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--- notes:physics:quantum-mechanics:angular-momentum [2026/10/01 16:51] – [Using Operators] chris
+++ notes:physics:quantum-mechanics:angular-momentum [2026/10/01 19:57] (current) – [Example calculations of different quantities] chris
@@ Line 408: / Line 408: @@
 ===== Example calculations of different quantities =====
+Calculating quantities is trivial if their operator can be described as
+a function of the raising and lowering operators. Take for example the
+$x$ and $y$ components of the angular momentum.
+$$\hat L_x=\frac{\hat L_++\hat L_-}{2},\quad\hat L_y=\frac{\hat L_+-\hat L_-}{2}$$
+For example below are the calculations for
+$\left<\hat L_x\right>,\left<\hat L_y\right>,\left<\hat L_x^2\right>, \left<\hat L_y^2\right>$.
+$$\begin{align}
+\left<\hat L_x\right>
+&= \left< l,m\left|\hat L_x\right|l,m\right> \\
+&\propto \left< l,m\left|\hat L_+\right|l,m\right> + \left< l,m\left|\hat L_-\right|l,m\right> \\
+&\propto \left< l,m|l,m+1\right> + \left< l,m|l,m-1\right> \\
+&= 0
+\end{align}$$
+The system is $\phi$-independent, since it has a central potential. This means symmetry around
+the $z$ axis and thus $\left<\hat L_x\right>=\left<\hat L_y\right> =0$. Similarly
+$\left<\hat L_x^2\right>=\left<\hat L_y^2\right>$. Then:
+$$
+\left<\hat L^2\right>
+= \left<\hat L_x^2+\hat L_y^2+\hat L_z^2\right>
+= \left<\hat L_x^2\right> + \left<\hat L_y^2\right> + \left<\hat L_z^2\right>
+= 2\left<\hat L_x^2\right> + \left<\hat L_z^2\right>
+$$
+$$\begin{align}
+\left<\hat L_x^2\right>
+= \left<\hat L_y^2\right>
+&= \frac12\left(\left<\hat L^2\right> - \left<\hat L_z^2\right> \right) \\
+&= \frac12\left(\left< l,m\left|\hat L^2\right|l,m\right> -\left< l,m\left|\hat L_z^2\right|l,m\right>\right) \\
+&= \frac12\Big(l(l+1)\left< l,m|l,m\right> - m^2\left< l,m|l,m\right>\Big) \\
+&= \frac{l(l+1)-m^2}{2}
+\end{align}$$
+This way we can find the dispersion around the expected value.
+$$\require{cancel}
+\Delta L_x
+= \Delta L_y
+= \sqrt{\left<\hat L_x^2\right> - \cancelto{0}{\left<\hat L_x\right>^2}}
+= \sqrt{\frac{l(l+1)-m^2}{2}}
+$$

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