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

Hi

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

regards
Frímannn
 

dextercioby

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What do you think ? On what variables does [itex] L^2 [/itex] depend ?
 
What do you think ? On what variables does [itex] L^2 [/itex] depend ?
Well we have that:

[tex]\underline{\hat L }^2 = \hat L_1^2+\hat L_2^2+\hat L_3^2
[/tex]

[tex]\hat L_1 = \hat x_2 \hat p_3 - \hat x_3 \hat p_2
[/tex]

[tex]\hat L_2 = \hat x_3 \hat p_3 - \hat x_1 \hat p_3
[/tex]

[tex]\hat L_3 = \hat x_1 \hat p_2 - \hat x_2 \hat p_1
[/tex]

[tex]\underline{r }^2 = \hat x_1^2+\hat x_2^2+\hat x_3^2
[/tex]

and

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

[tex][L_i,p_j] = 0, i = j [/tex]

[tex][L_i,p_j] \ne 0, i \ne j [/tex]

[tex][L_i,x_j] = 0, i = j [/tex]

[tex][L_i,x_j] \ne 0, i \ne j [/tex]

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:
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Are you sure about that? I get [tex][x_1,L_2] = i\hbar x_3[/tex]
 
Sorry that was a mistake. That should have been a [tex]\ne[/tex]. I fixed the original post.
 

dextercioby

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

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