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Physical Review C

Number of states for identical particles

Man Bao, Y. M. Zhao, A. Arima

Published Jan 12, 2016• 7 citations
Publisher

Abstract

In this paper we study the number of states with given spin for identical particles, denoted as ${D}_{I}^{n}(l)$ for bosons and as ${\mathcal{D}}_{I}^{n}(j)$ for fermions, where $I$ is the total angular momentum of the system, $n$ is the particle number, $l$ is the spin of a single boson, and $j$ is the spin of a single fermion. We derive relations of ${D}_{I}^{n}(l)$ and ${\mathcal{D}}_{I}^{n}(j)$; in particular, we obtain recursive and explicit formulas of ${D}_{I}^{n}(l)$ and ${\mathcal{D}}_{I}^{n}(j)$ in terms of those for $n\ensuremath{-}1$ particles and apply them to systems of two, three, and five identical particles.

DOI
10.1103/physrevc.93.014307

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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−1n\ensuremath{-}1n\ensuremath−1 particles and apply them to systems of two, three, and five identical particles.
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 quantum mechanics, elementary particles of the same type are considered identical. For example, an electron in an atom could be replaced by another electron and the atom would behave in exactly the same way. Nature has taken particular advantage of this, allowing only two types of elementary particles: bosons and fermions. They differ fundamentally in how a collection of identical particles interact. Identical fermions cannot occupy the same quantum state (only one per state!). Identical bosons can occupy the same quantum state. The statistics of identical particles (how they act in groups) has profound affects. For example, the rich and complex structure of the periodic table arises from the principle that only one electron can occupy each quantum state. The spin and the statistics of particles are closely connected: (fermions) bosons have (half-)integer spin. (Spin is an internal angular momentum measured in units of Planck’s constant.)
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.
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.