Solve Spherical Harmonics: Y_{1,1} Eigenfunction of L^2 & L_z

In summary, the conversation covers how to show that ##Y_{1,1}(\theta,\phi)## is an eigenfunction of ##\hat{L}^2## and how to find the eigenvalue and eigenfunction of ##\hat{L_z}##.
  • #1
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Homework Statement
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Relevant Equations
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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##.
 
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  • #2
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.
 
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  • #3
kuruman said:
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
 
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Related to Solve Spherical Harmonics: Y_{1,1} Eigenfunction of L^2 & L_z

1. What are spherical harmonics?

Spherical harmonics are a set of mathematical functions used to describe the distribution of a scalar or vector field on the surface of a sphere. They are commonly used in physics and engineering to solve problems involving spherical symmetry.

2. What is the significance of the Y_{1,1} eigenfunction?

The Y_{1,1} eigenfunction is significant because it represents the lowest non-zero eigenvalue of the spherical harmonics. This means it is the first non-trivial solution to the equations governing the behavior of a scalar field on a sphere. It is also important in quantum mechanics, where it represents the angular momentum of a particle in the x-y plane.

3. How is the Y_{1,1} eigenfunction related to L^2 and L_z?

The Y_{1,1} eigenfunction is an eigenfunction of both the L^2 and L_z operators. This means that when these operators act on the Y_{1,1} function, they produce a scalar multiple of the original function. In other words, the Y_{1,1} eigenfunction is a simultaneous eigenfunction of both L^2 and L_z.

4. How is the Y_{1,1} eigenfunction used in solving problems?

The Y_{1,1} eigenfunction is used in solving problems involving spherical symmetry, such as finding the electric potential around a charged sphere or the gravitational potential around a massive sphere. It is also used in quantum mechanics to describe the angular momentum of particles in spherical coordinates.

5. Are there other eigenfunctions of L^2 and L_z?

Yes, there are an infinite number of eigenfunctions of L^2 and L_z, each corresponding to a different eigenvalue. The Y_{1,1} eigenfunction is just one example, and it is the lowest non-zero eigenvalue. Other examples include Y_{1,0} and Y_{1,-1}, which have higher eigenvalues and correspond to different angular momentum states.

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