all about: light-matter interaction

Or: “How photons and electrons say hello”

Low energy — Photoelectric effect
This is the first one you learn: a photon knocks an electron out of its atomic orbit. It is most likely to occur at low energies… as you move up in energy it becomes more likely that the photon will be scattered rather than absorbed.

Medium energy — Compton scattering
While in the photoelectric effect the energy of the incoming photons is absorbed completely by the electrons, at higher energies the photon will instead bounce off the electron, leaving some of its energy/momentum behind in the recoil.

Using relativistic energy/momentum formulas, you can derive the wavelength shift \(\lambda’-\lambda = \frac{h}{m_e c}\left(1-\cos\theta\right)\) (higher wavelength ⇒ lower energy).

High energy — Pair production
γ → e- + e+ looks pretty reasonable, right? If the photon had enough energy, it could account for the mass of the created electron-positron pair, and charge is certainly conserved, so why not? Well, consider this: Given the conservation of momentum, output energy will be minimized by having the two electrons each take half the photon’s original momentum. But this gives \(E_\mathrm{out}=2\sqrt{\left(\frac{pc}{2}\right)^2+(m_e c^2)^2}>pc=E_\mathrm{in}\), so even in this best case we don’t have enough energy to support the electron’s/positron’s momentum.

All that means is that we need some other ingredient in the mix. One good option is an atomic nucleus… When the photon gets near, it can allow the nucleus to absorb some of its momentum, to make electron-positron pair production possible. This is why pair-production is a form of light-matter interaction, rather than just something light does on its own.

(And no matter what, this is definitely going to be a high-energy interaction: the incoming photon must have AT LEAST \(pc>2m_e c^2\).)

twitter by the #s

Some people have numbers as their Twitter names. Which ones?

Click above to see full-size

The full-size image is 1000 pixels high by 1000 pixels wide. Each pixel represents a number from 0 to 999,999. The row number gives the first three digits (0-999), the column number gives the last three digits (0-999)… so the pixels are ordered like letters on a page of English text. The pixel is black if the number is taken as a Twitter name, white if the number is still available.

Let me know if you’ve got any ingenious hypotheses, or if you want the data. (Although, in a very real sense, the image above is the data…)

[Secret bonus: Here's the image in the far more hierarchically-revealing Z-order (which I was delighted to discover has a name and Wikipedia page, since otherwise I would have had a hell of a time trying to explain it).]

all about: splittings

So the electron in the hydrogen atom is just a particle in a spherically-symmetric 1/r potential… you’ve got a ladder of energy eigenvalues indexed by a quantum number n. The nth eigenvalue has degeneracy n2, but that’s cool; picking an axis z, the total angular momentum operator L and the z-axis angular momentum operator Lz give a complete set of commuting observables (together with H), so you get yer n, l, ml eigenstates.

And you think everything’s cool, everything’s ok. Wrong, because physics gets in the way of all this math fun. A number of physical effects (in various environments & regimes) break spherical symmetry and perturb our energy levels from their sweetly degenerate state. Here are some of them.

Normal Zeeman effect
An orbital with \(L_z\neq 0\) has a non-zero magnetic moment about the z-axis (spinning electron ⇒ little loop of current ⇒ magnetic dipole). This means that it interacts with the z-component of a magnetic field. The potential of this interaction is \(V=-\mu B_z\) where μ is the dipole moment. How to calculate this? \(L_z = m_e v r\) and \(\mu = I A = \frac{-e v}{2\pi r} \pi r^2 = -\frac{e}{2} v r\): comparing these gives \(\mu = -\frac{e}{2 m_e} L_z\). (The coefficient in front is the “Bohr magneton”, within a factor of \(\hbar\).)

Since the spacing between Lz-values is \(\hbar\), the spacing between the split energy levels will be \((e\hbar B_z)/(2 m_e)\), which is the Lamor frequency times \(\hbar\).

Also interesting: Due to some symmetry magic, the eigenstates remain the same; only the eigenvalues change. Huh.

Anomalous Zeeman effect
Thing is, we’ve been talking about orbital angular momentum this whole time. There’s also the electron’s intrinsic “spin” angular momentum. This has can have magnitude \(\pm\hbar/2\) in the z direction. WEIRDLY enough, however, when computing a magnetic moment from this, you have to multiply by a “g-factor” of two. This shouldn’t surprise you, though, because spin was really weird in the first place.

The end result of all of this: Each state you get from normal Zeeman splitting splits \(\pm (e\hbar B_z)/(2 m_e)\), which was the spacing between these states in the first place. We get two extra energy levels.

Spin–orbit interaction / Fine structure
An orbiting electron sees a magnetic field produced by the movement of the nucleus. This magnetic field interacts with the spin dipole: when the field is parallel to the dipole (that is, anti-parallel to the spin), energy is minimized. Two possible values of spin ⇒ a new splitting. This produces a feature of atomic spectra known as fine structure doublets.

Stark effect
Just like an external magnetic field breaks symmetry and causes splitting in the Zeeman effect, an external electric field will cause a form of splitting known as the Stark effect. The reason why I put this in a second, subsidiary position is because it is way more complicated: our Hamiltonian now has a 1/r+z potential, which is going to completely screw up the old energy eigenstates. So you do perturbation theory, and get first/second-order corrections in the small-field limit. That’s quantum mechanics for you, I guess.

Hyperfine structure
This is real small. It includes things like electron-dipole/nuclear-dipole and electric-field/nuclear-quadrupole interactions.

Oh, and when it comes to computing spectra from these: have I mentioned selection rules? No? Oh, well those are pretty important too better look them up.

a quine in Python

import re
x="import re\"+nl+\"q='\"+q+\"'\"+nl+\"s='\"+s+s+\"'\"+nl+\"nl='\"+s+\"n'\"+nl+\"x=\"+q+re.sub('\"',s+'\"',x)+q+nl+\"print \"+q+x"
print "import re"+nl+"q='"+q+"'"+nl+"s='"+s+s+"'"+nl+"nl='"+s+"n'"+nl+"x="+q+re.sub('"',s+'"',x)+q+nl+"print "+q+x

Hey, I didn’t say it was pretty. Note how the syntax highlighter completely throws in the towel by line 3. :(

things I really gotta work on

Having taken GRE Physics practice test 8677, it is clear that there are some very specific holes in my knowledge… Three main things:

  • atomic physics (hydrogen atom: spectra? K series? Zeeman effect? Stark effect? selection rules? chemistry/orbitals?)
  • nuclear physics (decay, what with \(\beta\)s and \(\alpha\)s and antineutrinos; binding energies)
  • optics (diffraction, Rayleigh condition, indices of refraction, etc.)

And, with various degrees of relation to the above, some other things I should take a look at:

  • the particle zoo (muons, etc.)
  • solid-state (conductivity of metals, semiconductors)
  • J-lab experiments (Franck-Hertz, Compton)
  • E&M (dielectrics, induction, charges on conductors, Biot-Savart, impedance)

Fortunately, these topics are all fun & interesting, and I think I’ll gain a lot from reading about them. Woohoo studying.