Bosonic annihilation and creation operators commutators

[RicardoMP]

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 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.