Commutator relations of field operators

[QuantumRose]

Here is the question:
By using the equality (for boson)

---------------------------------------- (1)
Prove that

Background:
Currently I'm learning things about second quantization in the book "Advanced Quantum Mechanics"(Franz Schwabl).
Given the creation and annihilation operators(

), define field operators as

The following 3 commutator relations are for Boson.

-----------------------------------(2)

And here is my attempt (but it doesn't work):
First step, using equality (1) to expand the commutator:

-------------(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

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

So, by inserting those commutators in (2), I found

 

[strangerep]

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.

 

[QuantumRose]

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' !