a well-studied problem. The velocity distribution can be recovered by Fourier transforming the wave functions and if one is interested in the phase space distribution, one can apply the Wigner transform. This is, however, not part of the algorithm itself. This can be done in post-processing if necessary. The entire evolution of the system can be done at the “quantum level”, i.e. using the wave functions alone.

We stress that this is another approximation of the true underlying physical problem (equation \[eq:VP\]) and that this framework, as any other, will have limitations. Some of these limitations and their relevance to the case of structure formation studies will be discussed in this paper. We will address those in the context of the science we are interested in and demonstrate how alternative cosmologies, including non cold dark matter scenarios, could effectively be simulated.

The development of this framework has been pioneered by [@Widrow1993] and [@Davies1997] with the important difference that these authors use a single wavefunction and another way to map the distribution function in the quantum world. Their general procedure is very similar to ours: sample the wavefunction from the initial phase space distribution, evolve in time using the Schrödinger-Poisson equations 