Derivation of the ladder operators for the Orbital Angular Momentum, L | Quantum Science Philippines

## Derivation of the ladder operators for the Orbital Angular Momentum, L

Vanessa V. Destura, MSU-IIT

Let $\left|l,m\right>$ be an eigenvector of $L_{+}$ and $L_{-}$ with eigenvalues $A$ and $B$ respectively.

$\begin{array}{lclr} L_{+} \left|l,m\right> &=& A \left|l,m\right> & \textbf{(1)}\\ L_{-} \left|l,m\right> &=& B \left|l,m\right> & \textbf{(2)}\end{array}$

Now, we take the square of the norm of the eigenvalue equations for $L_{+}$ and $L_{-}.$

For $L_{+}$:

$\begin{array}{rclr} {\left| L_{+} \left|l,m\right>\right|}^{2} &=& \left \\ \left &=& {\left| A \right|}^{2} \left & \textbf{(3)}\end{array}$

For $L_{-}$:

$\begin{array}{rclr} {\left| L_{-} \left|l,m\right>\right|}^{2} &=& \left\\ \left &=& {\left| B \right|}^{2} \left & \textbf{(4)}\end{array}$

where $\left = \left = 1.$

Note that

$\begin{array}{lclr} L_{-}L_{+} &=& L^{2} - {L_{z}}^{2} -\hbar L_{z} & \textbf{(5)} \\ L_{+}L_{-} &=& L^{2} - {L_{z}}^{2} +\hbar L_{z} & \textbf{(6)} \end{array}$

We expand the left-hand side of the equations obtained for $L_{+}$ (equation $\textbf{(3)}$) and $L_{-}$ (equation $\textbf{(4)}$) and substitute equations $\textbf{(5)}$ and $\textbf{(6)}$ respectively, where

$\begin{array}{lcl} L^{2} &=& l(l+1){\hbar}^{2} \\ L_{z} &=& m\hbar\end{array}$

So, for $L_{+}$:

$\begin{array}{lcl} \left &=& \left\\ &=& \left \\&=& {\hbar}^{2} (l(l+1) - m(m+1)) \left \end{array}$

where $\left = 1.$ Going back to $\textbf{(3)}$, we have

$\begin{array}{lcl} \left| A \right|^{2} &=& {\hbar}^{2} (l(l+1) - m(m+1)) \\ A &=& \hbar \sqrt{l(l+1) - m(m+1)}\end{array}$

For $L_{-}$:

$\begin{array}{lcl} \left &=& \left\\ &=& \left \\&=& {\hbar}^{2} (l(l+1) - m(m-1)) \left \end{array}$

Going back to $\textbf{(4)}$, we have

$\begin{array}{lcl} \left| B \right|^{2} &=& {\hbar}^{2} (l(l+1) - m(m-1)) \\B &=& \hbar \sqrt{l(l+1) - m(m-1)}\end{array}$

With this, we obtain

$\begin{array}{lcl} L_{+} \left|l,m\right> &=&\hbar \sqrt{l(l+1) - m(m+1)} \left|l,m+1\right> \\ &=& \hbar \sqrt{l^2 - m^2 + l - m - lm + lm} \left|l,m+1\right> \\ &=& \hbar \sqrt{(l-m) (l+m+1)} \left|l,m+1\right>\end{array}$

$\begin{array}{lcl} L_{-} \left|l,m\right> &=&\hbar \sqrt{l(l+1) - m(m-1)} \left|l,m+1\right> \\ &=& \hbar \sqrt{l^2 - m^2 + l + m - lm + lm} \left|l,m-1\right> \\ &=& \hbar \sqrt{(l+m) (l-m+1)} \left|l,m-1\right>\end{array}$

Hence, the ladder operators for the Orbital Angular Momentum, $L$ are

$L_{+} \left|l,m\right> = \hbar \sqrt{(l-m) (l+m+1)} \left|l,m+1\right>$

$L_{-} \left|l,m\right> =\hbar \sqrt{(l+m) (l-m+1)} \left|l,m-1\right>$