Editing Appendix/CGH/QuantumTransitions (section)

==Characteristic Size and Spin==

What is the angular momentum of a spherical shell spinning with angular frequency, <math>~\omega</math>, whose radius is the Bohr radius <math>~(a_0 = 5.29 \times 10^{-11}~\mathrm{m})</math> and whose mass is one electron mass <math>~(m_e = 9.1\times 10^{-31} ~\mathrm{kg})</math>?  Compare this to <math>~\hbar = 6.6 \times 10^{-34}~\mathrm{kg} \cdot \mathrm{m}^2/\mathrm{s}</math>.  Drawing from [https://en.wikipedia.org/wiki/List_of_moments_of_inertia Wikipedia], the relevant moment of inertia is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~I</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{3} m_e a_0^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~L</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2}{3} m_e a_0^2 \omega \, .</math>
  </td>
</tr>
</table>

What is the spin period if we set this angular momentum to ''half'' of the Planck constant?
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P = \frac{2\pi}{\omega}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{8\pi}{3}\cdot \frac{m_e a_0^2}{\hbar} =  3.2 \times 10^{-17}~\mathrm{s} \, .</math>
  </td>
</tr>
</table>

For comparison, what is the light-travel time across the sphere?

<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~t</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2a_0}{c} = 3.5 \times 10^{-19} ~\mathrm{s}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{P}{t}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~92 \, .</math>
  </td>
</tr>
</table>