Spherical harmonics, angular momentum, quantum

[Chronos000]

Homework Statement

I have to construct 3, 3X3 matrices for L_z, L_x, L_y for the spherical harmonics Y(l,m) given l=1 and m = 1,0,-1

So I can determine the relevant harmonics for these values of l and m.

I act with L_z on Y to get

L Y(1,0) = 0

L Y(1,1) = hbar Y(1,1)

L Y(1,-1) = -hbar Y(1,-1)

I'm not sure what to do with this however

 

[fzero]

If you choose the basis

$$Y(1,1) = \begin{pmatrix} 1 \\ 0 \\ 0\end{pmatrix}, ~ Y(1,0) = \begin{pmatrix} 0 \\ 1 \\ 0\end{pmatrix},~ Y(1,-1) = \begin{pmatrix} 0 \\ 0 \\ 1\end{pmatrix},$$

then you should be able to confirm that you've shown that

$$L_z = \begin{pmatrix} \hbar & 0 & 0 \\ 0 & 0 & 0 \\ 0& 0 & -\hbar \end{pmatrix}.$$

You need to work out the action of $$L_x$$ and $$L_y$$ and rewrite them in terms of the basis vectors above. This will give you the matrix representations of the other generators.

 

[Chronos000]

so can I choose whatever basis I want? Y(1,1) could be (0,1,0)?

Do these vectors make up the vertical components of the matrix?

 

[fzero]

so can I choose whatever basis I want? Y(1,1) could be (0,1,0)?

You could choose another basis. The one I suggested is particularly nice if you ever work with the ladder operators

$$L_\pm = L_x \pm i L_y$$

Do these vectors make up the vertical components of the matrix?

The way to see the matrix is to write, say

$$L_z Y(1,1) = m_{11} Y(1,1) + m_{12} Y(1,0) + m_{13} Y(1,-1)$$

$$L_z Y(1,0) = m_{21} Y(1,1) + m_{22} Y(1,0) + m_{23} Y(1,-1)$$

$$L_z Y(1,-1) = m_{31} Y(1,1) + m_{32} Y(1,0) + m_{33} Y(1,-1)$$

Then the coefficients $$m_{ij}$$ are the entries of the corresponding matrix. You should be able to do this for the other components.

 

[paranormal]

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I would suggest breaking down the problem into smaller steps and utilizing known equations and principles to construct the matrices. First, it is important to understand the concept of spherical harmonics, which are mathematical functions used to describe the behavior of waves on a spherical surface. Next, we can use the equations for angular momentum and quantum numbers to determine the values for Lx, Ly, and Lz for the given values of l and m. From there, we can construct the matrices by using the appropriate mathematical operations and principles. Additionally, it may be helpful to consult with other scientists or resources to ensure the accuracy and completeness of the matrices.