Suppose to have a chain (of size $L$) with bosons, and $\hat{a}_i^\dagger$,$\hat{a}_i$ are the associated creation and annihilation operators at site $i$. A Fock state can be written as: \begin{equation} | n_1 \dots n_L \rangle = \prod_{i} \frac{1}{\sqrt{n_i!}} \left( \hat{a}_i^\dagger \right)^{n_i} |\rangle \end{equation} where $|\rangle$ is the empty state. Now we define a new set of bosons: \begin{align} \hat{b}_{k} &= \frac{1}{\sqrt{L}} \sum_{j} e^{-ikj} \hat{a}_j & \hat{b}_{k}^\dagger &= \frac{1}{\sqrt{L}} \sum_{j} e^{ikj} \hat{a}_j^\dagger \end{align} where $k$ are such that some boundary condition is fulfilled. Now a Fock state can be written as: \begin{equation} | \dots \tilde{n}_k \dots \rangle = \prod_{k} \frac{1}{\sqrt{\tilde{n}_k!}} \left( \hat{b}_k^\dagger \right)^{\tilde{n}_k} |\rangle \end{equation} The question is: is there a simple formula to express or compute the scalar product $\langle \dots \tilde{n}_k \dots | n_1 \dots n_L \rangle$?
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Define the operators $\hat a(f)=\sum f_j\hat a_j$ and $ |f_1,...,f_n\rangle:=\hat a(f_1)...\hat a(f_n)|vac\rangle$. Then $\langle g_1,...,g_m|f_1,...,f_n\rangle$ vanishes for $m\ne n$ and is a sum of the products $\langle g_1|f_{j_1}\rangle...\langle g_n|f_{j_n}\rangle$ for all possible permutations $(j_1,...,j_n)$ of $1,...,n$. Note that the $\langle g|f\rangle$ are easy to compute. The formula you requested is a special case of this. |
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I realise the question has already been answered, but just a suggestion to the topic starter: finite sums of the form $(\sum_i x_i)^N$ are expanded using objects called multinomial coefficients. They should make your life a little easier. |
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