29.5 Electric Dipole Approximation and Selection Rules
We can now expand thee−i
~k·~r
≈ 1 −i~k·~r+...term to allow us to compute matrix elements more
easily. Since~k·~r≈α 2 and the matrix element is squared, our expansion will be in powers ofα^2
which is a small number. The dominant decays will be those from the zeroth order approximation
which is
e−i
~k·~r
≈ 1.
This is called theElectric dipole approximation.
In this Electric Dipole approximation, we can make general progresson computation of the matrix
element. If the Hamiltonian is of the formH=p
2
2 m+Vand [V,~r] = 0, then[H,~r] =̄h
ip
mand we can write~p=im ̄h[H,~r] in terms of the commutator.
〈φn|e−i
~k·~r
ǫˆ·~pe|φi〉 ≈ ˆǫ·〈φn|~pe|φi〉=im
̄h
ˆǫ·〈φn|[H,~r]|φi〉=im
̄h
(En−Ei)ˆǫ·〈φn|~r|φi〉=
im(En−Ei)
̄h〈φn|ǫˆ·~r|φi〉This equation indicates the origin of the name Electric Dipole: the matrix element is of the vector
~rwhich is a dipole.
We can proceed further, with the angular part of the (matrix element) integral.
〈φn|ˆǫ·~r|φi〉 =
∫∞
0r^2 drR∗nnℓnRniℓi∫
dΩYℓ∗nmnˆǫ·~rYℓimi=
∫∞
0r^3 drR∗nnℓnRniℓi∫
dΩYℓ∗nmnˆǫ·rYˆℓimiˆǫ·rˆ = ǫxsinθcosφ+ǫysinθsinφ+ǫzcosθ=√
4 π
3(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
〈φn|ˆǫ·~r|φi〉 =
√
4 π
3∫∞
0r^3 drR∗nnℓnRniℓi∫
dΩYℓ∗nmn(
ǫzY 10 +−ǫx+iǫy
√
2Y 11 +
ǫx+iǫy
√
2Y 1 − 1
)
YℓimiAt this point, lets bring all the terms in the formula back together sowe know what we are doing.
Γtot =e^2 (Ei−En)
2 π ̄h^2 m^2 c^3∑
λ∫
dΩγ|〈φn|e−i
~k·~r
ǫˆ(λ)·~pe|φi〉|^2