多粒子量子流体力学：基本原理和基本应用
Many-particle quantum hydrodynamics: Basic principles and fundamental applications
摘要：
从多粒子薛定谔方程出发，详细推导了连续性方程、欧拉方程和能量平衡方程。粒子间的相互作用被明确地认为是库仑相互作用。我们给出了适用于低维系统的QHD方程。给出了柱坐标系和球坐标系下的QHD方程。本文提出了一种从QHD方程导出非线性薛定谔方程的方法。本文描述了一种计算二维和三维量子等离子体交换相互作用的方法。我们讨论了明确的形式和适用范围的状态方程的一维，二维和三维简并电子。本文明确地考虑了外磁场中简并电子气的状态方程。考虑柱坐标系下的QHD方程，我们着重研究了惯性力的量子部分以及量子玻姆势对量子柱面波谱的贡献。研究了球面二维电子气的集体激发。得到了球表面简并二维电子气的物态方程。我们将QHD方程应用于线性区域的小振幅集体激发。我们考虑了包含交换库仑相互作用的一维、二维和三维量子等离子体中的朗缪尔波。考虑到量子等离子体位于外磁场中，我们把注意力集中在垂直于外磁场传播的纵波上。我们提出了1D，2D和3D等离子体的考虑。我们讨论了交换相互作用对外磁场中自旋极化率的依赖关系。我们包括依赖于外部磁场的强度，通过在应用程序的外部磁场的量子态的占领的变化的简并电子的压力。
英文摘要：
We present a detailed derivation of the continuity, Euler, and energy balance equations from many particle Schrodinger equation. Interparticle interaction is explicitly considered as the Coulomb interaction. We show the QHD equations in a form suitable for low dimensional systems. The QHD equations in the cylindrical and spherical coordinates are presented as well. We present a method of non-linear Schrodinger equation derivation from the QHD equations. We describe a method of exchange interaction calculation for two- and three-dimensional quantum plasmas. We discuss explicit form and area of applicability of equations of state for 1D, 2D, and 3D degenerate electrons. We explicitly consider equations of state for degenerate electron gas in an external magnetic field. Considering QHD equations in cylindrical coordinates we focus our attention on the quantum part of the inertia forces and contribution of the quantum Bohm potential to the spectrum of quantum cylindrical waves. Collective excitations of 2DEG on spherical surface are considered. Equation of state for degenerate 2DEG on a spherical surface is obtained. We apply the QHD equations to small amplitude collective excitations in linear regime of QHD equations. We consider the Langmuir waves in 1D, 2D, and 3D quantum plasmas including the exchange Coulomb interaction. Considering quantum plasmas located in an external magnetic field we focus our attention on longitudinal waves propagating perpendicular to the external magnetic field. We present this consideration for 1D, 2D, and 3D plasmas. We discuss dependence of the exchange interaction on rate of polarization of spins in the external magnetic field. We include dependence of the pressure of degenerate electrons on the strength of the external magnetic field via the change of occupation of quantum states at application of an external magnetic field.
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