We begin by recalling the well-known  CE for a single particle.
The time evolution of a non-relativistic spin-
in a local potential is given by the Schrödinger equation,
Similarly, by multiplying Eq. (1) with
, summing up over
and , and taking the imaginary part, we obtain the CE for the
in terms of the spin current
It is interesting to note that when potential is parallel to the spin density (non-linear Schrödinger equation), all components of the spin density fulfill the CEs. In fact, this is exactly the case for the TDHF equation induced by a zero-range two-body interaction, see below. Another interesting case corresponds to the vector potential aligned along a fixed direction in space, say, along the axis, that is . In this case, the time evolutions of the spin-up and spin-down components decouple from one another, that is, implies , and the spin-up and spin-down components individually obey the corresponding CEs.
We also note here that for a nonlocal potential-energy term,