# Fermion density with Wick's theorem

I want to calculate the expectation value \begin{equation} \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle \end{equation} where $\lvert\textrm{F}\rangle$ is the ground state of a system of $N$ interacting fermions, and $\Psi_m(x)$ the field operator of spin $m$ and position $x$. Using Wick's theorem I can reduce the previous expression to \begin{equation} \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_1}(x_1)\lvert\textrm{F}\rangle \langle\textrm{F}\rvert\Psi^\dagger_{m_2}(x_2)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle + \langle\textrm{F}\rvert\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\lvert\textrm{F}\rangle \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle. \end{equation} Now, in the last term, I have calculated \begin{equation} \langle\textrm{F}\rvert\Psi^\dagger_{m_1}(x_1)\Psi_{m_2}(x_2)\lvert\textrm{F}\rangle= \delta_{m_1m_2}\frac1{2\pi^2r^3}\bigl[\sin(k_\textrm{F}r)-k_\textrm{F}r\cos(k_\textrm{F}r)\bigr] \end{equation} where $r$ is $\lVert x_1-x_2\rVert$ and $k_\textrm{F}$ is the Fermi impulse of the state.

My professor told me that the other factor $\langle\textrm{F}\rvert\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\lvert\textrm{F}\rangle$ of the last term can be reduced to the one I just calculated with an appropriate change of labels and other operations such as taking the adjoint, but I'm not able to arrive at his conclusion.

So I ask you, how can I reuse the previous results in order to avoid calculating the factor $\langle\textrm{F}\rvert\Psi_{m_1}(x_1)\Psi^\dagger_{m_2}(x_2)\lvert\textrm{F}\rangle$?

Hint: note that $$\langle\textrm{F}\rvert\color{red}{\Psi_{m_1}(x_1)}\color{blue}{\Psi^\dagger_{m_2}(x_2)}\lvert\textrm{F}\rangle=-\langle\textrm{F}\rvert\color{blue}{\Psi^\dagger_{m_2}(x_2)}\color{red}{\Psi_{m_1}(x_1)}\lvert\textrm{F}\rangle+\langle\textrm{F}\rvert\{\color{red}{\Psi_{m_1}(x_1)},\color{blue}{\Psi^\dagger_{m_2}(x_2)}\}\lvert\textrm{F}\rangle$$ where $$\{\Psi_{m_1}(x_1),\Psi^\dagger_{m_2}(x_2)\}=\delta_{m_1m_2}\delta(x_1-x_2)$$ denotes an anti-commutator.