(no subject)

[impossiblewizardry]

I finally get the intuition behind Bohr's derivation of the Rydberg constant. Not his original one; the one that I like, which he published a few years later.

It makes sense if I think about, what is the Rydberg constant, and what is Planck's constant.

Planck's constant tells us the scale where the world is quantum. In an oscillating electric field (such as being exposed to light) with frequency ν, electrons won't continuously gain energy as you'd expect classically, but will instead randomly acquire a quantum of energy h ν. To get some idea of what this implies physically: the way Millikan did his measurement of h was by illuminating a sample, and measuring the kinetic energy of the excited electrons that shot out of the sample: h is the slope of their kinetic energy as a function of ν.

The Rydberg constant, R, also tells us the scale where the world is quantum, but specifically for the hydrogen atom. You can define it in multiple ways, but I'm defining it as the Rydberg energy. It appears in Rydberg's law, which long predates quantum mechanics, but with quantum mechanics in mind you can interpret it as the work required to ionize hydrogen from its ground state: R = 13.6 eV.

Our goal is to figure out the relationship between them. We at least know that R has to be a decreasing function of h. Classically (h=0), R should be infinity, because the potential well of a hydrogen atom is infinitely deep. Since h>0, R is finite. So it'd make sense for h to be in the denominator of R, so that the h→0 limit gives you infinite R. h is a measure of how quantum the wold is. R is a measure of how non-quantum the hydrogen atom is.

The way we find this relationship is to consider a high-n (high energy level) limit of the hydrogen atom. The great thing about the hydrogen atom is that the high-n limit is classical: the spacing of energy levels approaches 0. This is not the case for example with the harmonic oscillator, where the energy levels are always equally spaced. In the classical high-n limit, the electron will be in an elliptical orbit around the proton, following Kepler's laws.

The key to relating R and h is the frequency. In the classical limit, two different frequencies have to be equal:

• The frequency of the electromagnetic radiation absorbed or emitted
• The frequency of rotation of the electron

Both depend on n and R through Rydberg's law. The radiation frequency of course is given by Rydberg's law (specifically, consider a transition between adjacent energy levels, with large n). And the frequency of rotation depends on the total energy of the system, which you can get by interpreting Rydberg's law.

They both depend on n in the same way, so n will cancel.

HOWEVER. ONLY the frequency of radiation has h in it (through change in energy = hν). The frequency of rotation is just from classical mechanics and h is not involved, just R.

By setting these equal to each other, you're equating a quantity depending on R, with a quantity depending on both h and R. So you're deriving a relationship between our two measures of quantum-ness, h and R.

And that's how you solve for R, the Rydberg constant, in terms of h and a bunch of constants related to protons, electrons, and electromagnetism. You're deriving how quantum the hydrogen atom is, from how quantum the world is, plus some facts about the hydrogen atom.

THAT'S how Bohr's derivation of the Rydberg constant really works, set out clearly in his later derivation, but obscured in his original derivation by a bunch of trivial algebra converting between the 1/n² formula for the energy levels and other equivalent specifications of the energy levels.