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The Ground State of a Strongly Interacting Fermion System (

J. C. Owe

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

44 ~0 57 ~7 60 62 63 64 65 66 0. Introduct ion. 1. Cor re l a t ed Jas t row-type t r i a l wave functions. 2. Monte Carlo calculation of the energy expectat ion value. 3. Observed propert ies of l iquid 3He and some results of Monte Carlo c a l culations. 4. The Jaekson-Feenberg expression of the kinet ic energy. 5. Diagrammat ic expansion of the norm of the Jas t row wave function (local correlat ion factors). 6. Diagrammat ic expansion of the single-particle density. 7. Pointirreducible graphs: el iminat ion of ar t iculat ion points. 8. Expansion of the norm in the presence of spin-dependent correlation factors, 9. Rules for Jas t row diagrams with spin correlation bars but involving at most second-order loops. 10. Diagrammat ic expansion of the energy. 11. The Euler equations. 12. The chain equations in the absence of spin correlations. 13. A simple example of chain equations. 14. An example of chain equations in the presence of spin correlations. 141. Chain equations in the spin-0 channel. 142. Chain equations in the spin-1 channel. 15. Spin correlations in l iquid aHe. 151. Renormalizat ion of the (dashed) central correlation line. 152. Renormalizat ion of the spin correlation bar. 153. Renormalizat ion of the (central) s ta t ic L i n d h a r d f u n c t i o n S~c. 154. Renormalizat ion o f the s ta t ic L indhard function SF in the spin-i channel. 66 155. Calculation of S ' and the Euler equation. 69 16, Sound and spin waves in the collective co-ordinate approximation.

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The origin of the wave-part ic le dual ism of quantum theory can be t raced back to Einstein 's analysis of the fluctuations of b lack-body radia t ion in 1909 (1-3). Based on this invest igat ion EINSTEIlq proposed the in te rpre ta t ion tha t the electromagnetic field in thermal equil ibrium has as well par t ic le as wave propert ies . In his analysis concerning the ideal quantum gas of 1925 he came to the conclusion tha t this dualist ic in te rpre ta t ion applies to the par t ic les of the gas as well (a). Recent ly we have shown tha t the fluctuations of b lack-body radia t ion can be explained from the assumption of s ta t i s t ica l ly dependent indist inguishable part icles (5). Moreover, i t tu rned out tha t the so-called wave contr ibut ions to the var iance are ent i re ly due to the s ta t is t ical dependence of these part icles. F rom this viewpoint Einstein 's dualist ic in te rpre ta t ion is incomprehensible. I t is the aim of this note to explain a weak point in Einstein 's analysis in detai l and to show tha t wave proper t ies emerge in the classical l imi t only. Planck 's formula gives the mean value of the energy densi ty g* (with respect to dco) of b lack-body rad ia t ion :

