in some cases), making the whole framework effectively 3D.

It is important to stress from the onset that we are not trying to solve the evolution of structure formation at the quantum level. Although we make use of quantum mechanics concepts, we merely use it as mathematical “trick” to solve the Vlasov-Poisson system (\[eq:VP\]). For this reason, the constant $\hbar$ appearing in our equations has to be understood as a computational parameter whose value bares no relation to the actual Planck constant $\hbar_{\rmn{phys}}= 1.0545 \cdot10^{-34}~\rmn{m}^2~\rmn{kg}~\rmn{s}^{-1}$.

Once the wavefunctions are built, they are evolved forward in time using [Schrödinger]{}’s equation. The density sourcing Poisson’s equation is obtained through equation (\[eq:2\]) and one can then solve for the potential at each time step using standard techniques. This potential enters the [Schrödinger]{}equation, closing the loop. We have, hence, built a closed system of equations using only a set of wave functions (which serve as a proxy for density) and the gravitational potential. $$\label{eq:SP}
\begin{array}{rcl}
 i\hbar \partial_t \psi_n &=& - \displaystyle\frac{\hbar^2}{2} \nabla^2_x \psi_n +
U\psi_n, \Big.\\
\nabla^2_x U &=& 4\pi G\delta\rho. \Big.
\end{array}$$

The study of structure formation then becomes an exercise in solving $n$ copies of the [Schrödinger]{}equation on a computer, which is 