- #1
Tom_12
- 6
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Homework Statement
Using the operator identity:
[tex]
\hat{L}^2=\hat{L}_-\hat{L}_+ +\hat{L}_z^2 + \hbar\hat{L}_z
[/tex] show explicitly:
[tex]
\hat{L}^2 = -\hbar^2 \left[
\frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\phi^2} +
\frac{1}{\sin\theta} \frac{\partial}{\partial\theta}
\left(\sin\theta\frac{\partial}{\partial\theta}\right)
\right]
[/tex](Note: all L are operators, i.e. L(hat))
Homework Equations
[tex]
\hat{L}_\pm = \hbar e^{\pm i\phi}\left(\pm \frac{\partial}{\partial\theta} + i\cot\theta\frac{\partial}{\partial\phi}\right) \\
\hat{L}_z = -i\hbar \frac{\partial}{\partial\phi}
[/tex]
The Attempt at a Solution
\begin{align*}
\hat{L}^2 &= \hat{L}_-\hat{L}_+ + \hat{L}_z^2 + \hbar \hat{L}_z \\
&= \hbar e^{-i\phi}\left(- \frac{\partial}{\partial\theta} + i\cot\theta\frac{\partial}{\partial\phi}\right) \times \hbar e^{+i\phi}\left(+ \frac{\partial}{\partial\theta} + i\cot\theta\frac{\partial}{\partial\phi}\right) + \left(-i\hbar \frac{\partial}{\partial\phi}\right)^2 + \hbar \left(-i\hbar \frac{\partial}{\partial\phi}\right) \\
&= \hbar^2\left[\left(- \frac{\partial}{\partial\theta} + i\cot\theta\frac{\partial}{\partial\phi}\right)\left( \frac{\partial}{\partial\theta} + i\cot\theta\frac{\partial}{\partial\phi}\right)\right] -\hbar^2\frac{\partial^2}{\partial\phi^2} - i\hbar^2\frac{\partial}{\partial\phi} \\
&= -\hbar^2\left[\left( \frac{\partial}{\partial\theta}\right)^2 + \left(\cot\theta\right)^2 + \left(\frac{\partial}{\partial\phi}\right)^2 + i\frac{\partial}{\partial\phi}\right] \\
&= -\hbar^2\left[\left( \frac{\partial}{\partial\theta}\right)^2 + \left(\frac{\partial}{\partial\phi}\right)^2\left(\cot^2\theta+1\right) + i\frac{\partial}{\partial\phi}\right] \\
&= -\hbar^2\left[\left( \frac{\partial}{\partial\theta}\right)^2 + \left(\frac{\partial}{\partial\phi}\right)^2\left(\frac{1}{\sin^2\theta}\right) + i\frac{\partial}{\partial\phi}\right]
\end{align*}
not sure how to procced from here, it's close to the required form but I do not know how to deal with the [itex] i\frac{\partial}{\partial\phi} [/itex] term or I might have made mistakes...
Hope someone can help, thanks
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