# Overlap between Many-Body States

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?

• Now expand out (3) and collect all the coefficients corresponding to the state $|\psi_\lambda\rangle$. The result is a Slater determinant. Mar 17, 2022 at 23:44
• It looks like following this approach I will get that the answer is the determinant of the matrix $\langle k | \lambda \rangle$. Is this correct?
– Ivan
Mar 17, 2022 at 23:49
• that is correct. Mar 18, 2022 at 0:01