The Bohr Model

We have spent the better part of several weeks to derive the solutions to a 3D model of a Hydrogen-like atom. The results so far include the bound state energy eigenvalues and the rough “radius of the electron orbit”.

\[E_n = -\dfrac{1}{2n^2}\left(\dfrac{Ze^2}{4\pi\varepsilon_0}\right)^2 \dfrac{\mu}{\hbar^2}\] \[a = \dfrac{4\pi\varepsilon_0\hbar^2}{\mu Z e^2}\]

Interestingly, these results are consistent with “proto-quantum” ideas. Consider the following:

  1. An electron is in a circular orbit around a nucleus of charge $Z$. Find the speed with which the electron orbits the nucleus at a given radius $r$.
  2. Write down the angular momentum $L$ for this system considering the nucleus as the origin of coordinates.
  3. Now, take a leap of faith: quantize it $L=n\hbar$. Use that to determine the possible radii of the orbits.
  4. Now find the kinetic energy of the electron in terms of what you know.
  5. Compare your results to the results above.