commutation relations

indigojoker

i need to find the commutation relation for:

[tex] [x_i , p_i ^n p_j^m p_k^l] [/tex]

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

[tex]=x_i p_i ^n p_j^m p_k^l g - p_i ^n p_j^m p_k^l x_i g [/tex]

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

indigojoker

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

indigojoker

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

Gokul43201

If you use the second relationship in post #2 recursively, you will discover a general form for the commutator [itex][x_i,p_i^n] [/itex].
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: [itex][x_i,p_i^n]=ni \hbar p_i ^{n-1} [/itex]

Gokul43201

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