excitonic insulator in the superconducting state. JETP, 2012, vol. 114, iss. 1, pp. 132-140.

[14] Meissner W., Ochsenfeld R. Ein neurer effekt bei eintritt der supraleitfahigkeit. Naturwissenschaften, 1933, vol. 21, iss. 44, pp. 787-788.

[15] Yurchenko S., Komarov K., Pustovoit V. Multilayer-graphene-based amplifier of surface acoustic waves. AIP advances, 2015, vol. 5, p. 057144.