Citation
Levine, Paul Hersh (1963) Phase space formulation of the quantum manybody problem. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:11062012103119433
Abstract
By means of a quantum mechanical phase space distribution function introduced by von Roos, the Schroedinger equation for a nonrelativistic system of N identical particles with scalar interactions is transformed into a quantum mechanical generalization of the Liouville equation, thereby formulating the problem in terms of a generalized density in phase space, a quantity of primary interest in most treatments of the corresponding classical system (or "plasma"). This transformation permits a parallel development of the theories of classical and quantum plasmas and thus allows the quantum manybody problem to be discussed virtually completely in classical terms. In particular, a kinetic theory of quantum plasmas is obtained by deriving the quantum analogue of the BBGKY hierarchy, and applying thereto approximation techniques similar to those of Rostocker and Rosenbluth, and Bogoliubov. The point of departure from similar previous studies based on the Wigner distribution function is that the proper exchange symmetry can be tractably introduced into the formalism.
Attention is first focused on the Hartree and HartreeFock approximations, in which case the quantum BBGKY system reduces to a simple quantum generalization of the Vlasov equation. This equation is used to study the response of spatially homogeneous systems to weak external forces, and the associated problems of plasmon and spinwave excitations. It is also used to derive the quantum and exchange corrected equations of inviscid hydrodynamical transport which are then applied to the problem of sound propagation in the degenerate electron gas.
The second part of the study is concerned with the theory of the manyelectron atom in the Hartree and HartreeFock approximations. The relevant quantum Vlasov equations lead naturally to a "statistical" theory of the atom which reduces to the ThomasFermiAmaldi and Thomas Fermi models (respectively) as ħ → 0. For ħ ≠ 0, the quantum and exchange corrections to these models are simultaneously generated. The quantum hydrodynamical theory developed earlier is used to determine the influence of these corrections on the boundary conditions of the model, and a theory of the compressed atom is consequently obtained. Considered in somewhat less detail are the effects of nonzero temperature, net orbital angular momentum, relativity and correlations, as well as time dependent processes.
The final part deals with the problem of the degenerate electron gas with a uniform neutralizing background. Going beyond the HartreeFock approximation, the pair correlation functions for particles with "parallel" and "antiparallel" spin are obtained by neglecting three particle correlations. From these functions, a quantummechanical collision integral is derived which differs from that obtained by Silin and Guernsey and conjectured by Wyld and Pines in that dynamical exchange effects are included. Also obtained from the pair correlation function is an expression for the "correlation energy" which reduces in the high density limit to the result of GellMann and Brueckner. At intermediate densities an additional term appears in the energy due to the screening of the exchange interaction by the dielectric properties of the medium. It is evaluated in the high density limit and found to be 0.151 r_s ln r_s Ryd/electron in marked disagreement with the corresponding value obtained by DuBois.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Physics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Physics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1963 
Record Number:  CaltechTHESIS:11062012103119433 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:11062012103119433 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7257 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  06 Nov 2012 19:39 
Last Modified:  23 Sep 2014 17:00 
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