As shown e.g. in ref. Zitterbewegung (trembling motion) of the free Dirac electron is generated if transitions between positive and negative energy states occur. Here we treat this effect in a single-mode configuration using a density matrix method. As compared with more elaborate conventional treatments, this method allows an easy estimate of the amplitude of the motion. The result is by predicted spreads of the free electron charge.
The density matrix calculation
We consider solutions of the Dirac equation of the form
with p a momentum in a given space direction. We then define a density matrix by the relation
where the symbol × designates a tensor product. The quantity
satisfies the Liouville-von Neumann equation,  natural units
involving the commutator with the free-electron Hamiltonian hD. The functions
obey the eigen value equations with E > 0.
specifying them as positive and negative energy solutions. Introducing the function of eq. (1) into the density matrix of eq. (2) we write
Here the commutators
vanish and we are left with the equations
yielding the solutions
For the time-independent factors u(p), v(p) we now introduce quantities corresponding to a Lorentz boost in the x3 direction and more explicitly those given by the following expressions 
Working out the tensor products we then find after some algebra and omitting for simplicity the superscript on p3.
So far the density matrix has not been normalized to unity. To achieve normalization we first notice that we have
from the definitions of these quantities. Noticing further that, according to eq.’s (8.1) and (8.2) we have
we arrive at the total trace.
Let us now consider the amplitude operator of the oscillatory electron motion, which we define as follows:
and calculate the average values
and . Using eq.(8.1) and (8.2) we then find
where the normalization factor of eq.(10) has been taken into account.
To describe the motion of the electron, we then add the time-dependent factors of eq.’s (7.1) and (7.2). and thus consider the quantity