Editing Bohr radius (section)

== Hydrogen atom and similar systems ==
The Bohr radius including the effect of [[reduced mass]] in the hydrogen atom is given by
: <math> \ a_0^* \ = \frac{m_\text{e}}{\mu}a_0 ,</math>
where <math display="inline">\mu = m_\text{e} m_\text{p} / (m_\text{e} + m_\text{p})</math> is the reduced mass of the electron–proton system (with <math>m_\text{p}</math> being the mass of proton). The use of reduced mass is a generalization of the [[two-body problem]] from [[classical physics]] beyond the case in which the approximation that the mass of the orbiting body is negligible compared to the mass of the body being orbited. Since the reduced mass of the electron–proton system is a little bit smaller than the electron mass, the "reduced" Bohr radius is slightly ''larger'' than the Bohr radius (<math>a^*_0 \approx 1.00054\, a_0 \approx 5.2946541 \times 10^{-11}</math> meters).

This result can be generalized to other systems, such as [[positronium]] (an electron orbiting a [[positron]]) and [[muonium]] (an electron orbiting an [[anti-muon]]) by using the reduced mass of the system and considering the possible change in charge. Typically, Bohr model relations (radius, energy, etc.) can be easily modified for these exotic systems (up to lowest order) by simply replacing the electron mass with the reduced mass for the system (as well as adjusting the charge when appropriate). For example, the radius of positronium is approximately <math> 2\,a_0</math>, since the reduced mass of the positronium system is half the electron mass (<math> \mu_{\text{e}^-,\text{e}^+} = m_\text{e}/2 </math>).

A [[hydrogen-like atom]] will have a Bohr radius which primarily scales as <math>r_{Z}=a_0/Z</math>, with <math>Z</math> the number of protons in the nucleus. Meanwhile, the reduced mass (<math>\mu</math>) only becomes better approximated by <math>m_\text{e}</math> in the limit of increasing nuclear mass. These results are summarized in the equation
: <math> r_{Z,\mu} \ = \frac{m_\text{e}}{\mu}\frac{a_0}{Z} .</math>

A table of approximate relationships is given below.

{| class="wikitable"
|-
! System !! Radius 
|-
| [[Hydrogen atom|Hydrogen]] || <math>a_0^*=1.00054\, a_0</math>
|-
| [[Positronium]] || <math>2 a_0</math>
|-
| [[Muonium]] || <math>1.0048\, a_0</math>
|-
| [[Helium|He<sup>+</sup>]] || <math>a_0/2</math>
|-
| [[Lithium|Li<sup>2+</sup>]] || <math>a_0/3</math>
|}