Editing Rydberg constant (section)

== Alternative expressions ==
The Rydberg constant can also be expressed as in the following equations.
: <math>R_\infty = \frac{\alpha^2 m_\text{e} c}{2h} = \frac{\alpha^2}{2 \lambda_{\text{e}}} = \frac{\alpha}{4\pi a_0}</math>
and in energy units
: <math>\text{Ry} = h c R_\infty = \frac{1}{2} m_{\text{e}} c^2 \alpha^2 = \frac{1}{2} \frac{e^4 m_{\text{e}}}{(4 \pi \varepsilon_0)^2 \hbar^2} = \frac{1}{2} \frac{m_{\text{e}} c^2 r_{\text{e}}}{a_0} = \frac{1}{2} \frac{h c \alpha^2}{\lambda_{\text{e}}} = \frac{1}{2} h f_{\text{C}} \alpha^2 = \frac{1}{2} \hbar \omega_{\text{C}} \alpha^2 = \frac{1}{2 m_{\text{e}}}\left(\dfrac{\hbar}{a_0}\right)^2 = \frac{1}{2}\frac{e^2}{(4\pi\varepsilon_0)a_0} ,</math>
where
* <math>m_\text{e}</math> is the [[electron rest mass]],
* <math>e</math> is the [[electric charge]] of the electron,
* <math>h</math> is the [[Planck constant]],
* <math>\hbar= h/2\pi</math> is the [[reduced Planck constant]],
* <math>c</math> is the [[speed of light]] in vacuum,
* <math>\varepsilon_0</math> is the [[electric constant]] (vacuum permittivity),
* <math>\alpha = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{\hbar c}</math> is the [[fine-structure constant]],
* <math>\lambda_{\text{e}} = h/m_\text{e} c</math> is the [[Compton wavelength]] of the electron,
* <math>f_{\text{C}}=m_{\text{e}} c^2/h</math> is the Compton frequency of the electron,
* <math>\omega_{\text{C}}=2\pi f_{\text{C}}</math> is the Compton angular frequency of the electron,
* <math>a_0={4\pi\varepsilon_0\hbar^2}/{e^2m_{\text{e}}}</math> is the [[Bohr radius]],
* <math>r_\mathrm{e} = \frac{1}{4\pi\varepsilon_0}\frac{e^2}{m_{\mathrm{e}} c^2} </math> is the [[classical electron radius]].

The last expression in the first equation shows that the wavelength of light needed to ionize a hydrogen atom is 4''π''/''α'' times the Bohr radius of the atom.

The second equation is relevant because its value is the coefficient for the energy of the atomic orbitals of a hydrogen atom: <math>E_n = -h c R_\infty / n^2 </math>.