Bookmark 1 & 2: Zitterbewegung motion of the electron CC (blue) and also of the CM (red) when there is a uniform magnetic field. It is also depicted the CM spin, at different CM points, which forms an angle of with OZ, the initial CM velocity is = 0.1 in natural units. The red spin is the last value of the spin at the end of the integration time. We also show the projection of the CM trajectory on the XOY plane and the projection of the CM spin. The spin of value 1/2 in natural units has been rescaled in the picture.
o
30
v
y
7. Magnetic moment μCM of the electron
7. Magnetic moment of the electron
μ
CM
8. Electric dipole moment dCM of the electron
8. Electric dipole moment of the electron
d
CM
9. Experimental measurement of μCM and dCM
9. Experimental measurement of and
μ
CM
d
CM
10. Electron in a uniform electric field
10. Electron in a uniform electric field
The dynamical equation (3) in natural units and in the presence of a uniform electric field along OX axis = E is
E
e
x
2
2
t
q
AE
γ(v)
e
x
v
v
x
And to express A in natural units we use the expressions derived in Part I.Section14 using the tilde in the variables expressed in natural units : = 2 t = = 2 2 =
q
R
0
q
τ
0
t
τ
0
R
0
c
R
0
ℏ
mc
2
2
t
q
2
R
0
2
2
R
0
c
2
2
t
q
2
c
2
R
0
2
2
t
q
and using (3) and =:
v
v
c
2
2
t
q
2
R
0
2
c
2
2
t
q
2
R
0
2
c
e
mγ(v)
E
1
2
c
v
E
v
2
2
t
q
2
R
0
2
c
e
mγ(v)
E
v
E
v
2
2
t
q
ℏ
2
m
3
c
e
γ(v)
E
v
E
v
A
γ(v)
E
v
E
v
with :
A==1n.u.
eℏ
2
m
3
c
If E is expressed in V/m, we have to use the conversion factor (Part I.Section14) and thus the parameter A E is dimensionless.
With a field of intensity 1 V/m the factor a = A E = A · 1 =7.55676 · in natural units.
In natural units, the system of differential equations is transformed into the system of first order
With a field of intensity 1 V/m the factor a = A E = A · 1 =7.55676 ·
−19
10
In natural units, the system of differential equations is transformed into the system of first order
t
r
u
t
u
1-
v
u
2
(-)
q
r
q
r
t
q
v
t
v
aE
γ(v)
e
x
v
v
x
(
6
)This last equation written by components
t
v
x
aE
γ(v)
2
v
x
t
v
y
-aE
γ(v)
v
x
v
y
t
v
z
-aE
γ(v)
v
z
v
x
where the value of E is the intensity of the electric field in V/m.
In the dynamics of the electron we have two different scales:
The scale of the separation between the CM an CC, of order of ≈ 1m, to show the relative motion between these two points, and the macroscopic scale at the laboratory level.
The other is the time scale. The CC is turning around the CM at the speed of light and in natural units the time taken during a turn is π γ(v) where v is the velocity of the CM, and this period is of order of 1s, for low velocity Dirac particles and multiplied by the γ(v) factor for high energy particles.
To depict in the same figure the global and the internal motion we have needed to use very high external fields. The numerical analysis has been performed with very high hypothetical fields that are difficult to produce at the laboratory. Since the dynamical equations are relativistically invariant, they can be solved for arbitrary high velocities. If we use laboratory fields we will not distinguish at that scale a difference between the CC and CM motions, but we can always analyze the spin evolution.
In the dynamics of the electron we have two different scales:
The scale of the separation between the CM an CC, of order of
R
o
−13
0
The other is the time scale. The CC is turning around the CM at the speed of light and in natural units the time taken during a turn is π γ(v) where v is the velocity of the CM
q
−21
0
To depict in the same figure the global and the internal motion we have needed to use very high external fields. The numerical analysis has been performed with very high hypothetical fields that are difficult to produce at the laboratory. Since the dynamical equations are relativistically invariant, they can be solved for arbitrary high velocities. If we use laboratory fields we will not distinguish at that scale a difference between the CC and CM motions, but we can always analyze the spin evolution.