130_notes.dvi

considered. Applying periodic boundary conditions in a cubic volumeV, the integral over final
states can be done as indicated below.

kxL= 2πnx dnx= 2 Lπdkx

kyL= 2πny dny= 2 Lπdky

kzL= 2πnz dnz= 2 Lπdkz

d^3 n= L

3

(2π)^3 d

(^3) k= V
(2π)^3 d
(^3) k
Γtot=

∫

Γi→nd^3 n

With thisphase space integraldone aided by the delta function, the general formula for the decay
rate is

Γtot=

e^2 (Ei−En)

2 π ̄h^2 m^2 c^3

∑

λ

∫

dΩγ|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2.

This decay rate still contains the integral over photon directions and a sum over final state polar-
ization.

Computation of the atomic matrix element is usually done in the ElectricDipole approximation
(See section 29.5)

e−i

~k·~r

≈ 1

which is valid if the wavelength of the photon is much larger than the size of the atom. With the
help of some commutation relations, the decay rate formula becomes

Γtot=

αωin^3

2 πc^2

∑

λ

∫

dΩγ|ǫˆ·〈φn|~r|φi〉|^2.

The atomic matrix element of the vector operator~ris zero unless certain constraints on the angu-
lar momentum of initial and final states are satisfied. The selection rules for electric dipole (E1)
transitions are:
∆ℓ=±1 ∆m= 0,±1 ∆s= 0.

This is the outcome of the Wigner-Eckart theorem which states that the matrix element of a vector
operatorVq, where the integerqruns from -1 to +1, is given by

〈α′j′m′|Vq|αjm〉=〈j′m′|j 1 mq〉〈α′j′||V||αj〉

Hereαrepresents all the (other) quantum numbers of the state, not the angular momentum quantum
numbers. In the case of a simple spatial operator like~r, only the orbital angular momentum is
involved.

Γ 2 p→ 1 s=

2 αω^3 in

3 c^2

(2)(4π)

∣

∣

∣

∣

∣

4

√

6

(

2

3

) 5

a 0

∣

∣

∣

∣

∣

2

1

12 π

=

4 αω^3 in

9 c^2

∣

∣

∣

∣

∣

4

√

6

(

2

3

) 5

a 0

∣

∣

∣

∣

∣

2

We derive a simple result for the total decay rate of a state, summed over final photon polarization
and integrated over photon direction.

Γtot=

4 αωin^3

3 c^2

|~rni|^2