Summations Homework: Is Rewrite Correct?

[Niles]

Hi guys

Is the following way of rewriting the sum correct?

$$\sum\limits_{k,k',k'',k'''} {c_k^\dag c_{k'}^{} c_{k''}^\dag c_{k'''}^{} \sum\limits_n {e^{ - ir_n \left( {k - k'} \right)} e^{ - ir_n \left( {k'' - k'''} \right)} } } = \sum\limits_{k,k',k'',k'''} {c_{k + q}^\dag c_{k'}^{} c_{k'' - q}^\dag c_{k'''}^{} \sum\limits_n {e^{ - ir_n \left( {k - k'} \right)} e^{ - ir_n \left( {k'' - k'''} \right)} } }$$

If yes, then the next step in my calculation is to use that the last sum on the RHS is a delta-function in k,k' and k'',k'''.

 

[mathman]

Off hand I don't see it. Where did q come from and how do you justify the index shifts?
(I'm looking at in purely mathematical terms - you should give the ranges for the indices - it would clarify the analysis).

 

[kanato]

The sum on the right hand side can only be a single delta-function. You should get a delta function in k-k'+k''-k''', which will eliminate one k variable.

 

[Niles]

Thanks guys. Ok, so looking at the LHS, then I have the condition that k''' = k-k'+k'', and the k''' can be removed. What would be the next step from here (I assume some change in indices, but I cannot quite see which one)?

I think we can disregard what I wrote in the OP. It doesn't seem correct.

 

[kanato]

Well you could do something like q = k'' - k'. But since indices are summed over, they are just dummy indices so there is no real advantage to do this. What is your goal with manipulating this quantity?

 

[Niles]

My goal is merely to simplify. I have Fourier Transformed from real-space. If I define q = k'' - k', then do I have the two k'', k' sums into one q-sum?

 

[znbhckcs]

I don't really see why you needed to go to the right hand side of the equation in the first place... You could have gotten that delta function right from the start.

And all you get eventually is a conservation of momentum: k-k'+k''-k'''=0 .
There is no further simplification... so I would write:
$$N \sum c_k _1 ^\dag c _k _2 c_k _3 ^\dag c _k _4 \delta _{k1-k2+k3-k4,0}$$

 

[kanato]

Yeah, then you would have sum over k',k'' and q. You might get something with nice looking indices like $$c_{k'}^\dagger c_{k'+q} c_{k''}^\dagger c_{k''-q}$$ but I think that's about as far as you could go.