What is the commutation between Xk and Xl?

[Chronos000]

Homework Statement

The question asks for [Xi^2, Lj]

I can get to the line:

ih(bar) Xk ekjl Xl + ih(bar) ekjl Xk Xl

this line must be zero but I don't see how it is.

It looks like an expansion of a commutation between Xk and Xl but not quite right

 

[tiny-tim]

Hi Chronos000!

(try using the X² and X₂ icons just above the Reply box )

ih(bar) Xk ekjl Xl + ih(bar) ekjl Xk Xl

That's just 2ih(bar) X_k e_kjl X_l.

Maybe the second one should be ih(bar) X_k e_ljk X_l ?

 

[Chronos000]

I done it again

got ih X_o e_jmo X_m + ih e_jmo X_m X_o

can I switch the order of the last X operators and put a negative in front? which would give me a commutation

why did u say the previous solution was 2ih(bar) Xk ekjl Xl when the order was different in the last expression?

 

[vela]

That is equal to 0 because X_oX_m is symmetric while e_jmo is antisymmetric.

 

[Chronos000]

I'm sure I've seen something like this before but I really don't have a clue how you can see which is symmetric or not

 

[vela]

X_oX_m is symmetric because you can swap the subscripts without changing anything, i.e. X_oX_m=X_mX_o. The tensor e_jmo on the other hand is antisymmetric because it changes sign when you exchange a pair of subscripts, i.e. e_jmo=-e_jom.

 

[tiny-tim]

Hi Chronos000!

sorry, I wasn't thinking straight earlier …

vela is right, e_jmo is antisymmetric, so if you multiply it by something symmetric (in 2 of the 3 indices in e_jmo), you get 0

(when you do all the adding, for example you add e_j12X₁X₂ and e_j21X₂X₁ … the latter is -e_j12X₂X₁ = -e_j12X₁X₂)

 

[Chronos000]

thanks for your replies. So my expression is just 0 + 0. I don't do anything like you have done in your final comment tiny-tim?

Are you allowed to change the position of the epsilon yes? as it does really "act" on anything but just tells you what the cross product is.

 

[tiny-tim]

thanks for your replies. So my expression is just 0 + 0. I don't do anything like you have done in your final comment tiny-tim?

That's right …

I only did that to prove that it works, but you can just say "it's obvious!"

Are you allowed to change the position of the epsilon yes? as it does really "act" on anything but just tells you what the cross product is.

You can change the position of any whole thing …

what you can't do is change the position of indices (unless they're symmetric indices, in which case you can, or unless they're anti-symmetric indices, in which case you can if you multiply by -1).

 

[Chronos000]

thanks I'm pretty happy with this now