Problem following a demonstration

[Frank Einstein]

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.

Σ_nml ∈_jnm∈_kln iħ X_lP_m - Σ_nmk ∈_jnm∈_kmh iħ X_nP_h = iħ(Σ_ml (-δ_jkX_lP_l)+Σ_m∈_jkmX_jP_k+Σ_mkX_nP_nδ_jk-Σ_n∈_jkmX_jP_k).

What I don't get is how we transform the levi civita symbols (∈)

Homework Equations

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https://wikimedia.org/api/rest_v1/media/math/render/svg/649e6209e5af520ca1a5ea07c33b58591565ab3a

The Attempt at a Solution

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In this case, we can transform the symbols of the left part of the equal to get four deltas; ∈_jmn∈_kln = -δ_jkδ_ml + δ_jlδ_mk & ∈_jnm∈_kmh = -δ_jkδ_nh + δ_jhδ_mk.

Then, we can use δml on XlPm and δmk on PmXl to obtain what is seen at the left of the equal, Σ_ml (-δ_jkX_lP_l) & Σ_mkX_nP_nδ_jk

But I don't know how can I calculate the other two terms, Σ_m∈_jkmX_jP_k & -Σ_n∈_jkmX_jP_k.

If anyone can tell me how to calculate them, I would be very thankfull

 

[jedishrfu]

Can you check your indices again?

It seems that the -Σn∈jkmXjPk has no n to index with the summation.

The other ∈jnm∈kmh= -δjkδnh + δjhδmk. looks off too as the righthand side sums over m, right?
and so there should not be an m index on the lefthand side.

CAVEAT: I'm no expert so wait until a math advisor comments here (@micromass or @Mark44)

 

[Frank Einstein]

Can you check your indices again?

It seems that the -Σn∈jkmXjPk has no n to index with the summation.

The other ∈jnm∈kmh= -δjkδnh + δjhδmk. looks off too as the righthand side sums over m, right?
and so there should not be an m index on the lefthand side.

CAVEAT: I'm no expert so wait until a math advisor comments here (@micromass or @Mark44)

You are right, it is ∈_jkn.
On the other hand, I wanted to post an image from wikipedia in which the product of two levi civita symbols is described, but I can't copy the image, what I want to do is to write the symbols as a matrix of deltas, then grab the δ_ii multiplied by the substraction of two products of deltas.

 

[NascentOxygen]

I wanted to post an image from wikipedia

It seems to not be a image. So you'd need to take a screenshot, then attach to your post that screenshot image.

EDIT: If you surround wikipedia's SVG link with IMG tags, you could request that viewers click on Reply (whether or not they may intend replying), and they will see that the image does display as intended in the pop-up editor pane.

https://api.viglink.com/api/click?format=go&jsonp=vglnk_147089323842911&key=6afc78eea2339e9c047ab6748b0d37e7&libId=irpvisre010009we000MAfj4chclj&loc=https%3A%2F%2Fwww.physicsforums.com%2Fthreads%2Fproblem-following-a-demonstration.881099%2F&v=1&out=https%3A%2F%2Fwikimedia.org%2Fapi%2Frest_v1%2Fmedia%2Fmath%2Frender%2Fsvg%2F649e6209e5af520ca1a5ea07c33b58591565ab3a&ref=https%3A%2F%2Fwww.physicsforums.com%2Fwatched%2Fthreads%2Fall&title=Problem%20following%20a%20demonstration%20%7C%20Physics%20Forums%20-%20The%20Fusion%20of%20Science%20and%20Community&txt=https%3A%2F%2Fwikimedia.org%2Fapi%2Frest_v1%2Fmedia%2Fmath%2Frender%2Fsvg%2F649e6209e5af520ca1a5ea07c33b58591565ab3a