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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

By using the equality (for boson)

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(

The following 3 commutator relations are for Boson.

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

First step, using equality (1) to expand the commutator:

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

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