Commutation relation for [x_i, p_i^n p_j^m p_k^l]

[indigojoker]

i need to find the commutation relation for:

$$[x_i , p_i ^n p_j^m p_k^l]$$

I could apply a test function g(x,y,z) to this and get:

$$=x_i p_i ^n p_j^m p_k^l g - p_i ^n p_j^m p_k^l x_i g$$

but from here I'm not sure where to go. any help would be appreciated.

 

[Gokul43201]

You don't need a test function. All you need are the following:

(i) $[x_i,p_j] = i \hbar \delta_{i,j}$
(ii) $[AB,C]=A[B,C]+[A,C]B$

 

[indigojoker]

should that be $[x_i,p_j] = i \hbar \delta_{i,j}$?

 

[indigojoker]

i guess a more reasonable question would i expand $[x_i,p_i^n]$

 

[Gokul43201]

If you use the second relationship in post #2 recursively, you will discover a general form for the commutator $[x_i,p_i^n]$.
Try p^2 and p^3 first - you'll see what I mean.

PS: Yes, there was a "bad" minus sign which I've now fixed.

 

[indigojoker]

how about: $[x_i,p_i^n]=ni \hbar p_i ^{n-1}$

 

[Gokul43201]

Looks good. Now you're just a step or two away from the answer to the original question.