for the bosonic and fermionic fields can be written as $$\begin{aligned}
i\hbar\frac{\partial \hat\psi_b}{\partial t}= \hat H_b^0
\hat\psi_b(r)+ \int dr'
\hat\psi_b^\dagger(r')U_{bb}(r-r')\hat\psi_b(r')\hat\psi_b(r)
\nonumber
\\+ \int
dr'\hat\psi_f^\dagger(r')U_{bf}(r-r')\hat\psi_f(r')\hat\psi_b(r),
\nonumber \\
i\hbar\frac{\partial \hat\psi_f}{\partial t}= \hat H_f^0
\hat\psi_f(r)+ \int dr'
\hat\psi_b^\dagger(r')U_{bf}(r'-r)\hat\psi_f(r')\hat\psi_f(r).\nonumber\end{aligned}$$ For dilute mixtures one can replace the interaction potentials with the effective interactions $$U_{bb}(r-r')=g_{b} \delta(r-r'),\;\; U_{bf}(r-r')=g_{bf}
\delta(r-r'),$$ where the coupling constants $g_{b}, g_{bf}$, are related to the s-wave scattering lengths $a_{b}, a_{bf}$, by the relations: $g_{b}= 4\pi \hbar^2 a_{b}/m_b$, $g_{bf}=2\pi \hbar^2
a_{bf}/m_{bf}$, where $m_{bf}$ denotes the reduced mass $m_{bf}=m_b m_f/ (m_b+m_f)$. Using these potentials, the above equations reduce to $$\begin{aligned}
 && i \hbar \frac{\partial \hat\psi_b(r,t)}{\partial t}=
\left(\hat H_b^0 +g_{b} \hat\rho_b(r,t) + g_{bf}
\hat\rho_f(r,t)\right) \hat\psi_b(r), \nonumber
\\&& i \hbar\frac{\partial \hat\psi_f(r,t)}{\partial t} =\left( \hat
H_f^0 + g_{bf} \hat\rho_b(r,t)\right) \hat\psi_f(r),
\label{heiseqn2}\end{aligned}$$ where $\hat\rho_b=\hat\psi_b^\dagger\hat\psi_b$ and $\hat\rho_f=\hat\psi_f^\dagger \hat\psi_f$, are the bosonic and fermionic particle densities, respectively. Expanding Bose and Fermi fields in terms of the usual creation and annihilation operators $\hat a^\dagger, \hat a$ and $\hat c^\dagger, \hat c$, for bosons and fermions, respectively, $$\hat \psi_b(r,t)= \sum \psi^b_n \hat a_n, \;\;\; \hat \psi_f(r,t)=
\sum \psi^f_n \hat c_n,$$ and replacing all field operators appearing in the bosonic equation with their expectation values on the ground state (mean field approximation) $$\begin{aligned}
\hat\psi_b 