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Journal of Physics A Mathematical and General

Completely integrable N-body quantum systems in three dimensions. II. N identical spin-1/2 fermions

A. O. Barut, Y Kitagawara

Published Jan 1, 1982• 0 citations
Publisher

Abstract

For pt.I see ibid., vol.14, p.2581 (1981). If one begins with N non-interacting fermions, the Pauli principle can be easily incorporated by the use of Slater determinants. This is not the case for N interacting fermions. The authors consider exactly soluble 'Coulomb-type' quantum systems in three dimensions of N interacting identical spin- 1 / 2 -fermions. A systematic procedure for constructing Pauli antisymmetry adapted wavefunctions is given. The resulting antisymmetric wavefunctions are labelled by conserved 'good' quantum numbers. In particular, for N=2, all the physically acceptable states are obtained. For N=3, they present a class of antisymmetric states which consists of all the ground states, all the first excited states and the states obtained by the hyperradial excitations of these. For N=2 and 3, the ground states of the authors' model systems are found to be 1 S(s 2 ) and 2 P(s 2 p), respectively, in the quantum numbers of the interacting system.

DOI
10.1088/0305-4470/15/1/021

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For pt.I see ibid., vol.14, p.2581 (1981). If one begins with N non-interacting fermions, the Pauli principle can be easily incorporated by the use of Slater determinants. This is not the case for N interacting fermions. The authors consider exactly soluble 'Coulomb-type' quantum systems in three dimensions of N interacting identical spin- 1 / 2 -fermions. A systematic procedure for constructing Pauli antisymmetry adapted wavefunctions is given. The resulting antisymmetric wavefunctions are labelled by conserved 'good' quantum numbers. In particular, for N=2, all the physically acceptable states are obtained. For N=3, they present a class of antisymmetric states which consists of all the ground states, all the first excited states and the states obtained by the hyperradial excitations of these. For N=2 and 3, the ground states of the authors' model systems are found to be 1 S(s 2 ) and 2 P(s 2 p), respectively, in the quantum numbers of the interacting system.
Given a function w completely antisymmetric in n variables, there may exist a set of n functions of one variable such that the given function is a Slater determinant in the latter. The first problem considered is that of obtaining a criterion for this to be the case for a given function. This problem is solved by considering the function w as a mapping of the space of functions in n − 1 variables onto the space of functions of one variable. A necessary and sufficient condition for the initial function to be a Slater determinant is then shown to be that the image space be n dimensional. This criterion is converted into practical algorithms which can be employed for the determination. The application of one of these yields the theorem that an arbitrary linear combination of the n + 1 Slater determinants in n variables formed from n + 1 one-variable functions can always be written as a single Slater determinant. It is further proved that if the image space of the mapping is m(>n) dimensional, the original function can be expressed as a linear combination of m!/(m − n)!n! Slater determinants in n variables formed from m one-variable functions. Playing an important role in the analysis is the product of the mapping described above by its adjoint (the product is simply related to Dirac's density matrix for a quantum mechanical system of identical particles) as well as the eigenvectors and eigenvalues of this Hermitian positive semidefinite mapping. The latter form a basis for a systematic approximation procedure for representing a given function by a single Slater determinant or by sums of Slater determinants formed from a particular number of one-variable functions, which yields results obtained previously by Löwdin. Problems of simultaneous approximation of sets of antisymmetric functions and possible physical applications to many-fermion systems are briefly discussed.
The exact energies and wavefunctions for a system of (N − 1) one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large volume, finite density limit, when the particles interact via an attractive delta function potential. It is found that the attractive potential gives rise to a bound state, but, in spite of the presence of this bound state, all of the physical properties which are calculated (ground-state energy, effective mass of a certain class of excitations, etc.) are analytic continuations in the coupling constant of the corresponding results in the repulsive case. In addition, it is possible to have eigenstates which do not have the bound state present. These excited states are also discussed and are found to exhibit a negative effective mass and to modify the particle density at very large distances from the different particle.
Introduction to integrable many-body systems I This is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/
Abstract “Identical particles and the helium atom” introduces bosons and fermions. Fermion states are expressed in terms of Slater determinants and the Pauli Principle. Helium is presented in such a way as to show what properties are and are not due to electron identity. Quantum states are described according as the space wave function is symmetric or antisymmetric under interchange of labels attached to the electrons. These in turn form singlet and triplet spin states when the electrons’ fermion identity is taken into account. Helium is an example of LS coupling, but a rather stunted example.
Enforcing the Pauli principle in many-body systems of fermions to ensure an antisymmetric wave function is typically a numerically expensive task. Numerical methods, such as coupled-cluster, correlated basis function, Monte Carlo, and traditional configuration interaction, scale with powers of NNN that make the determination of the energy spectrum of these systems a challenge. In a recent paper, we successfully obtained energies for large systems of fermions in the unitary regime in which we applied the Pauli principle ``on paper'' without obtaining an explicit wave function. This approach, an extension of the symmetry invariant perturbation method to fermions, applies the Pauli principle by imposing restrictions on the quantum numbers used to obtain the energy. As a followup to this study, we perform an explicit test of the validity of this application of the Pauli principle by comparing our energy results against the energies corresponding to explicitly antisymmetrized wave functions for an exactly solvable system of harmonically confined, harmonically interacting fermions. Our results show that our simple method of applying the Pauli principle selects from a spectrum of possible many-body energies only those energies that correspond to explicitly antisymmetrized wave functions. Our results were tested for values of NNN through $\ensuremath{\approx}10 000$ and for weak and strong interactions both repulsive and attractive.