How Do Angular Momentum Operators Satisfy Algebraic Equations?

[atomicpedals]

Homework Statement

Using matrix representations find $L^{3}_{x},L^{3}_{y},L^{3}_{z}$ and from these show that $L_{x},L_{y},L_{z}$ satisfy the same algebraic equations. What are the roots of the algebraic equations?

2. The attempt at a solution

My problem is that I'm not sure what this question is asking me. I know what matrix reprsentations are; for example

$$L_z=\hbar \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{array} \right)$$
But I've never before come upon reference to $L^{3}_{x},L^{3}_{y},L^{3}_{z}$; what would matrix representations for these look like? Then there's the question about finding the roots of the algebraic equation; when talking about $L_x$ I would tend to think of $L_x=yp_z-zp_y$ as being the algebraic equations. If I'm correct then there is some analog for $L^{3}_{x}$? How does that flow from the matrix representation?

I'm horribly confused here, any help is greatly appreciated.

 

[mark726]

Thank you for your question. It seems like the problem is asking you to find the matrix representations for L^{3}_{x},L^{3}_{y},L^{3}_{z} and then use those matrices to show that L_{x},L_{y},L_{z} satisfy the same algebraic equations. Let me explain in more detail.

The matrices for L^{3}_{x},L^{3}_{y},L^{3}_{z} represent the third components of the angular momentum operators in the x, y, and z directions, respectively. They can be written as:

L^{3}_{x}=\hbar \left( \begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -1 \\
0 & 1 & 0 \end{array} \right)

L^{3}_{y}=\hbar \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0 \end{array} \right)

L^{3}_{z}=\hbar \left( \begin{array}{ccc}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0 \end{array} \right)

Now, using these matrices, you can show that L_{x},L_{y},L_{z} satisfy the same algebraic equations by calculating L^{3}_{x}L^{3}_{y}-L^{3}_{y}L^{3}_{x}, L^{3}_{y}L^{3}_{z}-L^{3}_{z}L^{3}_{y}, and L^{3}_{z}L^{3}_{x}-L^{3}_{x}L^{3}_{z}. You should find that these expressions are equal to i\hbar L_{z}, -i\hbar L_{x}, and i\hbar L_{y}, respectively. This shows that L_{x},L_{y},L_{z} satisfy the same algebraic equations as L^{3}_{x},L^{3}_{y},L^{3}_{z}.

As for the roots of the algebraic equations, they correspond to the eigenvalues of the matrices L^{3}_{x},L^{3}_{y},L^{3}_{z}.