Commutator relations of field operators

In summary, in learning second quantization from the book "Advanced Quantum Mechanics" by Franz Schwabl, we are given the creation and annihilation operators and define field operators. The 3 commutator relations for Boson are also provided. To prove the equality (1), we first expand the commutator using (1) and then apply ##\nabla'## to each of the three commutator equations in (2) to simplify our calculations.
  • #1
QuantumRose
11
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:
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  • #2
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
strangerep said:
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' !
 

FAQ: Commutator relations of field operators

1. What are commutator relations of field operators?

Commutator relations of field operators are mathematical relationships that describe how two field operators, which are mathematical representations of physical quantities, behave when they are applied in a specific order. They determine whether the operators commute or anti-commute with each other.

2. Why are commutator relations important?

Commutator relations are important because they help us understand the fundamental properties and behavior of physical systems. They are also crucial in quantum mechanics, where they are used to derive important equations such as the Heisenberg uncertainty principle.

3. How do commutator relations affect measurements?

The commutator relations of field operators directly affect the measurement process in quantum mechanics. This is because the order in which the operators are applied can change the outcome of the measurement, making the results of quantum measurements inherently unpredictable.

4. Can commutator relations be used to simplify calculations?

Yes, commutator relations can be used to simplify calculations in quantum mechanics. By knowing the commutator relations of certain operators, we can determine which terms in an equation will cancel out, making the calculations more manageable.

5. How are commutator relations related to the uncertainty principle?

The uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. Commutator relations play a crucial role in the uncertainty principle, as they determine the limits of our ability to measure these quantities simultaneously.

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