Central Forces 2022
For \(\ell=1\), the operators that measure the three components of angular momentum
in matrix notation are given by:
\begin{align}
L_x&=\frac{\hbar}{\sqrt{2}}\left(
\begin{matrix}
0&1&0\\
1&0&1\\
0&1&0
\end{matrix}
\right)\\
L_y&=\frac{\hbar}{\sqrt{2}}\left(
\begin{matrix}
0&-i&0\\
i&0&-i\\
0&i&0
\end{matrix}
\right)\\
L_z&=\;\;\;\hbar\left(
\begin{matrix}
1&0&0\\
0&0&0\\
0&0&-1
\end{matrix}
\right)
\end{align}
Show that:
- Find the commutator of \(L_x\) and \(L_y\).
- Find the matrix representation of \(L^2=L_x^2+L_y^2+L_z^2\).
- Find the matrix representations of the raising and lowering operators
\(L_{\pm}=L_x\pm iL_y\).
- Show that \([L_z, L_{\pm}]=\lambda L_{\pm}\). Find \(\lambda\). Interpret this expression as an eigenvalue equation. What is the operator?
- Let \(L_{+}\) act on the following three states given in matrix representation.
\begin{equation}
\left|{1,1}\right\rangle =\left(
\begin{matrix}
1\\0\\0
\end{matrix}
\right)\qquad
\left|{1,0}\right\rangle =\left(
\begin{matrix}
0\\1\\0
\end{matrix}
\right)\qquad
\left|{1,-1}\right\rangle =\left(
\begin{matrix}
0\\0\\1
\end{matrix}
\right)
\end{equation}
Why is \(L_{+}\) called a “raising operator”?