Derivation of an expression involving boson operators

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The discussion revolves around the derivation of an expression involving boson operators, specifically addressing the expectation value of the identity operator in an n-boson state. Participants clarify that the partial derivative in the expression likely stems from a general formula relating to commutation relations. The term "big N" is suggested to refer to the number operator, which counts the bosons. There is uncertainty about the correctness of the proposed expression, with advice to properly write out the equations and apply the relevant rules. The original poster seeks clarification on specific equations from a non-open access paper, indicating a need for further assistance in understanding the derivation.
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Homework Statement
derive the following expression involving boson operator
Relevant Equations
B=\sum_{i}\alpha_{i}b_{i}
Hi all
I found this expression in a paper that concerns the derivation of some relations about boson operators but it is not very clear to me how the results were obtained. The derivation starts as, let B be an operator as a linear combination of different boson operators:
$$
B=\sum_{i}\alpha_{i}b_{i}
$$
then the expectation value of the identity operator in the n-boson state is :
$$
\bra{B^{n}}\hat{1}\ket{(B^{\dagger})^{n}}=\bra{B^{n-1}}\sum_{i}\alpha_{i}\frac{\partial}{\partial b^{\dagger}_{i}}\ket{(B^{\dagger})^{n}}=n\alpa^{2}N_{n-1}
$$
where the partial derivative came from? and what is big N,the paper doesn't mention that, shouldn't the expression be :
$$
\bra{B^{n}}\hat{1}\ket{(B^{\dagger})^{n}}=\bra{B^{n-1}}B\ket{(B^{\dagger})^{n}}=\bra{B^{n-1}}\sum_{i}\alpha_{i}b_{i}\ket{(B^{\dagger})^{n}}
$$
can any one clarify, I will appreciate any help.
Thanks in advance
 
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patric44 said:
Homework Statement: derive the following expression involving boson operator
Relevant Equations: B=\sum_{i}\alpha_{i}b_{i}
Did you forget some hash hash symbols?

patric44 said:
I found this expression in a paper
It's usually a good idea to include a link to your source, just in case you've mistyped or misunderstood something.

patric44 said:
where the partial derivative came from?
It's possible to prove a general formula like $$[a, f(a^\dagger)] ~=~ i\hbar \, \partial_{a^\dagger} f(a^\dagger) ~.$$The constant ##i\hbar## factor might be different depending on what conventions you're using for the canonical commutation relations. (Exercise: use induction to prove this formula for simple functions like ##f(x) = x^n##, then use linearity of the commutator to generalize the formula to polynomials.)
patric44 said:
and what is big N,
I'm guessing it's the number operator, something involving ##\sum_i b_i^\dagger b_i##.
patric44 said:
the paper doesn't mention that, shouldn't the expression be :
$$
\bra{B^{n}}\hat{1}\ket{(B^{\dagger})^{n}}=\bra{B^{n-1}}B\ket{(B^{\dagger})^{n}}=\bra{B^{n-1}}\sum_{i}\alpha_{i}b_{i}\ket{(B^{\dagger})^{n}}
$$
Without seeing the paper, it's impossible to say for sure. But my guess is "no". Write out the expression properly and apply the rule I described above.
 
strangerep said:
Did you forget some hash hash symbols?It's usually a good idea to include a link to your source, just in case you've mistyped or misunderstood something.It's possible to prove a general formula like $$[a, f(a^\dagger)] ~=~ i\hbar \, \partial_{a^\dagger} f(a^\dagger) ~.$$The constant ##i\hbar## factor might be different depending on what conventions you're using for the canonical commutation relations. (Exercise: use induction to prove this formula for simple functions like ##f(x) = x^n##, then use linearity of the commutator to generalize the formula to polynomials.)

I'm guessing it's the number operator, something involving ##\sum_i b_i^\dagger b_i##.

Without seeing the paper, it's impossible to say for sure. But my guess is "no". Write out the expression properly and apply the rule I described above.
the paper isn't open access so I thought I would write the question separably, here is the link of the paper:
the paper, the commutation relation is included in the paper but i am not interested in proving them, rather my concern is equations 4a,4b,4c
 
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