Why is positronium unstable compared to the stable hydrogen atom?

[StateOfTheEqn]

I have a question about the stability of positronium.

Positronium consists of an electron and a positron whereas hydrogen consists of an electron and a proton. The energy levels of positronium, ignoring fine structure, are -6.8eV/n² whereas those for hydrogen are -13.6eV/n².

My question is, why is positronium unstable resulting in the annihilation of the positron-electron pair while hydrogen is stable? Why does positronium not simply remain in its lowest energy level -6.8eV?

 

[SpectraCat]

I have a question about the stability of positronium.

Positronium consists of an electron and a positron whereas hydrogen consists of an electron and a proton. The energy levels of positronium, ignoring fine structure, are -6.8eV/n² whereas those for hydrogen are -13.6eV/n².

My question is, why is positronium unstable resulting in the annihilation of the positron-electron pair while hydrogen is stable? Why does positronium not simply remain in its lowest energy level -6.8eV?

Because the ground state wavefunction has non-zero probability density at the nucleus. So there is always a finite change that the electron and positron will come close enough together to annihilate.

EDIT: Of course which particle you consider the "nucleus" is arbitrary in positronium, since both have identical masses. However their respective wavefunctions still overlap, which provides non-zero probability of annihilation.

 

[mathman]

There is one interesting factor prolonging the life of positronium. If the spins are in the same direction, they can't end up as 2 photons, since the spins of the positron and electron add up to 1, while 2 photons must have an even number for total spin.