We begin by recalling the well-known [12] CE for a single particle.
The time evolution of a non-relativistic spin-
particle moving
in a local potential is given by the Schrödinger equation,
(3) | |||
(4) |
Similarly, by multiplying Eq. (1) with
, summing up over
and , and taking the imaginary part, we obtain the CE for the
spin density
in terms of the spin current
,
(6) | |||
(7) |
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,