Sommerfeld's Constant and Bohr's Quantisation Condition I was solving a problem which had a different solution than what I found. This question doesn't emphasise on that problem but instead finds the motive for certain rearrangement in a formula:-
According to a classical result in the Bohr Model:-
$$
\begin{aligned}
&F_{e}=F_{q}\\
&\frac{m v^{2}}{r}=\frac{Z k e^{2}}{r^{2}}
\end{aligned}
$$
where,
$$
k=\frac{1}{4 \pi \varepsilon_{0}}.
$$
Using Bohr's Quantisation condition and a little bit of rearrangement, one can conveniently obtain,
$$
v=\left(\frac{Z e^{2}}{2 \varepsilon_{0} n h}\right)
$$
substituting the values to simplify it into one constant, we get: 
$$
v=\frac{(2187691.254 \times Z)}{n}
$$
which has a surprising different simplification and is given by: 
$$
v=\left(\frac{Z c}{137 n}\right)
$$
where $c$ is the speed of light.
While researching into why this seems intriguing, I found a little info. (and I'm relatively new to this topic) on the infamous Sommerfeld's Constant.
Why does this constant arise so much in various problems in Modern Physics? What is the motive for including it without any mention on its derivation?
 A: The short answer is that the fine structure constant arises everywhere because of QED. Quantum electrodynamics describes how the EM-field and the fermions interact, $\alpha$ is the coupling constant between the two fields. QED calculations are very important in particle physics and are also very useful in material physics, condensed matter physics, photonics, and such since, in many ways, we are interested in how photons interact with matter, specially with electrons. The fine structure constant sets the scale upon which electromagnetic interactions are prevale. 
As an example, this is the QED lagrangian 
$$ \mathcal{L} = \bar{\psi}(i\not D -m)\psi -\frac{1}{4}F^{\mu\nu}F_{\mu\nu} = \bar{\psi}(i\not \partial -m)\psi -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-e\bar{\psi}\not A_\mu\psi $$
and with this we can evaluate scattering processes using Wick's theorem and the $S$-matrix. What we find is that the perturbative parameter on which we expand the $S$-matrix is, in natural units $$\frac{e^2}{4\pi} = \alpha $$
As an example we can take a second order process in which a fermion and a photon interact, like Compton scattering
$$M = \bar{u}(p^\prime)\epsilon_{\mu}^{(\lambda)}(k^\prime)(-ie\gamma^{\mu})\frac{1}{\not q -m+i\eta}\epsilon^{(\lambda)}_\nu(k)(-ie\gamma^\mu)u(p)$$
leaving behind the complicate details of the formula, you can easily see this depends on $e^2\propto\alpha$. The probability amplitude is given by the modulus square of this quantity making it clear that this is a second order process which in fact will depend on $\alpha^2$
Clearly there's much more to say about the fine structure constant and how it comes to be, but i hope that this will give you sort of an idea for why it pops up so much in physics. Moreover, QED is one of the most precise theory we know of, it gives results which are perfectly in accordance to the experiments up to some incredible precision. 
