# True muonium energy levels [closed]

In a true muniom(muon-antimuon atom) do the energy levels match with that of the hydrogen's?

The core is lighter for about 100 times but the muon is 100 times heavier than the electron so it would make sense the energy levels of a true muonium and a hydrogen atom to match?

Or is this wrong?

• I'm not sure, but it seems the opposite - it's even more different. – Mithoron Mar 31 '20 at 21:01
• This isn’t a chemistry question, really. But since the reduced mass of the system is different, the energy levels will be different. – Jon Custer Mar 31 '20 at 21:08
• Core mass is not all that important. Think of hydrogen vs deuterium, they barely differ at all. – Ivan Neretin Mar 31 '20 at 22:03

The theory for a muon-antimoun "atom" is essentially the same as any hydrogen-like atom, just with masses changed. See this Wikipedia page for the relevant formulae for relativistic (Dirac) and non-relativistic treatments.

The non-relativistic energy levels are given by :

$$E_n = \frac { \mu c^2 \alpha^2}{2n^2}$$

where $$alpha$$ is the fine structure constant and $$\mu$$ is the reduced mass, in this case because the masses are equal, it's $$\mu=\frac m 2$$ where $$m$$ is the muon mass.

Note that the muon mass is about $$206.8$$ times that of the electron and the reduced mass is then $$103.4$$ times than of the electron making the energy levels for the non-relativistic state about $$103.4$$ times that of the Hydrogen energy levels.

As the Wikipedia page explain (in more detail that you asked for), a more detailed relativistic calculation using the Dirac equation includes corrections for the spin-orbit interaction.

The core is lighter for about 100 times but the muon is 100 times heavier than the electron

This is wrong.

Both the muon and antimoun have the same mass, hence the effect of reduced mass being a factor of one-half.

The reduced mass of two particles is given by :

$$\mu = \frac {m_1m_2}{m_1+m_2}$$

Strictly speaking there is no "core" as both the muon and antimoun are of the same mass whereas in Hydrogen the electron is of much smaller mass than the proton.

• So a bound muon-antimuon pair would actually be almost exactly described by the mathematics of the electron-positron pair (positronium), up to a scaling factor equal to the electron/muon mass ratio. Is that correct? – Nicolau Saker Neto Apr 1 '20 at 3:44
• @NicolauSakerNeto A scaling factor of half the ratio of the electron and muon masses ! That's the reduced mass factor coming into play. – StephenG Apr 1 '20 at 9:47
• @StephenG But in positronium you also have this factor, so Nicolau is right. – Mithoron Apr 1 '20 at 16:27