Eigenstates in a 3-state spin 1 system

[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

 

[DrClaude]

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

 

[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 .

 

[DrClaude]

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 $L_-$.

 

[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

 

[dyn]

I've got it now !