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The Journal of Chemical Physics

Quantum mechanics of the ground state of four identical fermions

Ludwig W. Bruch

Published May 15, 1980• 9 citations
Publisher

Abstract

Wave functions for four identical spin-one-half fermions with total spin 0, 1, and 2 are constructed. Lower bounds on the ground state energies of these spin states are derived. The results are illustrated with an analysis of 4-fermions interacting via Hooke’s law potentials in three spatial dimensions. Other bounds on the ground state energy of four identical particles are also discussed.

DOI
10.1063/1.438968

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Wave functions for four identical spin-one-half fermions with total spin 0, 1, and 2 are constructed. Lower bounds on the ground state energies of these spin states are derived. The results are illustrated with an analysis of 4-fermions interacting via Hooke’s law potentials in three spatial dimensions. Other bounds on the ground state energy of four identical particles are also 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.
We derive a lower bound for the ground-state energy EF(N,S){E}^{F}(N,S)EF(N,S) of N fermions with total spin S in terms of binding energies ${E}^{F}(N\ensuremath{-}1,S\ifmmode\pm\else\textpm\fi{}1/2)$ of $(N\ensuremath{-}1)$ fermions. Numerical examples are provided for some simple short-range or confining potentials.
Lower bounds to the ground-state energy of a system composed of N identical particles interacting by two-body forces (and possibly also with an external potential) are derived. They are expressed in terms of the ground-state energies of (simpler) systems composed of a smaller number of particles having masses and/or interactions different from the original ones. This reduction process may be continued all the way down to systems involving one and two particles only. The generalization to systems containing some identical and some distinguishable particles is also discussed.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.