PX262 - G8 - photon emission from atoms

light emitted from atoms can be understood as emissions from electron deexcitation
photons can be seen as a periodic perturbation of the electromagnetic field

light polarized in $z$

electromagnetic field in the form of a plane wave, aligned along the $z$ -axis can be described as:

ε = ε_{0} z \cos (\vec{k} \cdot \vec{r} - ω t) \approx ε_{0} z \cos ω t

where, $\vec{k} \cdot \vec{r}$ is small in terms of atomic size

the wavelength is much lager than the atom, so the field can be considered to be uniform across the atom
the perturbation hamiltonian can be written as:

{\hat{H}}^{'} = e ε_{0} z \cos ω t = e ε z

- this changes the state of the electron

the variations are slow, so it can be treated as time-independent:

{\hat{H}}^{″} = e ε_{0} z

the probability of electron transition between two states in the hydrogen atom is given by the overlap integral:

\begin{aligned} H_{12}^{″} & = e ε_{0} \int ϕ_{n_{1} l_{1} m_{1}}^{*} z ϕ_{n_{2} l_{2} m_{2}} d τ \\ \begin{matrix} = e ε \int_{0}^{2 π} \int_{0}^{π} \int_{0}^{\infty} R_{n_{1} l_{1}}^{*} (r) P_{l_{1}}^{| m_{1} |} (\cos θ) e^{- i m_{1} φ} \\ r \cos θ R_{n_{2} l_{2}} (r) P_{l_{2}}^{| m_{2} |} (\cos θ) e^{i m_{2} φ} r^{2} \sin θ d r d θ d φ \end{matrix} \\ \begin{matrix} = e ε \int_{0}^{2 π} e^{i (m_{2} - m_{1}) φ} d φ \\ \int_{0}^{π} P_{l_{1}}^{| m_{1} |} (\cos θ) P_{l_{2}}^{| m_{2} |} (\cos θ) \cos θ \sin θ d θ \\ \int_{0}^{\infty} R_{n_{1} l_{1}}^{*} (r) R_{n_{2} l_{2}} (r) d r \end{matrix} \end{aligned}

the integral must be non-zero for transmissions to occur
the integral cannot be fully evaluated generically, but the quantum numbers can be worked out
considering the integral over $φ$ , there is the product of two eigenfunctions of ${\hat{L}}_{z}$ , which will be non-zero only if $m_{1} = m_{2}$
considering the integral over $θ$ , the relationships between legendre polynomials:

(2 l + 1) \cos θ P_{l}^{| m |} = (l - | m | + 1) P_{l + 1}^{| m |} + (l + | m |) P_{l - 1}^{| m |}

using this relation for $P_{2}^{| m_{2} |}$ , the integral can be rewritten as:

A \int_{0}^{π} P_{l_{1}}^{| m_{1} |} P_{l_{2 + 1}}^{| m_{2} |} \sin θ d θ + B \int_{0}^{π} P_{l_{1}}^{| m_{1} |} P_{l_{2 - 1}}^{| m_{2} |} \sin θ d θ

where, $A$ and $B$ contain constants that can be supressed for simplification

using the orthogonality of the legendre polynomials, the integral will be non-zero only if: $l_{1} = l_{2} + 1$ or $l_{1} = l_{2} - 1$
therefore, it can be concluded that the electron in a hydrogen atom can move between two states for which $l_{1} = l_{2} \pm 1$ and $m_{1} = m_{2}$

light polarized in $x$

the above discussion is only for light polarized in the $z$ -direction, but light can be polarized in all directions
considering light polarized in the $x$ -direction:

{\hat{H}}^{″} = e ε_{0} r \sin θ \cos φ

where, $x = r \sin θ \cos φ$

considering the integral over $φ :$

\frac{1}{2} \int_{0}^{2 π} [e^{i (m_{2} - m_{1} + 1) φ} + e^{i (m_{2} - m_{1} - 1) φ}] d φ

where, $\cos φ = \frac{1}{2} (e^{i φ} + e^{- i φ})$

for this to be non-zero: $m_{1} = m_{2} \pm 1$
considering the integral over $θ :$ similar to above, this will only be non-zero if $l_{1} = l_{2} \pm 1$

summary

transitions are only possible between states which differ in orbital quantum number by $1$ $(Δ l = \pm 1)$ , and the magnetic quantum number remains the same or changes by $1$ $(Δ m = 0, \pm 1)$
these transitions are called electric dipole transitions
at this approximation, all other possibilities are forbidden
such conditions are called selection rules