# For the quasispin model of Problem 31.1 , find the eigenvalues of $s_{0}^{(m)}$ for the levels labeled

## Question:

For the quasispin model of Problem 31.1 , find the eigenvalues of $s_{0}^{(m)}$ for the levels labeled by $m$. Show that the system has a total quasispin $S$ that is the vector sum of quasispins for each level $m$, which is the mathematical analog of total spin.

**Data from Problem 31.1**

Shell models are important in various fields of physics. Consider a shell of fermions consisting of $(2 j+1)$ degenerate levels of angular momentum $j$, with each level labeled by a projection quantum number $m$ (this is often called a "single $j$-shell model"). Define the fermion operators for a fixed angular momentum $j$ as

\[ s_{+}^{(m)}=a_{m}^{\dagger} a_{-m}^{\dagger} \quad s_{-}^{(m)}=a_{-m} a_{m} \quad s_{0}^{(m)}=\frac{1}{2}\left(a_{m}^{\dagger} a_{m}+a_{-m}^{\dagger} a_{-m}-1\right), \]

where $a_{m}^{\dagger}$ and $a_{m}$ are the usual fermion creation and annihilation operators, respectively, obeying anticommutation relations of the form (3.23). Show that these operators (called quasispin or pseudospin operators) close under

\[ \left[s_{+}^{(m)}, s_{-}^{(m)}\right]=2 s_{0}^{(m)} \quad\left[s_{0}^{(m)}, s_{ \pm}^{(m)}\right]= \pm s_{ \pm}^{(m)} \]

which is the $\mathrm{SU}(2)$ commutator algebra. Quasispin models are of physical relevance for fermion shells exhibiting a strong tendency for pairs of particles to couple to zero total angular momentum. Such models and their generalizations have found substantial application in nuclear, atomic, and condensed matter physics.

**Data from 3.23**

## Step by Step Answer:

**Related Book For**

## Symmetry Broken Symmetry And Topology In Modern Physics A First Course

**ISBN:** 9781316518618

1st Edition

**Authors:** Mike Guidry, Yang Sun