Let's say we have the two many-body states $$ |\psi_k\rangle=\left(\prod_{k=1}^n c_k^\dagger\right)|0\rangle ,\qquad |\psi_\lambda\rangle=\left(\prod_{\lambda=1}^n c_\lambda^\dagger\right)|0\rangle \tag{1}$$ where $k$ and $\lambda$ label the states in different basis. These are related by $$ c_k=\sum_\lambda \langle k| \lambda \rangle c_\lambda ,\qquad c_k^\dagger=\sum_\lambda \langle \lambda |k\rangle c_\lambda^\dagger\tag{2}$$ Are there techniques to evaluate $\langle\psi_k|\psi_\lambda\rangle$? My first approach is to write
$$ |\psi_\lambda \rangle = \left(\prod_{\lambda'} \sum_k \langle \lambda'|k\rangle c_k^\dagger\right)|0\rangle\tag{3}$$ and then try to expand the sums and match operators in the same basis. This, however, seems to be a nightmare! Is there a different approach?
