Spherical Harmonics

[docnet]

Homework Statement

psb

Relevant Equations

psb

To show $Y_{1,1}(\theta,\phi)$ is an eigenfunction of $\hat{L}^2$ we operate on $Y_{1,1}(\theta,\phi)$ with $\hat{L}^2$
\begin{equation}
\hat{L}^2Y_{1,1}(\theta,\phi)=\hat{L}^2\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big)
\end{equation}
\begin{equation}
=-\hbar^2\Big[\frac{1}{sin\theta}\frac{\partial}{\partial\theta}\Big(sin\theta\frac{\partial}{\partial\theta}\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big)\Big)+\frac{1}{sin^2\theta}\frac{\partial^2}{\partial\phi^2}\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big)\Big]
\end{equation}

\begin{equation}
=\hbar^2\sqrt{{\frac{3}{8\pi}}}\Big[ \frac{1}{sin\theta}\frac{\partial}{\partial\theta}\Big(sin\theta cos\theta\Big)e^{i\phi}+\frac{1}{sin\theta}\frac{\partial^2}{\partial\phi^2}e^{i\phi}\Big]
\end{equation}
\begin{equation}
=\hbar^2\sqrt{{\frac{3}{8\pi}}}\Big[\frac{1}{sin\theta}(cos^2\theta-sin^2\theta)-\frac{1}{sin\theta}\Big]e^{i\phi}
\end{equation}
\begin{equation}
=\hbar^2\sqrt{{\frac{3}{8\pi}}}\Big[\frac{cos^2\theta-sin^2\theta-1}{sin\theta}\Big]e^{i\phi}
\end{equation}
\begin{equation}
\hat{L}^2Y_{1,1}(\theta,\phi)=2\hbar^2\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big)=2\hbar^2Y_{1,1}(\theta,\phi)
\end{equation}
so $Y_{1,1}(\theta,\phi)$ is an eigenfunction of $\hat{L}^2$ with a corresponding eigenvalue of $2\hbar^2$. Next we work out how $\hat{L_z}$ operates on $Y_{1,1}(\theta,\phi)$
\begin{equation}
\hat{L_z}Y_{1,1}(\theta,\phi)=-i\hbar\frac{\partial}{\partial\phi}(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi})
\end{equation}
\begin{equation}
=i\hbar\sqrt{{\frac{3}{8\pi}}}sin\theta\frac{\partial}{\partial\phi}e^{i\phi}=\hbar\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big)=\hbar Y_{1,1}(\theta,\phi)
\end{equation}
and we find that $Y_{1,1}(\theta,\phi)$ is an eigenfunction of $\hat{L_z}$ with a corresponding eigenvalue of $\hbar$.

 

[kuruman]

This is all correct, nice job. A minor point is that (2)-(5) are not separate equations. Usually, equations are numbered only if it is necessary to refer to them in the text. Numbered equations are usually "bottom lines" after all the algebraic manipulation has been completed.

 

[docnet]

This is all correct, nice job. A minor point is that (2)-(5) are not separate equations. Usually, equations are numbered only if it is necessary to refer to them in the text. Numbered equations are usually "bottom lines" after all the algebraic manipulation has been completed.

that is a great and valid point that I will be sure to remember on my next assignment. Thank you