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Commutator relations of field operators

  • #1
Here is the question:
By using the equality (for boson)
ABC.png
---------------------------------------- (1)
Prove that
equality.png


Background:

Currently I'm learning things about second quantization in the book "Advanced Quantum Mechanics"(Franz Schwabl).
Given the creation and annihilation operators(
a+ and a.png
), define field operators as
field operators.png

The following 3 commutator relations are for Boson.
commutators of field operators.png
-----------------------------------(2)

And here is my attempt (but it doesn't work):

First step, using equality (1) to expand the commutator:
step 1.png
-------------(3)
since the nabla operator is an operator, so I think the first term of (3)'s right-hand-side can be expressed as following
step 2.png

also, I expressed the second term of (3)'s right-hand-side by using the same method
step 3.png

So, by inserting those commutators in (2), I found
step 4.png
 
Last edited:

Answers and Replies

  • #2
strangerep
Science Advisor
3,062
884
Your line after (3) looks wrong to me.

Try doing this first: apply ##\nabla'## to each of the three commutator equations in (2). That will give you some extra utility formulas that you can use to simply (3) more correctly, and quicker.
 
  • #3
Your line after (3) looks wrong to me.

Try doing this first: apply ##\nabla'## to each of the three commutator equations in (2). That will give you some extra utility formulas that you can use to simply (3) more correctly, and quicker.
That helps me a lot! Thanks! Indeed, my calculations are wrong after (3). And I also forgot that ##\nabla'## only acts on x' !!
 

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