# Angular momentum operator in different bases

The Eigenvectors of $$L_3$$ (for spin 1) are $$\left| m \right>$$ with $$m=1,0,-1$$. One can compute the matrix

$$D_i=\begin{pmatrix}\left< 1 \middle| L_i \middle| 1 \right> & \left< 1 \middle| L_i \middle| 0 \right> & \left< 1 \middle| L_i \middle| -1 \right>\\\left< 0 \middle| L_i \middle| 1 \right> & \left< 0 \middle| L_i \middle| 0 \right> & \left< 0 \middle| L_i \middle| -1 \right> \\\left< -1 \middle| L_i \middle| 1 \right> & \left< -1 \middle| L_i \middle| 0 \right> & \left< -1 \middle| L_i \middle| -1 \right>\end{pmatrix}$$

for $$i=+,-,1,2,3$$.

As an exercise I was supposed to calculate

$$D_i'=\begin{pmatrix}\left< x \middle| L_i \middle| x \right> & \left< x \middle| L_i \middle| y \right> & \left< x \middle| L_i \middle| z \right>\\\left< y \middle| L_i \middle| x \right> & \left< y \middle| L_i \middle| y \right> & \left< y \middle| L_i \middle| z \right> \\\left< z \middle| L_i \middle| x \right> & \left< z \middle| L_i \middle| y \right> & \left< z \middle| L_i \middle| z \right>\end{pmatrix}$$

In the basis $$\left| x \right>=\frac{1}{\sqrt{2}}(\left| -1 \right>-\left| 1 \right>)$$, $$\left| y \right>=\frac{i}{\sqrt{2}}(\left| -1 \right>+\left| 1 \right>)$$, $$\left| z \right>=\left| 0 \right>$$.

I did that by writing

$$\left< x \middle| L_i \middle| x \right> = \frac{1}{\sqrt{2}}(\left< -1 \right| - \left< 1 \right|)L_i(\left| -1 \right> - \left| 1 \right>)=\frac{1}{\sqrt{2}}(\left< -1 \middle| L_i \middle| -1 \right>-\left< -1 \middle| L_i \middle| 1 \right>-\left< 1 \middle| L_i \middle| -1 \right> + \left< 1 \middle| L_i \middle| 1 \right>)$$

For all elements of $$D_i'$$.

What I am wondering is, what does each base physicly mean? I guess in $$\left| x \right>,\left| y \right>,\left| z \right>$$ base the corresponding matrix $$D_i'$$ is the angular momentum operator in the position space while $$D_i$$ is the angular momentum in the Eigenbasis - but how can I imagine it/get an intuitive idea of the difference of using two bases?

• but how can I imagine it/get an intuitive idea of the difference of using two bases? I am not sure if this question is clear enough. Can you elaborate? What type of difference are you looking for? Can your question be boiled down to "What are the reasons we choose to work in some particular basis?"? Jul 2 '19 at 18:49
• The closest you can probably get is en.wikipedia.org/wiki/Vector_model_of_the_atom , but this is quite an outdated approach.
– Cryo
Jul 3 '19 at 0:45
• @AaronStevens Yes, "What are the reasons we choose to work in some particular basis?" might be better phrasing of my question, sorry. Jul 3 '19 at 5:45