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44 ~0 57 7 60 62 63 64 65 66 0. Introduct ion. 1. Cor re l a t ed Jas t row-type t r i a l wave functions. 2. Monte Carlo calculation of the energy expectat ion value. 3. Observed propert ies of l iquid 3He and some results of Monte Carlo c a l culations. 4. The Jaekson-Feenberg expression of the kinet ic energy. 5. Diagrammat ic expansion of the norm of the Jas t row wave function (local correlat ion factors). 6. Diagrammat ic expansion of the single-particle density. 7. Pointirreducible graphs: el iminat ion of ar t iculat ion points. 8. Expansion of the norm in the presence of spin-dependent correlation factors, 9. Rules for Jas t row diagrams with spin correlation bars but involving at most second-order loops. 10. Diagrammat ic expansion of the energy. 11. The Euler equations. 12. The chain equations in the absence of spin correlations. 13. A simple example of chain equations. 14. An example of chain equations in the presence of spin correlations. 141. Chain equations in the spin-0 channel. 142. Chain equations in the spin-1 channel. 15. Spin correlations in l iquid aHe. 151. Renormalizat ion of the (dashed) central correlation line. 152. Renormalizat ion of the spin correlation bar. 153. Renormalizat ion of the (central) s ta t ic L i n d h a r d f u n c t i o n Sc. 154. Renormalizat ion o f the s ta t ic L indhard function SF in the spin-i channel. 66 155. Calculation of S ' and the Euler equation. 69 16, Sound and spin waves in the collective co-ordinate approximation.
The hypotheticai system of nucleons known as nuclear matter provides a very interesting field of study with relevance to both nuclear physics and astrophysics. The task of finding nucleon-nucleon potentials that explain the empirical properties of nuclear matter, as well as the two-body data, has been hampered by the lack of an adequate many-body theory for treating the complicated forces involved. Realistic nucleon-nucleon potentials have strong spin, isospin, tensor and spin-orbit components which induce correspondingly strong correlations in nuclear matter. In this thesis we present a variational theory designed to cope with these complicated correlations. To place the theory in perspective we give an overview of several different many-body theories and compare their predictions in the somewhat simpler systems of the 4He and 3He liquids, and for a simple central potential model of nuclear matter. The variational theory is then developed, starting with the choice of wave function. A generalized Jastrow product of two-body correlation functions containing central, spin, isospin, tensor, and spin-orbit operators is used. The correlation function is generated by a series of two-body Schrodinger equations with boundary conditions which require the wave function to heal at a distance d. The wave function is parameterized by d and by the magnitudes of the non-central correlations. Expectation values with variational wave functions of the Jastrow type may be studied conveniently with generalized Mayer diagrams. Diagram rules are given and a general diagrammatic cluster expansion is derived. It is an expansion in powers of commutators of the non-central operators, and in their absence reduces to the usual Mayer cluster expansion. Evaluation of diagrams in the cluster expansion is complicated by the presence of the non-commuting operators. Some simple rules and useful methods for calculating the contribution due to the central, spin, isospin, and tensor operators are given. Spin-orbit operators are neglected beyond this point, but the approach used is extendible to them. Large classes of diagrams contributing to an expectation value can be summed by means of integral equations. The development of single chain, hypernetted chain, and Fermi hypernetted chain (FHNC) equations for central correlations is reviewed. The chain summation methods are extended to sum single operator chains (SOC) for the non-central operators. Methods are also given for treating the leading commutator corrections in the expansi0n by simple vertex factors. The energy expectation value for potentials with six operators are evaluated using the FHNC/SOC functions, with an exact treatment of the commutators involved. The energy is found to have a minimum with respect to variations in all parameters. Results of calculations with model potentials based on the Reid, Bethe-Johnson, Hamada-Johnston, and Gammel-Thaler potentials are reported. A crude estimate of the effect of the neglected spin-orbit potentials on the nuclear matter binding energy and saturation density indicates that it could be significant. A correct treatment of the spin-orbit potentials is the next major step that should be taken.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.
The origin of the wave-part ic le dual ism of quantum theory can be t raced back to Einstein 's analysis of the fluctuations of b lack-body radia t ion in 1909 (1-3). Based on this invest igat ion EINSTEIlq proposed the in te rpre ta t ion tha t the electromagnetic field in thermal equil ibrium has as well par t ic le as wave propert ies . In his analysis concerning the ideal quantum gas of 1925 he came to the conclusion tha t this dualist ic in te rpre ta t ion applies to the par t ic les of the gas as well (a). Recent ly we have shown tha t the fluctuations of b lack-body radia t ion can be explained from the assumption of s ta t i s t ica l ly dependent indist inguishable part icles (5). Moreover, i t tu rned out tha t the so-called wave contr ibut ions to the var iance are ent i re ly due to the s ta t is t ical dependence of these part icles. F rom this viewpoint Einstein 's dualist ic in te rpre ta t ion is incomprehensible. I t is the aim of this note to explain a weak point in Einstein 's analysis in detai l and to show tha t wave proper t ies emerge in the classical l imi t only. Planck 's formula gives the mean value of the energy densi ty g* (with respect to dco) of b lack-body rad ia t ion :