1.8 The Zeeman effect 15The eigenvaluesω^2 are found from the following determinant:
∣ ∣ ∣ ∣ ∣ ∣
ω^20 −ω^2 −2iωΩL 0
2iωΩL ω^20 −ω^20
00 ω 02 −ω^2
∣ ∣ ∣ ∣ ∣ ∣
=0. (1.39)
This gives
{
ω^4 −(
2 ω 02 +4Ω^2 L)
ω^2 +ω^40}
(ω^2 −ω^20 ) = 0. The solution
ω=ω 0 is obvious by inspection. The other two eigenvalues can be found
exactly by solving the quadratic equation forω^2 inside the curly brackets.
For an optical transition we always have ΩLω 0 so the approximate
eigenfrequencies areωω 0 ±ΩL. Substituting these values back into
eqn 1.38 gives the eigenvectors corresponding toω=ω 0 −ΩL,ω 0 and
ω 0 +ΩL, respectively, as
Fig. 1.5A simple model of an atom
as an electron that undergoes simple
harmonic motion explains the features
of the normal Zeeman effect of a mag-
netic field (along thez-axis). The
three eigenvectors of the motion are:
̂ezcosω 0 tand cos ({ω 0 ±ΩL}t)̂ex±
sin ({ω 0 ±ΩL}t)̂ey.r=
cos (ω 0 −ΩL)t
−sin (ω 0 −ΩL)t
0
,
0
0
cosω 0 t
and
cos (ω 0 +ΩL)t
sin (ω 0 +ΩL)t
0
The magnetic field does not affect motion along thez-axis and the angu-
lar frequency of the oscillation remainsω 0. Interaction with the magnetic
field causes the motions in thex-andy-directions to be coupled together
(by the off-diagonal elements±2iωΩLof the matrix in eqn 1.38).^34 The
(^34) The matrix does not have off-
diagonal elements in the last column
or bottom row, so the x-andy-
components are not coupled to thez-
component, and the problem effectively
reduces to solving a 2×2matrix.
result is two circular motions in opposite directions in thexy-plane, as
illustrated in Fig. 1.5. These circular motions have frequencies shifted
up, or down, fromω 0 by the Larmor frequency. Thus the action of the
external field splits the original oscillation at a single frequency (actu-
ally three independent oscillations all with the same frequency,ω 0 )into
three separate frequencies. An oscillating electron acts as a classical
dipole that radiates electromagnetic waves and Zeeman observed the
frequency splitting ΩLin the light emitted by the atom.
This classical model of the Zeeman effect explains the polarization
of the light, as well as the splitting of the lines into three components.
The calculation of the polarization of the radiation at each of the three
different frequencies for a general direction of observation is straight-
forward using vectors;^35 however, only the particular cases where the
(^35) Some further details are given in Sec-
tion 2.2 and in Woodgate (1980).
radiation propagates parallel and perpendicular to the magnetic field
are considered here, i.e. the longitudinal and transverse directions of
observation, respectively. An electron oscillating parallel toBradiates
an electromagnetic wave with linear polarization and angular frequency
ω 0 .Thisπ-component of the line is observed in all directionsexcept
along the magnetic field;^36 in the special case of transverse observation^36 An oscillating electric dipole pro-
portional tôezcosω 0 tdoes not radi-
ate along thez-axis—observation along
this direction gives a view along the
axis of the dipole so that effectively the
motion of the electron cannot be seen.
(i.e. in thexy-plane) the polarization of theπ-component lies along
̂ez. The circular motion of the oscillating electron in thexy-plane at
angular frequenciesω 0 +ΩLandω 0 −ΩLproduces radiation at these
frequencies. Looking transversely, this circular motion is seen edge-on
so that it looks like linear sinusoidal motion, e.g. for observation along