Anguluar momentum Commutation Identity

[decerto]

Homework Statement

Given that $[A_i,J_j]=i\hbar\epsilon_{ijk}Ak$ where A_i is not invariant under rotation

Show that $[J^2,Ai]=-2i\hbar\epsilon_{ijk}J_jAk-2\hbar^2A_i$

Homework Equations

$[AB,C]=A[B,C]+[A,C]B$

$[A,B]=-[B,A]$

The Attempt at a Solution

[itex][/itex]
$[J^2,Ai]=[J_x^2,Ai]+[J_y^2,Ai]+[J_z^2,Ai]$
$=J_x[J_x,Ai]+[J_x,Ai]J_x+J_y[J_y,Ai]+[J_y,Ai]J_y+J_z[J_z,Ai]+[J_z,Ai]J_z$
$=-J_x\epsilon_{ixk}Ak-\epsilon_{ixk}AkJ_x-J_y\epsilon_{iyk}Ak-\epsilon_{iyk}AkJ_y-J_z\epsilon_{izk}Ak-\epsilon_{izk}AkJ_z$

Not sure where to go from here

 

[TSny]

You left out a factor of $i\hbar$ in getting to your last line.

Note that your final line can be written compactly as $-\epsilon_{ijk}(J_jA_k + A_kJ_j)$

 

[decerto]

You left out a factor of $i\hbar$ in getting to your last line.

Note that your final line can be written compactly as $-\epsilon_{ijk}(J_jA_k + A_kJ_j)$

Thanks that looks a lot easier to deal with, I guess I use that $A_kJ_j= J_jA_k-[A_k,J_j]$?

 

[TSny]

Thanks that looks a lot easier to deal with, I guess I use that $A_kJ_j= J_jA_k-[A_k,J_j]$?

That's the right idea, but there's a sign error in your expression $A_kJ_j= J_jA_k-[A_k,J_j]$.

 

[decerto]

That's the right idea, but there's a sign error in your expression $A_kJ_j= J_jA_k-[A_k,J_j]$.

I have the exact same thing written?

 

[TSny]

I have the exact same thing written?

Yes, I was just rewriting the same expression that you wrote. But the expression is incorrect due to sign errors.

 

[decerto]

Yes, I was just rewriting the same expression that you wrote. But the expression is incorrect due to sign errors.

Ah right sorry, I had the right expression on the page and it worked so I was confused why you were correcting me

 

[decerto]

Yes, I was just rewriting the same expression that you wrote. But the expression is incorrect due to sign errors.

To prove the full identity $[J^2,[J^2,A]]=2\hbar^2(J^2A+AJ^2)-4\hbar^2(A\cdot J)J$ can I just use a nested expression of what I just proved, as in let $[J^2,Ai]=Ai$ in my original identity

 

[TSny]

To prove the full identity $[J^2,[J^2,A]]=2\hbar^2(J^2A+AJ^2)-4\hbar^2(A\cdot J)J$ can I just use a nested expression of what I just proved, as in let $[J^2,Ai]=Ai$ in my original identity

Yes, but of course $[J^2,Ai] \neq Ai$. I managed to get the result, but only after a couple of pages of tedious index manipulations. I suspect there is a more elegant way to get to the result, but I don't see it.

 

[decerto]

Yes, but of course $[J^2,Ai] \neq Ai$. I managed to get the result, but only after a couple of pages of tedious index manipulations. I suspect there is a more elegant way to get to the result, but I don't see it.

Its a bonus problem, and the last bonus problem was about 6 pages of tedious trigonometric stuff so it wouldn't surprise me if there wasn't, the identity is from dirac but I can't find his original derivation. Thanks for your help though