  It is well known that any given density rho(x)can be realized by a
determinantal wave function for N particles. The question addressed here is
whether any given density rho(x) and current density j(x) can be simultaneously
realized by a (finite kinetic energy) determinantal wave function. In case the
velocity field v(x) =j(x)/rho(x) is curl free, we provide a solution for all N,
and we provide an explicit upper bound for the energy. If the velocity field is
not curl free, there is a finite energy solution for all N\geq 4, but we do not
provide an explicit energy bound in this case. For N=2 we provide an example of
a non curl free velocity field for which there is a solution, and an example
for which there is no solution. The case $N=3 with a non curl free velocity
field is left open.
