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Bosonic annihilation and creation operators commutators

  • #1
47
2

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
 

Answers and Replies

  • #2
nrqed
Science Advisor
Homework Helper
Gold Member
3,605
206

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.
 

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