# Eigenstates in a 3-state spin 1 system

1. May 6, 2015

### dyn

I have been looking at the solution to a question and I don't understand how the eigenstates are calculated. The question concerns a 3-state spin-1-system with angular momentum l=1. The 3 eigenstates of L3 are given as $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ , $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ , $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ which leads to the z-component of angular momentum as L3 = $\hbar$ $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}$. When I try to calculate the eigenvalues and eigenvectors of the L3 matrix using determinants I get no answer. Can anybody tell me how to get the eigenvalues and eigenvectors ? Thanks

2. May 6, 2015

### Staff: Mentor

Could you post the original question? It will help to know what is the assumed starting point.

3. May 6, 2015

### dyn

The original question is - A quantum particle is known to have total angular momentum one ie. l = 1. Use the eigenstates of L2 and L3 , denoted by | l , m > = | 1 , m > as a basis and find the matrix representation of operators L1 , L2 and L2 in this 3-D subspace. Use | 1 , 1> = $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ = | 1 > , | 1 , 0 > = $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ = | 2 > , | 1 , -1 > = $\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$ as our basis. You may find it useful to compute the matrix representation of operators L+ , L- and L3 .

4. May 6, 2015

### Staff: Mentor

Ok, so the idea is to use your knowledge of how operators like $L_3$ work on the basis states to build the matrix. For instance, you (should) know that
$$L_3 | 1, -1 \rangle = -\hbar | 1, -1 \rangle$$
so you build the matrix accordingly. Since you are using the eigenstates of $L_3$ to build the matrix, you know that the resulting matrix is diagonal, with the eigenvalues as the diagonal elements. You do the same for $L^2$. As for $L_1$ and $L_2$, you will indeed find it useful to use the operators $L_+$ and $L_-$.

5. May 6, 2015

### dyn

The bit that I don't understand is that if I was given the L3 matrix I can't work out the eigenvalues and eigenvectors from it

6. May 6, 2015

### dyn

I've got it now !