130_notes.dvi

〈n|Hint(abs)|i〉 = −

e

mc

1

√

V

√

̄hc^2

2 ω

〈ψn;n~k,α− 1 |ǫˆ(α)·~p

(

ak,α(0)eikρxρ

)

|ψi;n~k,α〉

= −

e

m

1

√

V

√

̄h

2 ω

〈ψn;n~k,α− 1 |ˆǫ(α)·~p

√

n~k,αeikρxρ|ψi;n~k,α− 1 〉

= −

e

m

1

√

V

√

̄hn~k,α

2 ω

〈ψn|ei

~k·~r

ˆǫ(α)·~p|ψi〉e−iωt

Similarly, for theemission of a photonthe matrix element is.

〈n|Hint|i〉 = 〈ψn;n~k,α+ 1|−

e

mc

A~·~p|ψi;n~

k,α〉

〈n|H

(emit)

int |i〉 = −

e

mc

1

√

V

√

̄hc^2

2 ω

〈ψn;n~k,α+ 1|ˆǫ(α)·~p a†k,α(0)e−ikρxρ|ψi;n~k,α〉

= −

e

m

1

√

V

√

̄h(n~k,α+ 1)

2 ω

〈ψn|e−i

~k·~r

ˆǫ(α)·~p|ψi〉eiωt

These give the same result as our earlier guess to put ann+ 1 in the emission operator (See Section
29.1).

33.12Review of Radiation of Photons

In the previous section, we derived the same formulas for matrix elements (See Section 29.1) that
we had earlier used to study decays of Hydrogen atom states with no applied EM field, that iszero
photons in the initial state.

Γi→n =

(2π)^2 e^2

m^2 ωV

|〈φn|e−i

~k·~r

ˆǫ·~p|φi〉|^2 δ(En−Ei+ ̄hω)

With the inclusion of thephase space integralover final states this became

Γtot=

e^2 (Ei−En)

2 π ̄h^2 m^2 c^3

∑

λ

∫

dΩp|〈φn|e−i

~k·~r

ˆǫ(λ)·~pe|φi〉|^2

The quantity~k·~ris typically small for atomic transitions

Eγ=pc= ̄hkc≈

1

2

α^2 mc^2

r≈a 0 =

̄h

αmc

kr≈

1

2

α^2 mc

̄h

̄h

αmc

=

α

2

Note that we have take the full binding energy as the energy difference between states so almost all
transitions will havekrsmaller than this estimate. This makes~k·~ran excellent parameter in which
to expand decay rate formulas.