Dielectric permittivity of quantum plasma. Part II
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LLATIONS AND WAVES IN PLASMA
Dielectric Permittivity of Quantum Plasma. Part II Yu. V. Bobylev and M. V. Kuzelev Faculty of Physics, Moscow State University, Moscow, 119992 Russia email: [email protected] Received October 16, 2013; in final form, November 20, 2013
Abstract—The transverse and longitudinal dielectric permittivities of isotropic quantum plasma are calcu lated in the quantum plasma models based on the Dirac and Pauli equations. The dispersion relations for transverse–longitudinal waves in quantum particle beams are derived. Relativistic longitudinal and transverse waves in cold isotropic quantum plasma in models based on the Klein–Gordon and Dirac equations, as well as spin waves in the model based on the Pauli equation, are considered. Conditions for wave–particle reso nance interactions in relativistic quantum plasma are analyzed. DOI: 10.1134/S1063780X14050043
1. INTRODUCTION In the second part of our work, devoted to the properties of quantum plasma, the dielectric permit tivities are calculated with allowance for the spin of plasma particles. To describe collisionless quantum plasma, we use the same approach as in Part I (see Plasma Phys. Rep. 40, 343 (2014)), where the dielec tric permittivities were calculated without allowance for the particle spin. We remind that there are two widespread methods of analyzing such plasmas. The first method is based on the integration of the kinetic equation for the oneparticle distribution function. In the classical case, it is the Vlasov kinetic equation, whereas in the quantum theory, it is the Wigner equa tion for the oneparticle density matrix. The second method is based on the integration of equations of motion for individual plasma particles in the selfcon sistent electromagnetic field. In the classical case, these are conventional equations of motion for the coordinate, velocity, and acceleration, whereas in the quantum theory, these are equations for wave func tions of plasma particles. In the quantum theory, the current density of α species particles is calculated by the formula
∫
j α(t, r) = eαn0α f 0α(p)ψ*α(t, r; p)ˆvψ α(t, r; p)dp,
(1.1)
where f 0α(p) is the distribution function over particle momenta, ψ α(t, r; p) is the particle wave function in the state with the momentum p in the absence of the self consistent field, and ˆv is the velocity operator. In this work, the dielectric permittivity of collisionless quan tum plasma is calculated using the second method, based on the direct integration of quantum equations of motion. In the first part of this work, the quantum plasma models based on the Schrödinger and Klein–Gordon
equations were used to find the dielectric permittivi ties. In this second part, we focus on the model based on the Dirac equation. For the sake of completeness of our analysis, the model based on the Pauli equation is also considered. The presentation is rather laconic, because the main features of the method were dis cussed in detail in Part I. 2. RELATIVISTIC QUANTUM PLASMA IN THE MODEL BASED ON THE DIRAC
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