The quantity B called magnetic flux density is related to H as B = μ 0 H, where
μ 0 is permeability of vacuum; see (7.10) and (7.11) of Sect. 7.1. In (4.88) E and B are
measured at a position where the electron is situated at a certain time t. Equation
(4.88) universally describes the motion of a charged particle in the presence of
electromagnetic fields. We consider another related example in Chap. 15.
Equation (4.86) can be rewritten as
E =
1
ffiffi ffi
2
p E 0 e 1 cos kz À ωt
ð
Þ2 e 2 sin kz À ωt
ð
Þ
½
Š
þi
1
ffiffi ffi
2
p E 0 e 2 cos kz À ωt
ð
Þþe 1 sin kz À ωt
ð
Þ
½
Š :
ð4:89Þ
Suppose that the electron exerts the circular motion in a region narrow enough
around the origin and that the said electron motion is confined within the xy-plane
that is perpendicular to the light propagation direction. Then, we can assume that
z % 0 in (4.89). Ignoring kz in (4.89) accordingly and taking a real part, we have
E =
1
ffiffi ffi
2
p E 0 e 1 cos ωt þ e 2 sin ωt
ð
Þ :
Thus, a force F exerting the electron is described by
F = eE,
ð4:90Þ
where e is an elementary charge (e < 0). Accordingly, an equation of motion of the
electron is approximated such that
m€ x = eE,
ð4:91Þ
E
H
x
y
electron motion
electron
O
Fig. 4.2 Synchronized
motion of an electron under
a left-circularly polarized
light
148
4 Optical Transition and Selection Rules