Assume the potential $V$ is not just a function of position. I'm trying to compute $[V, L_i]$. This is what I have so far:

$$ [V, L_i] = [V, \epsilon_{ijk}x_jp_k] = \epsilon_{ijk}(x_j[V,p_k]+[V,x_j]p_k).$$

Now, I could also do the following; (let $i=x$)

$$[V, L_x] = [V, yp_z - zp_y] = y[V,p_z] - z[V, p_y].$$

In which case I wouldn't get the same answer as before, where I used the implicit summation convention. This bothers me, as the second term in the first calculation doens't seem to vanish, since $V$ could well depend on the momentum.

I know both expressions are correct, and I am tempted to simply assume that the potential $V$ commutes with the coordinates $x_i$, but I need to motivate that assumption.

  • 1
    $\begingroup$ Your second expression is equal to the first term of the r.h.s. of your first expression for i=x. You dropped the second term. In the second part of your question, you treat A and B as numbers in 1) but as operators in 2). This may give confusion. $\endgroup$
    – my2cts
    Jun 17, 2018 at 14:37
  • $\begingroup$ I'm still stuck with computing the commutator though. The second term (in the very first equation) needs to dissapear, or so it seems to me at least. I don't know how to do that. $\endgroup$ Jun 17, 2018 at 15:43
  • $\begingroup$ Do I simply assume that $V$ and the $x_i$'s commute? Because then I do get what I expected to get. $\endgroup$ Jun 17, 2018 at 15:54
  • $\begingroup$ If you begin assuming the potential is not just a function of position then I doubt there is more that you can do. $\endgroup$
    – Triatticus
    Jun 17, 2018 at 22:00


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