# 臺灣博碩士論文加值系統

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 We review some properties of many electrons in a constant magnetic field at zero temperature and at finite temperature. Following the work of Weitao Yang, we get some ideas to deal with many-electron systems by use of the path integral formulation. In this thesis, we extend the integral formulation at zero temperature to electron systems at finite temperature. We also discuss the defect of the path integral formulation when it is applied to 2 dimensional system in a constant field. Furthermore, we discuss the mathematical techniques and phenomena when the many-electron system is subject to an oscillating magnetic field.
 1 Introduction 2 1.1 Units and universal constants . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Wave Functions in Coordinate Space . . . . . . . . . . . . . . . . . 5 1.2.2 Wave functions in momentum space . . . . . . . . . . . . . . . . . . 6 2 Density matrix formulation at zero temperature 8 2.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The Path Integral of the System . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Ground state energy and other properties . . . . . . . . . . . . . . . . . . . 16 2.3.1 The electron density in weak B field, ! << 1 . . . . . . . . . . . . . 17 2.3.2 Total energy in weak B field . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Pressure in a weak B field , magnetic dipole momment density and susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.4 The electron density in a strong B field . . . . . . . . . . . . . . . . 22 2.3.5 Total energy in a strong B field . . . . . . . . . . . . . . . . . . . . . 23 2.4 The exact solution of the system . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 The electron density . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.2 The total energy of the system . . . . . . . . . . . . . . . . . . . . . 25 2.4.3 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Landau diamagnetism, Pauli Paramagnetism, and two dimensional sys-tem in a constant magnetic field 29 3.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The mathematical formulation of canonical ensemble and grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 High temperature behavior of the diamagnetism . . . . . . . . . . . . . . . 34 3.3.1 Analogy to path integral . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Diamagnetism at high temperature . . . . . . . . . . . . . . . . . . . 36 3.4 Fermi-Dirac, Bose-Einstein, and Maxwell distributions . . . . . . . . . . . . 39 3.4.1 The derivations of Ferm-Dirac, Bose-Einstein, and Maxwell distribu-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Integral formulations of Fermi-Dirac statistics . . . . . . . . . . . . . 42 3.5 Pauli paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.1 Derivation of the equation of motion . . . . . . . . . . . . . . . . . . 44 3.5.2 Low temperature behavior . . . . . . . . . . . . . . . . . . . . . . . . 45 3.6 Two dimensional system at zero temperature . . . . . . . . . . . . . . . . . 51 3.6.1 Density and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4 Time-dependent system and linear response theory at zero temperature 56 4.1 The formulations of many body time-dependent perturbation and linear re-sponse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.1 Time-dependent perturbation about one particle . . . . . . . . . . . 57 4.1.2 Density correlation function . . . . . . . . . . . . . . . . . . . . . . . 60 4.1.3 First order energy correction . . . . . . . . . . . . . . . . . . . . . . 61 4.1.4 Density correlation function of homogeneous free electrons in momen-tum space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2 Interaction with magnetic wave in free electrons . . . . . . . . . . . . . . . . 64
 [1] Yang, W. (1988). Ab initio approach for many-electron systems without invoking orbitals: An integral formulation of density functional theory. Phys. Rev. A 38:5494-5503[2] Yang, W. (1988). Thermal properties of many electron system: An integral formulation of density functional theory. Phys. Rev. A 38:5504-5511[3] Yang,W. (1988). Dynamic linear response of many-electron systems: An integral formulation of density functional theory. Phys. Rev. A 38:5512-5519[4] Mermin, N.D. (1965). Thermal properties of the inhomogeneous electron gas. Phys. Rev. 137:A1441-A1443[5]Kohn, W. and Sham, L.J. (1965).Self-consistent equations including exchange and correlation effects. Phys. Rev. 140: A1133-A1138[6]Gross, E.K.U. and Kohn, W. (1985). Local density-functional theory of frequency-dependent linear response. Phys. Rev. Lett. 55:2850-2852[7]Hohenberg, P. and Kohn, W. (1964). Inhomogeneous electron gas . Phys. Revs. 136:B864-B871[8] Feynman, R.P. (1972). Statistical Mechanics. Reading, Mass.: Benjamin.[9] Feynman, R.P. and Hibbs, A.R. (1965) Quantum Mechanics and Path Integrals. New York: McGraw-Hill[10]Zangwill, A. and Soven, P. (1980). Density functional approach to local-field effects in finite systems: Photoabsorption in the rare gases. Phys. Rev. A 21: 1561-1572[11] Hoffman, G.G., Pratt, L.R., and Harris, R.A. (1988). Monte Carlo integration of density-functional theory: fermions in a harmonic well. Chem. Phys. Lett. 148:313-316[12] Parr, G.R. and Yang, W. (1989). Density-Functional Theory of Atoms and Molecules. New York: Oxford[13] Fetter, A.L. and Walecka, J.D. (1995). Quantum Theory of Many-Particle Systems. New York: McGraw-Hill[14] Mattuck, R.D. (1976). A Guide to Feynman Diagrams in the Many-Body Problem. New York: McGraw-Hill[15] Sakurai, J.J. (1994). Modern Quantum Mechanics Revised Edition. Addison-Wesley[16] Huang, K. (1987). Statistical Mechanics Second Edition. John Wiley & Sons
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