130_notes.dvi

All photon energies contribute to the real part. Onlyphotons that satisfy the delta
function constraint contribute to the imaginary part. Moreover, there will only be an
imaginary part if there is a lower energy state into which the state in question can decay. We can
relate this width to those we previously calculated.

−

2

̄h

ℑ(∆En) =

∑

~k,α

∑

j

2 π|Hnj|^2

̄h

δ(En−Ej− ̄hω)

The right hand side of this equation is just what we previously derivedfor the decay rate of state
n, summed over all final states.

−

2

̄h

ℑ(∆En) = Γn

Thetime dependence of the wavefunction for the statenis modified by the self energy
correction.

ψn(~x,t) =ψn(~x)e−i(En+ℜ(∆En))t/ ̄he

−Γnt

2

This also gives us theexponential decay behaviorthat we expect,keeping resonant scattering
cross sections from going to infinity. So, the width just goes into the time dependence as
expected and we don’t have to worry about it anymore. We can nowconcentrate on the energy
shift due to the real partof ∆En.

∆En≡ℜ(∆En) =

∑

~k,α

∑

j

|Hnj|^2

̄h(ωnj−ω)

Hnj = 〈n|H~k,αabs|j〉

Habs = −

√

̄he^2

2 m^2 ωV

ei

~k·~x

~p·ˆǫ(α)

∆En =

̄he^2

2 m^2 V

∑

~k,α

∑

j

|〈n|ei

~k·~x

~p·ˆǫ(α)|j〉|^2

̄hω(ωnj−ω)

=

e^2

2 m^2 V

∫

V d^3 k

(2π)^3

∑

α

∑

j

|〈n|ei~k·~x~p·ˆǫ(α)|j〉|^2

ω(ωnj−ω)

=

e^2

(2π)^32 m^2

∑

α

∑

j

∫

dΩ

k^2 dk

ω

|〈n|ei

~k·~x

~p·ˆǫ(α)|j〉|^2

(ωnj−ω)

=

e^2

(2π)^32 m^2 c^3

∑

j

∑

α

∫

dΩ

∫

ω|〈n|ei~k·~x~p|j〉·ˆǫ(α)|^2

(ωnj−ω)

dω

In our calculation of the total decay rate summed over polarizationand integrated over photon
direction (See section 29.7), we computed the cosine of the angle between each polarization vector
and the (vector) matrix element. Summing these two and integrating over photon direction we got
a factor of^83 πand the polarization is eliminated from the matrix element. Thesame calculation
applieshere.

∆En =

e^2

(2π)^32 m^2 c^3

∑

j

8 π

3

∫

ω|〈n|ei~k·~x~p|j〉|^2

(ωnj−ω)

dω