What Are the Commutation Relations of \( \hat{R}^2 \) with \( \hat{L} \)?

[spacetimedude]

Homework Statement

Deduece the commutation relations of position operator (squared) $\hat R^2$ with angular momentum $\hat L$

Homework Equations

[xi,xj]=0, L_j= ε_ijkx_jP_k, [x_i, P_l]=ih, [xi,Lj]=iℏϵ_ijkx_k

The Attempt at a Solution

The previous question related R and L and the result was $$[\hat R,\hat L_j]=i \hbar \epsilon _{ijk}x_k$$ after setting up the commutator as $$\epsilon _{jkl}[x_i,x_kP_l]$$ where I did not include the i in the epsilon.

Now, I did the same with with $[\hat R^2,\hat L_j]$ and set it up as $$[\hat R^2,\hat L_j]=[x_ix_i,L_j]=\epsilon_{jkl}[x_i,P_l]x_kx_i+x_i\epsilon_{jkl}[x_i,P_l]x_k$$, in which I simplified using the commutator property, and which is then equal to $$i\hbar\epsilon_{jkl}x_kx_i+i\hbar x_i\epsilon_{jkl}x_k$$. I don't think I can reduce it any further.
The solution has the i included in the epsilon in the setup and I don't know why that is.

Any help will be appreciated

 

[PeroK]

I can't follow your use of the epsilon symbol. Why not try calculating:

$[x^2, p_x]$

And from there:

$[R^2, L_x]$

Before you do the calculation, though, what do you think the answer will be?