Bookmark 4. Motion of the electron in an oscillating electric field along the direction of the OX axis
e
x
. We see the zitter motion of the CC (blue) and the motion of the CM (red). It is also depicted the CM spin, at different CM points, which forms an angle of
o
45
with the direction of the initial CM velocity, of value
v
y
= 0.02 in natural units. The natural frequency of the external field ω = 0.056. The integration time corresponds to a value of 310 units, approximately to 99 turns of the zitter motion. We also show the projection of the CM trajectory on the XOY plane and the projection of the CM spin. ​Note that the spin orientation seems not to change in the presence of an electric field during this short integration time.

12. Electron in a uniform and oscillating magnetic field

We are going to describe the motion of the Spinning Particle model, with the mass and charge of an electron, in a uniform magnetic field. Again, in order to draw in the same figure the cyclotron motion of the center of mass, that has a radius of order of
−6
10
m in a magnetic field of 5 Teslas, and the zitterbewegung motion of the CC around the CM of much smaller radius
R
o
≃
−13
10
m, we have to consider a high velocity electron and a very large magnetic field, to appreciate both trajectories in the same figure.Equation (3) in natural units and with
E
=0 becomes:
2


2
t
q
=
A
γ(v)

u
×
B
c-
v

u
×
B
c
v
​​​​A=
eℏ
2
2
m
3
c
=
1
2
n.u.
(
7
)
with B in the international system of units. If the uniform magnetic field is along OZ axis,
B
= B
e
z
, these equations become
2


2
t
q
=
KB
γ(v)
(
u
×
e
z
-
v
(
u
×
e
z
)
v
)​​​​K=Ac=
eℏ
2
2
m
2
c
=1.13272
-1
T
(
8
)
The system of dynamical equations is transformed into a system of first order differential equations:

t
r
=
u
​​

t
u
=
1-
v
u
2
(
q
-
r
)
(
q
-
r
)​​

t
q
=
v
​​

t
v
=
KB
γ(v)

u
×
e
z
-
v

u
×
e
z

v

This last equation is explicitly written as :

t
v
x
=
KB
γ(v)
(
u
y
-
v
x
(
v
x
u
y
-
v
y
u
x
))​​

t
v
y
=
KB
γ(v)
(-
u
x
-
v
y
(
v
x
u
y
-
v
y
u
x
))​​

t
v
z
=
KB
γ(v)
(-
v
z
(
v
x
u
y
-
v
y
u
x
))
in natural units and the magnetic field is given in Teslas, so that the parameter KB is dimensionless.