Summations Homework: Is Rewrite Correct?

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Discussion Overview

The discussion revolves around the correctness of rewriting a summation involving creation and annihilation operators in the context of a physics problem. Participants explore the implications of index shifts and the resulting delta functions, focusing on mathematical justification and simplification techniques.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the introduction of the variable q and the justification for index shifts in the summation.
  • Another participant asserts that the right-hand side should yield a single delta function related to the difference of the indices, suggesting that it eliminates one variable.
  • A later reply indicates that the condition k''' = k - k' + k'' allows for the removal of k''' and prompts further discussion on the next steps.
  • One participant proposes defining q = k'' - k' to simplify the expression, questioning the advantage of this manipulation given that indices are dummy variables.
  • Another participant expresses skepticism about the necessity of the right-hand side transformation, suggesting that the delta function could have been derived directly from the original equation.
  • There is a suggestion that the final expression reflects conservation of momentum, leading to a specific form involving delta functions.
  • Participants discuss potential simplifications and the appearance of indices in the final expression, indicating that while some transformations may yield cleaner forms, the overall simplification may be limited.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of the initial rewriting and the necessity of certain transformations. There is no consensus on the best approach to simplify the expression or the validity of the proposed steps.

Contextual Notes

Some participants note the importance of specifying index ranges for clarity, and there is an acknowledgment of the complexity involved in manipulating the summation and delta functions without resolving the underlying assumptions.

Niles
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Hi guys

Is the following way of rewriting the sum correct?

<br /> \sum\limits_{k,k&#039;,k&#039;&#039;,k&#039;&#039;&#039;} {c_k^\dag c_{k&#039;}^{} c_{k&#039;&#039;}^\dag c_{k&#039;&#039;&#039;}^{} \sum\limits_n {e^{ - ir_n \left( {k - k&#039;} \right)} e^{ - ir_n \left( {k&#039;&#039; - k&#039;&#039;&#039;} \right)} } } = \sum\limits_{k,k&#039;,k&#039;&#039;,k&#039;&#039;&#039;} {c_{k + q}^\dag c_{k&#039;}^{} c_{k&#039;&#039; - q}^\dag c_{k&#039;&#039;&#039;}^{} \sum\limits_n {e^{ - ir_n \left( {k - k&#039;} \right)} e^{ - ir_n \left( {k&#039;&#039; - k&#039;&#039;&#039;} \right)} } } <br />

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'''.
 
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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).
 


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.
 


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.
 
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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?
 


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?
 


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}
 
Last edited:


Yeah, then you would have sum over k',k'' and q. You might get something with nice looking indices like c_{k&#039;}^\dagger c_{k&#039;+q} c_{k&#039;&#039;}^\dagger c_{k&#039;&#039;-q} but I think that's about as far as you could go.
 

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