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I tried to ask a 'homework'-like question and I ran into a problem with the style of it because I have never seen a question like that. The general problem was that I didn't know how to express an operator which is in terms of creation and annihilation operators, as an operator in terms of the field operator.

I think this question would be helpful to many people as the issue isn't specific to this one computation and can be applied in many circumstances. Are there any edits I can make to get this question reopened?

The question: How can I find a operator originally expressed in terms of raising and lowering operators in terms of the field operators?

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  • $\begingroup$ you have an accepted answer so is this just to get experience at writing good questions or do you want to get additional answers? $\endgroup$
    – ZeroTheHero
    Aug 21, 2022 at 15:11
  • $\begingroup$ @ZeroTheHero Well, it's both. The question got closed by others, however, I think it should remain open as I couldn't find anywhere online or on StackExchange how to convert a operator originally expressed in terms of creation and annihilation operators to an operator just dependent of the field and its canonical momentum $\endgroup$
    – Joshua Pasa
    Aug 21, 2022 at 15:18

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I was one of those who VTCd so here are some minimal steps for me.

  1. Remove the screenshot as per this guideline.
  2. As you point out, there is an identity involving $\delta$-functions and exponentials. What is this identity and explain why can't you use it. It's a pretty standard identity so find it.
  3. momentum operators involve derivative, so what is this identity and why can't you use it.
  4. Do you need to use commutation relations. Does this help you or not?

I suspect that these last two points are where the conceptual struggle is located.

Maybe the following can help to identify the conceptual hurdle. Can you compare with ordinary 1d momentum operators or a harmonic oscillator with given frequency? In other words, if you started with $a^\dagger a$ and wanted to covert that to $x$ and $p$, how would you go about doing this? What is the dispersion relation and how does this come into play in relating $k$ and $\omega$? Is it important?

If you can re-edit your question so as to make it broader, and include the above elements so you clearly identify where the (conceptual) problem lies, then maybe your question will be reopened.

Remember that writing goods questions is hard, and writing good homework-like questions is harder. There's a thread for discussion, and another thread containing good examples, so be inspired by this.

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