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Physical Review C

Solvable model for one-dimensional nuclear matter: Simultaneous eigenstates of spin, isospin, and energy

T. Kebukawa

Published Feb 1, 1987• 4 citations
Publisher

Abstract

The eigenvalue problem is solved exactly for a one-dimensional system composed of many protons and neutrons interacting via delta-function potentials which conserve the total spin and isospin of protons and neutrons. It is shown that simultaneous eigenstates of spin, isospin, and energy, and energy eigenvalues are determined by the solutions of some transcendental coupled equations for given quantum numbers.

DOI
10.1103/physrevc.35.794

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The eigenvalue problem is solved exactly for a one-dimensional system composed of many protons and neutrons interacting via delta-function potentials which conserve the total spin and isospin of protons and neutrons. It is shown that simultaneous eigenstates of spin, isospin, and energy, and energy eigenvalues are determined by the solutions of some transcendental coupled equations for given quantum numbers.
The eigenvalue problem in a one-dimensional system of many fermions interacting via a delta-function potential is solved exactly in a field theoretical framework. Simultaneous eigenstates of spin and energy are represented in a systematic and compact form, that is, in such a form that the orbital and spin eigenfunctions are unified in harmony without the aid of Young’s tableau. The eigenstate and energy eigenvalue are specified by two sets of quantum numbers, particle momenta and spin momenta, besides two eigenvalues of a total spin angular momentum and its z-component. It is shown that for given particle momenta and spin momenta the energy eigenvalue is given by the solutions of coupled equations.
Equations for all the seniority-zero eigenstates of 2N2N2N nucleons in an arbitrary charge- and spin-independent potential well and interacting through charge- and spin-independent pairing forces are derived. These equations are solved exactly for a large number of states of this system. The interaction in this Hamiltonian is effective in the L=0L=0L=0 states of the two-nucleon system, and its strength is independent of the remaining quantum numbers of the two nucleons. We solve our equations exactly for those states whose wave functions are totally symmetric functions of the spin-isospin coordinates of the NNN L=0L=0L=0 pairs of nucleons in the state. The wave functions of these states factor into a spin-isospin-dependent part and a spatially dependent part. The spin-isospin-dependent part of one of these wave functions is an eigenvector of three tridiagonal matrices which insure that the state is a spin, isospin, and supermultiplet eigenstate, respectively. Explicit expressions are given for the eigenvalues of these three matrices in terms of the quantum numbers of the state. The spatially-dependent part of one of these wave functions is given explicitly in terms of NNN parameters which we call pair energies. These pair energies are shown to satisfy NNN coupled algebraic equations which depend parametrically upon the supermultiplet quantum numbers of the state. An expression for the occupation probabilities of the levels of the single-particle well is given. Throughout this work, an arbitrary splitting of the single-particle levels is treated exactly.