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Commutator relations of field operators

  1. Apr 23, 2016 #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: Apr 24, 2016
  2. jcsd
  3. Apr 24, 2016 #2

    strangerep

    User Avatar
    Science Advisor

    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.
     
  4. Apr 24, 2016 #3
    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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