- #1
Frank Einstein
- 170
- 1
Homework Statement
Good morning or afternoon everybody. I am trying to calculate the commutator between two components of angular momenta by following somebody's steps. During the process, I arrive to one step which I can't follow. It's between before and after the equal.
Σnml ∈jnm∈kln iħ XlPm - Σnmk ∈jnm∈kmh iħ XnPh = iħ(Σml (-δjkXlPl)+Σm∈jkmXjPk+ΣmkXnPnδjk-Σn∈jkmXjPk).
What I don't get is how we transform the levi civita symbols (∈)
Homework Equations
[/B]
https://wikimedia.org/api/rest_v1/media/math/render/svg/649e6209e5af520ca1a5ea07c33b58591565ab3a
The Attempt at a Solution
[/B]
In this case, we can transform the symbols of the left part of the equal to get four deltas; ∈jmn∈kln = -δjkδml + δjlδmk & ∈jnm∈kmh = -δjkδnh + δjhδmk.
Then, we can use δml on XlPm and δmk on PmXl to obtain what is seen at the left of the equal, Σml (-δjkXlPl) & ΣmkXnPnδjk
But I don't know how can I calculate the other two terms, Σm∈jkmXjPk & -Σn∈jkmXjPk.
If anyone can tell me how to calculate them, I would be very thankfull
Last edited by a moderator: