# Simple question: Do L^2 and r^2 commute?

#### dreamspy

Hi

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

regards
Frímannn

#### dextercioby

Homework Helper
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 :)

Last edited:

#### 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

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

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