How to derive eq.3.34 in Lancaster's QFT book for beginners?

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Homework Help Overview

The discussion revolves around deriving equation 3.34 from Lancaster's book "QFT for the Gifted Amateur." The context involves quantum field theory, specifically focusing on the Hamiltonian operator in different forms.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants compare equation 3.34 to equation 2.41, suggesting that 3.34 may represent an integral version of the discrete case presented in 2.41. There are inquiries about the necessity of knowing the form of \(E_p\) in the integral.

Discussion Status

The discussion is ongoing, with participants expressing a desire for more context and detail to facilitate understanding. Some participants indicate a lack of access to the book, which may hinder their ability to contribute effectively. There is an acknowledgment of the potential for further questions to arise as the discussion progresses.

Contextual Notes

Participants note the challenge of deriving the equation without direct access to the text, which may limit their ability to provide informed responses. There is a suggestion that more detailed context could help clarify the derivation process.

anklimekruk
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Homework Statement


How to derive eq.3.34 in Lancaster's book QFT for the gifted amateur

Homework Equations


##\hat H= \int d^3 p E_p\hat a\dagger_p\hat a_p##

The Attempt at a Solution


Comparing
 
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How to derive the relevant equation ?
By comparing it to eq 2.41 in the book:
##\hat H = \sum_{k=1}^N\hbar\omega_k (\hat a\dagger_k\hat a_k + 1/2)##
I think 3.34 is the integral version of 2.41
 
I would like to know why my question is left without an answer nor a comment
 
anklimekruk said:
I would like to know why my question is left without an answer nor a comment
If they are like me, they do not have easy access to that book and so would have nothing useful to say. Perhaps if you wrote out more context and detail some of us could, indeed, figure out what was happening. (Not guaranteed, but just maybe possible.)
 
Ray Vickson said:
If they are like me, they do not have easy access to that book and so would have nothing useful to say. Perhaps if you wrote out more context and detail some of us could, indeed, figure out what was happening. (Not guaranteed, but just maybe possible.)
I wrote the question so that there is no need for the book
 
anklimekruk said:
I wrote the question so that there is no need for the book

You mean we don't need to know what is the form of ##E_p## in the integral ##\int d^3 p E_p a_p^+ a_p##?
 
Ray Vickson said:
You mean we don't need to know what is the form of ##E_p## in the integral ##\int d^3 p E_p a_p^+ a_p##?
Maybe ##E_p## is the Energie in the momentum which replaces ##\hbar w_k## in 2.41.There is no more explanation in the book for as much as I can see.
In my eyes it must be the notation for the integral version of the discrete case of eq 2.41.I must of course convince myself that it is mathematicaly correct.
More questions will surely arise.I am willing to offer the book to somebody who would like to help me on a regular basis.
 

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