Do L² and r² Commute in Quantum Mechanics?

[dreamspy]

Hi

The subject says it all. I'm wondering if $$[L^2,r^2] = 0$$ is true?

regards
Frímannn

 

[dextercioby]

What do you think ? On what variables does $L^2$ depend ?

 

[dreamspy]

What do you think ? On what variables does $L^2$ depend ?

Well we have that:

$$\underline{\hat L }^2 = \hat L_1^2+\hat L_2^2+\hat L_3^2$$

$$\hat L_1 = \hat x_2 \hat p_3 - \hat x_3 \hat p_2$$

$$\hat L_2 = \hat x_3 \hat p_3 - \hat x_1 \hat p_3$$

$$\hat L_3 = \hat x_1 \hat p_2 - \hat x_2 \hat p_1$$

$$\underline{r }^2 = \hat x_1^2+\hat x_2^2+\hat x_3^2$$

and

$$[x_i,p_j] = i\hbar\delta_{i,j}$$

$$[L_i,p_j] = 0, i = j$$

$$[L_i,p_j] \ne 0, i \ne j$$

$$[L_i,x_j] = 0, i = j$$

$$[L_i,x_j] \ne 0, i \ne j$$

So it seems to me that they don't commute. But I'we been told otherwize so I'm trying to figure it out :)

 

[kanato]

Are you sure about that? I get $$[x_1,L_2] = i\hbar x_3$$

 

[dreamspy]

Sorry that was a mistake. That should have been a $$\ne$$. I fixed the original post.

 

[dextercioby]

I'll give you a further hint: use spherical coordinates.