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arXiv:nucl-th/0501070 · Physical Review C

Number of spinIstates of identical particles

Y. M. Zhao, A. Arima

Published Apr 18, 2005• 16 citations
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Abstract

In this article we study the enumeration of number (denoted as ${D}_{I}$) of spin I states for fermions in a single-j shell and bosons with spin l. We show that ${D}_{I}$ can be enumerated by the reduction from SU$(n+1)$ to SO(3). New regularities of ${D}_{I}$ are discerned. As an example of our new algorithm, we obtained analytical expressions of ${D}_{I}$ for four particles.

DOI
10.1103/physrevc.71.047304

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In this article we study the enumeration of number (denoted as DI{D}_{I}DI​) of spin I states for fermions in a single-j shell and bosons with spin l. We show that DI{D}_{I}DI​ can be enumerated by the reduction from SU(n+1)(n+1)(n+1) to SO(3). New regularities of DI{D}_{I}DI​ are discerned. As an example of our new algorithm, we obtained analytical expressions of DI{D}_{I}DI​ for four particles.
In this paper we study the number of states with given spin for identical particles, denoted as DIn(l){D}_{I}^{n}(l)DIn​(l) for bosons and as DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) for fermions, where III is the total angular momentum of the system, nnn is the particle number, lll is the spin of a single boson, and jjj is the spin of a single fermion. We derive relations of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j); in particular, we obtain recursive and explicit formulas of DIn(l){D}_{I}^{n}(l)DIn​(l) and DIn(j){\mathcal{D}}_{I}^{n}(j)DIn​(j) in terms of those for $n\ensuremath{-}1$ particles and apply them to systems of two, three, and five identical particles.
A discussion of the seniority quantum number in many-body systems is presented. The analysis is carried out for bosons and fermions simultaneously but is restricted to identical particles occupying a single shell. The emphasis of the paper is on the possibility of partial conservation of seniority which turns out to be a peculiar property of spin-9/2 fermions but prevalent in systems of interacting bosons of any spin. Partial conservation of seniority is at the basis of the existence of seniority isomers, frequently observed in semi-magic nuclei, and also gives rise to peculiar selection rules in one-nucleon transfer reactions. Key words: quantum mechanics, many-body systems, seniority, nuclear shell model, interacting bosons
In this article, we provide an alternative and general way to construct the result of the action of any particle-conserving bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators or even its elements. This approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) that are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and the evolution dynamics of a system become much more efficient, and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.
For identical particles, consisting of fermions, the upper bound for the number of particles that can occupy a single state is determined. In the macroscopic case, it is proportional to the square root of the number of possible ways of formation of particles of a given composition of all fermions present in the mixture (the normalization constant of the respective density matrix). Particles capable of accumulating in macroscopic quantities in one state can consist only of an even number of fermions of different kinds. In the case of atoms in a trap, this bound can approach arbitrarily close to the total number of atoms. Since the state of the centers of mass of the atoms is described by a symmetric wave function, they, like elementary bosons, can form a condensate, the coherence properties of the components of which are characterized by an antisymmetric wave function of a single atom in relative coordinates.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
For single-jjj shells with j=7/2,j=7/2,j=7/2, 9/2, and 11/2,11/2,11/2, we relate the large probability of I+{I}^{+}I+ ground states to the largest (smallest) coefficients ${\ensuremath{\alpha}}_{I(v\ensuremath{\beta})}^{J}=〈\mathrm{nv}\ensuremath{\beta}I|{A}^{J\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\cdot}{A}^{J}|\mathrm{nv}\ensuremath{\beta}I〉,$ where n is the particle number, vvv is the seniority, \ensuremath{\beta} is an additional quantum number, and I is the angular momentum of the state. Interesting regularities of the probabilities of I+{I}^{+}I+ ground states are noticed and discussed for four-particle systems. Several counter examples of the 0+{0}^{+}0+ ground state (0g.s.) predominance are noticed for the first time.