130_notes.dvi

We have performed that radial integration which will be unchanged. Assume that we start in a
polarized state withmi= 1. We then look at our result for the angular integration in the matrix
element
∣
∣
∣
∣

∫

dΩY 00 ∗

(

ǫzY 10 +

−ǫx+iǫy

√

2

Y 11 +

ǫx+iǫy

√

2

Y 1 − 1

)

Y 1 mi

∣

∣

∣

∣

2

=

1

4 π

(

ǫ^2 zδmi 0 +

1

2

(ǫ^2 x+ǫ^2 y)(δmi(−1)+δmi 1 )

)

=

1

4 π

(

1

2

(ǫ^2 x+ǫ^2 y)

)

where we have setmi= 1 eliminating two terms.

Lets study the rate as a function of the angle of the photon from the z axis,θγ. The rate will be
independent of the azimuthal angle. We see that the rate is proportional toǫ^2 x+ǫ^2 y. We still must
sum over the two independent transverse polarizations. For clarity, assume thatφ= 0 and the
photon is therefore emitted in the x-z plane. One transverse polarization can be in the y direction.
The other is in the x-z plane perpendicular to the direction of the photon. The x component is
proportional to cosθγ. So the rate is proportional toǫ^2 x+ǫ^2 y= 1 + cos^2 θγ.

If we assume thatmi= 0 then only theǫzterm remains and the rate is proportional toǫ^2 z. The
angular distribution then goes like sin^2 θγ.

29.9 Vector Operators and the Wigner Eckart Theorem

There are some general features that we can derive about operators which are vectors, that is,
operators that transform like a vector under rotations. We haveseen in the sections on the Electric
Dipole approximation and subsequent calculations that the vector operator~rcould be written as
its magnituderand the spherical harmonicsY 1 m. We found that theY 1 mcould change the orbital
angular momentum (from initial to final state) by zero or one unit. This will be true for any vector
operator.

In fact, because the vector operator is very much like adding an additionalℓ= 1 to the initial state
angular momentum, Wigner and Eckart proved that all matrix elements of vector operators can be
written as areduced matrix elementwhich does not depend on any of them, and Clebsch-Gordan
coefficients. The basic reason for this is that all vectors transform the same way under rotations, so
all have the same angular properties, being written in terms of theY 1 m.

Note that it makes sense to write a vectorV~in terms of the spherical harmonics using

V±=∓

Vx±iVy

√

2

and
V 0 =Vz.

We have already done this for angular momentum operators.

Lets consider our vectorVqwhere the integerqruns from -1 to +1. The Wigner-Eckart theorem
says
〈α′j′m′|Vq|αjm〉=〈j′m′|j 1 mq〉〈α′j′||V||αj〉