This is another small derivation that I liked, starting from the Bohr postulates and classical mechanics, we get to the formulas for the Bohr radius and the Rydberg energy.

The Bohr postulates are:

  • Electrons revolve around the nucleus in stationary orbits that do not decay, or radiate energy.
  • Angular momentum is quantized: $L = n\hbar$ ($n$ being an integer $\ge 0$)
  • Electrons do not absorve or emmit energy continuously, they only do so in “packets” jumping from one stable orbit to the next (absorving a photon) or to the previous (emmiting a photon).

With that in mind, we can consider an electron going around the nucleus in a circular orbit, balanced by the Coulomb force, so the centripetal acceleration has to match it:

$$ m\frac{v^2}{r} = \frac{ke^2Z}{r^2} $$

Note that we are considering only hydrogenic atoms, this is, atoms with only one electron, but $Z$ protons (thus the $e\dot Ze$ factor in the Columb force). Also, $e$ here is the electron charge, $m$ is the mass of the electron and $k=\frac{1}{4\pi\epsilon_0}$ is the Columb constant.

With this, we can then use the second Bohr postulate to replace the velocity $v$:

$$ \begin{align*} L &= mvr = n\hbar \\ v &= \frac{n\hbar}{rm} \\ m\frac{v^2}{r} &= \frac{m}{r}\left(\frac{n\hbar}{rm}\right)^2 = \frac{n^2\hbar^2}{mr^3} = \frac{ke^2Z}{r^2} \\ n^2\hbar^2 &= ke^2Zrm \\ r &= a_n = \frac{\hbar^2}{ke^2m}\frac{n^2}{Z} \\ \end{align*} $$

And this is the Bohr radius \o/

From here, we can then get the total energy ($E_t$) of the electron ($T$ is the kinetic energy, and $V$ the potential one, in this case, the Coulomb potential):

$$ \begin{align*} E_t &= T + V \\ T &= \frac{1}{2}mv^2 = \frac{1}{2}\frac{ke^2Z}{r} \\ V &= -\frac{ke^2Z}{r} \\ E_t &= \frac{1}{2}\frac{ke^2Z}{r} - \frac{ke^2Z}{r} = -\frac{ke^2Z}{2r} = -\frac{1}{2}\left(\frac{ke^2}{\hbar}\right)^2\frac{Z^2}{n^2} \\ &= -E_R\frac{Z^2}{n^2} \\ E_R &= \frac{1}{2}\left(\frac{ke^2}{\hbar}\right)^2 \end{align*} $$

So here we have it! This $E_R$ factor is the Rydberg energy!

This was really useful as alternative to remembering the formula for it during exams xd