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Progress of Theoretical Physics

One-Dimensional Many-Fermion System. II: Spin Eigenstates and Their Structure

T. Kebukawa

Published Mar 1, 1986• 3 citations
Publisher

Abstract

For a given eigenvalue of the square of the total spin operator, the spin eigenvectors with various eigenvalues of the z-component are obtained systematically by successive operations of the spin lowering operator on the eigenvector proposed in a previous paper for a one-dimensional many-fermion system with delta-function potential. The property of the spin eigenvector under the operation of the spin raising operator is investigated to clarify the mechanism how it has the given eigenvalue of the square of the total spin operator. It is shown that the spin eigenvector can be modified in various forms under some condition presented in the previous paper.

DOI
10.1143/ptp.75.506

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For a given eigenvalue of the square of the total spin operator, the spin eigenvectors with various eigenvalues of the z-component are obtained systematically by successive operations of the spin lowering operator on the eigenvector proposed in a previous paper for a one-dimensional many-fermion system with delta-function potential. The property of the spin eigenvector under the operation of the spin raising operator is investigated to clarify the mechanism how it has the given eigenvalue of the square of the total spin operator. It is shown that the spin eigenvector can be modified in various forms under some condition presented in the previous paper.
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.
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.
Recently, we have shown how the interpretation of quantum mechanics due to Lande' can be used to derive from first principles generalized formulas for the operators and some eigenvectors for spin 1/2 Though we gave the operators for all the components of the spin, we did not give the eigenvectors of the operators for the x and y components of the spin. We now give these vectors. In addition, we present a new and simple method of deriving the operators for the x and y components of the spin as well as their vectors from those for the z component. We give a general proof that the operator for the square of the spin is the unit matrix multiplied by the value of the square of the spin.
A unitary operator, which transforms free states into the exact eigenstates for a one-dimensional many boson system with repulsive delta-function potential, is explicitly constructed. By the unitary operator, original bare creation and annihilation operators are transformed into new operators which are shown to create or annihilate the exactly dressed bosons with the interaction cloud. The total Hamiltonian, total number and total momentum in the system are expressed in the diagonalized form in terms of the dressed operators. It is shown that the exact ground state is the condensed state of all exactly dressed bosons with zero momentum.
A generalization to all spin magnitudes of the drone-fermion representation (previously known for spin \textonehalf{}), is presented, and the resulting eigenstates investigated. Linear combinations of products of fermion operators acting on a vacuum state can always be found which correspond to spin states which are eigenvectors of S2{\mathrm{S}}^{2}S2 and Sz{S}^{z}Sz. It is then shown that for the case of spin 1 (units of $\ensuremath{\hbar}$) it is possible to construct a Wick theorem which is analogous to the usual temperature form for standard fermion creation and destruction operators.