Number of states with fixed angular momentum for identical fermions and bosons

Abstract

We present in this paper empirical formulas for the number of angular momentum I states for three and four identical fermions or bosons. In the cases with large I we prove that the number of states with the same $\mathcal{M}$ and n but different J is identical if $I\ge (n-2)J-1/2(n-1)(n-2)$ for fermions and $I\ge (n-2)J$ for bosons, and that the number of states is also identical for the same $\mathcal{M}$ but different n and J if $\mathcal{M}\le$min(n, 2J+1 - n) for fermions and for $\mathcal{M}\le$min(n, 2J) for bosons. Here $\mathcal{M}=I_{max}-I$, n is the particle number, and J refers to the angular momentum of a single-particle orbit for fermions, or the spin L carried by bosons.

Comments: 9 pages, no figures