How to go from one equation to another?

Question

I am starting to studying quantum physics, but this part of the theory catch me, how do we go from the $-e^2/2a_0$ to the next expression?

$$ E=\frac{1}{2}\left(\frac{e^2}{a_0}\right)-\frac{e^2}{a_0}=-\frac{e^2}{2a_0}=-\frac{1}{2}\alpha^2mc^2=-R_\infty$$

This is the energy of an electron in the hydrogen Bohr atom: $e$ is the electric charge

$\alpha$ is the fine structure constant.

$a_0$ is the bohr radius.

I think maybe the book the following equation for the first Bohr radius: $$a_0=\frac{\hbar^2}{me^2}=\alpha^{-1}\hbar_e=(5.29167\pm 0.00002) \times 10^{-9}~\rm cm$$

If so, how to derive this equation for the Bohr radius?

Answer 1

The Bohr radius is $a_0= \frac{\hbar^2}{me^2}$ so that

$\frac{-e^2}{2a_0} = \frac{-me^4}{2\hbar^2}$

$ = \frac{-m(e^2)^2}{2\hbar^2} \frac{c^2}{c^2} $

where I have just multiplied by $\frac{c^2}{c^2}$ (or one), and since the fine structure constant is defined as $\alpha = \frac{e^2}{\hbar c}$

$\frac{-e^2}{2a_0} = -\frac {m}{2}\alpha^2c^2$

For a full derivation of the Bohr radius, go to this article https://en.wikipedia.org/wiki/Bohr_model#Derivation