Commutation relations

  • #1
246
0
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.
 

Answers and Replies

  • #2
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
7,051
18
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]
 
Last edited:
  • #3
246
0
should that be [itex] [x_i,p_j] = i \hbar \delta_{i,j} [/itex]?
 
  • #4
246
0
i guess a more reasonable question would i expand [itex][x_i,p_i^n][/itex]
 
  • #5
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
7,051
18
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.
 
  • #6
246
0
how about: [itex][x_i,p_i^n]=ni \hbar p_i ^{n-1} [/itex]
 
  • #7
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
7,051
18
Looks good. Now you're just a step or two away from the answer to the original question.
 

Related Threads on Commutation relations

  • Last Post
Replies
19
Views
2K
  • Last Post
Replies
7
Views
787
  • Last Post
Replies
4
Views
3K
  • Last Post
Replies
3
Views
908
  • Last Post
Replies
1
Views
862
  • Last Post
Replies
1
Views
852
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
778
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
1
Views
278
Top