# Bosonic annihilation and creation operators commutators

## 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

nrqed
Homework Helper
Gold Member

## 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

## 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.