  In this work, we derive the time-dependent Hartree(-Fock) equations as an
effective dynamics for fermionic many-particle systems. Our main results are
the first for a quantum mechanical mean-field dynamics for fermions; in
previous works, the mean-field limit is usually either coupled to a
semiclassical limit, or the interaction is scaled down so much, that the system
behaves freely for large particle number $N$. We consider the fermionic Hartree
equations (i.e., the Hartree-Fock equations without exchange term) in this
work, since the exchange term is subleading in our setting. The main result is
that the fermionic Hartree dynamics approximates the Schr\"odinger dynamics
well for large $N$. We give explicit values for the rates of convergence. We
prove two types of results. The first type is very general and concerns
arbitrary free Hamiltonians (e.g., relativistic, non-relativistic, with
external fields) and arbitrary interactions. The theorems give explicit
conditions on the solutions to the fermonic Hartree equations under which a
derivation of the mean-field dynamics succeeds. The second type of results
scrutinizes situations where the conditions are fulfilled. These results are
about non-relativistic free Hamiltonians with external fields, systems with
total kinetic energy bounded by $const \cdot N$ and with long-range
interactions of the form $|x|^{-s}$, with $0 < s < \frac{6}{5}$ (sometimes, for
technical reasons, with a weaker or cut off singularity). We prove our main
results by using a new method for deriving mean-field dynamics developed by
Pickl in [Lett. Math. Phys., 97(2):151-164, 2011]. This method has been applied
successfully in quantum mechanics for deriving the bosonic Hartree and
Gross-Pitaevskii equations. Its application to fermions in this work is new.
Finally, we show how also the recently treated semiclassical mean-field limits
can be derived with this method.
