Bosonic annihilation and creation operators commutators

• RicardoMP
In summary, the individual has already proven two relations and now wants to prove a third one. They have attempted to apply the commutator and use the closure relation, but have not been successful and are seeking advice on where to start from.

Homework Statement

After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.

2. The attempt at a solution

I tried to apply the commutator to a generic state ##\mid i_1,i_2,...,i_N \rangle## in the one-particle basis, but failed to arrived to the desired result.
I also tried to use the closure relation, but couldn't find a way to fit the indices ##j## and ##k## in a kronecker delta ##\delta_{jk}##
I would like to ask for a clue from where to start and proceed from there. Thank you

RicardoMP said:

Homework Statement

After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.

2. The attempt at a solution

I tried to apply the commutator to a generic state ##\mid i_1,i_2,...,i_N \rangle## in the one-particle basis, but failed to arrived to the desired result.
I also tried to use the closure relation, but couldn't find a way to fit the indices ##j## and ##k## in a kronecker delta ##\delta_{jk}##
I would like to ask for a clue from where to start and proceed from there. Thank you
RicardoMP said:

Homework Statement

After proving the relations ##[\hat{b}^{\dagger}_i,\hat{b}^{\dagger}_j]=0## and ##[\hat{b}_i,\hat{b}_j]=0##, I want to prove that ##[\hat{b}_j,\hat{b}^{\dagger}_k]=\delta_{jk}##, however I'm not sure where to begin.

2. The attempt at a solution

I tried to apply the commutator to a generic state ##\mid i_1,i_2,...,i_N \rangle## in the one-particle basis, but failed to arrived to the desired result.
I also tried to use the closure relation, but couldn't find a way to fit the indices ##j## and ##k## in a kronecker delta ##\delta_{jk}##
I would like to ask for a clue from where to start and proceed from there. Thank you
It should work by applying to a generic state. Show what you did and we can help you complete it.

1. What is the significance of bosonic annihilation and creation operators?

Bosonic annihilation and creation operators are fundamental mathematical tools used in quantum mechanics to describe the behavior of bosons, which are particles with integer spin. These operators allow us to model the creation and destruction of bosons, and are essential for understanding the properties and interactions of bosonic systems.

2. How do bosonic annihilation and creation operators relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Bosonic annihilation and creation operators are related to this principle because they represent the operators for position and momentum, respectively, in the quantum mechanical description of a boson. This means that their commutator, which represents their relationship, is related to the uncertainty principle.

3. What is the commutator of bosonic annihilation and creation operators?

The commutator of bosonic annihilation and creation operators is a mathematical expression that describes how these operators behave when applied to a quantum mechanical system. Specifically, it represents the order in which the operators are applied, and how this affects the outcome of the system. The commutator is given by the difference between the operators multiplied in a specific order, and is an important quantity in the study of bosonic systems.

4. How are bosonic annihilation and creation operators used in quantum field theory?

In quantum field theory, bosonic annihilation and creation operators are used to describe the creation and destruction of particles in a field. This field is a mathematical construct that represents the space in which particles exist, and the operators are used to model the behavior of these particles. By applying these operators to the field, we can calculate the probability of finding a particle in a particular state, and understand how the particles interact with each other.

5. Can bosonic annihilation and creation operators be applied to all types of bosons?

Yes, bosonic annihilation and creation operators can be applied to all types of bosons, regardless of their specific properties. This is because these operators are based on the fundamental principles of quantum mechanics, which apply to all particles in the universe. However, the specific values and behaviors of these operators may vary depending on the type of boson being studied.