Derivation of the ladder operators for the Orbital Angular Momentum, L | Quantum Science Philippines
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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<l,m+1\right| A^{*}A \left|l,m+1\right> \\ \left<l,m\right| L_{-} L_{+} \left|l,m\right> &=& {\left| A \right|}^{2} \left<l,m+1|l,m+1\right> & \textbf{(3)}\end{array}

For L_{-}:

\begin{array}{rclr} {\left| L_{-} \left|l,m\right>\right|}^{2} &=& \left<l,m-1\right| B^{*}B \left|l,m-1\right>\\ \left<l,m\right| L_{+} L_{-} \left|l,m\right> &=& {\left| B \right|}^{2} \left<l,m-1|l,m-1\right> & \textbf{(4)}\end{array}

 

where \left<l,m+1|l,m+1\right> = \left<l,m-1|l,m-1\right> = 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<l,m\right| L_{-} L_{+} \left|l,m\right> &=& \left<l,m\right| L^{2} - {L_{z}}^{2} - \hbar L_{z} \left|l,m\right>\\ &=& \left<l,m\right| l(l+1){\hbar}^{2} - m^2{\hbar}^{2} - m{\hbar}^2 \left|l,m\right> \\&=& {\hbar}^{2} (l(l+1) - m(m+1)) \left<l,m|\l,m\right> \end{array}

where \left<l,m|\l,m\right> = 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<l,m\right| L_{+} L_{-} \left|l,m\right> &=& \left<l,m\right| L^{2} - {L_{z}}^{2} + \hbar L_{z} \left|l,m\right>\\ &=& \left<l,m\right| l(l+1){\hbar}^{2} - m^2{\hbar}^{2} + m{\hbar}^2 \left|l,m\right> \\&=& {\hbar}^{2} (l(l+1) - m(m-1)) \left<l,m|\l,m\right> \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>

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