130_notes.dvi

The approximation thate−i~k·~r≈1 is a very good one and is called theelectric dipole or E1
approximation. We previously derived the E1 selection rules (See Section 29.5).

∆ℓ = ± 1.

∆m = 0,± 1.

∆s = 0.

Thegeneral E1 decay result depends on photon direction and polarization. If information
about angular distributions or polarization is needed, it can be pried out of this formula.

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

≈

αωin^3

2 πc^2

∑

λ

∫

dΩγ

∣ ∣ ∣ ∣ ∣ ∣

√

4 π

3

∫∞

0

r^3 drR∗nnℓnRniℓi

∫

dΩYℓ∗nmn

(

ǫzY 10 +

−ǫx+iǫy

√

2

Y 11 +

ǫx+iǫy

√

2

Y 1 − 1

)

Yℓimi

∣ ∣ ∣ ∣ ∣ ∣

2

Summing over polarization and integrating over photon direction, we get a simpler formula
that is quite useful to compute the decay rate from one initial atomic state to one final atomic state.

Γtot=

4 αωin^3

3 c^2

|~rni|^2

Here~rniis the matrix element of the coordinate vector between final and initial states.

For single electron atoms, we cansum over the final stateswith differentmand get a formula
only requires us to do a radial integral.

Γtot=

4 αωin^3

3 c^2

{ℓ+1

2 ℓ+1

ℓ

2 ℓ+1

}

∣ ∣ ∣ ∣ ∣ ∣

∫∞

0

R∗n′ℓ′Rnℓr^3 dr

∣ ∣ ∣ ∣ ∣ ∣

2

for ℓ′=

{

ℓ+ 1

ℓ− 1

The decay rate does not depend on themof the initial state.

33.12.1Beyond the Electric Dipole Approximation

Some atomic states have no lower energy state that satisfies the E1 selection rules to decay to.
Then, higher order processes must be considered. The next order term in the expansion ofe−i~k·~r=
1 −i~k·~r+...will allow other transitions to take place but at lower rates. We will attempt to
understand theselection ruleswhen we include thei~k·~rterm.

The matrix element is proportional to−i〈φn|(~k·~r)(ˆǫ(λ)·~pe)|φi〉which we willsplit up into two
terms. You might ask why split it. The reason is that we will essentially becomputing matrix
elements of at tensorand dotting it into two vectors that do not depend on the atomic state.

~k·〈φn|~r~pe)|φi〉·ˆǫ(λ)