Understanding the Instability of Positronium: A Closer Look at K-Capture Decay

[TomCurious]

The 'reason' electrons do not collapse into the nucleus of atom is due to the standing wave nature of the electron. This is all fine and dandy.

My question, essentially, is why the same does not hold true for positronium (in which an electron orbits a positron). Shouldn't the electron demonstrate here some form of standing wave character here as well?

Considering the Schrodinger equation for both systems, the only difference is the reduced mass, which thus leads to a different energy spectrum. Of course, the real result would make use of the Bethe equation, etc..

But I fail to see why this should lead to the instability of positronium. An explanation would be greatly appreciated.

Thank you in advance.

 

[Bloodthunder]

There is a partial wave function overlap between the two particles. (The electron has wave functions around the positron, just as the positron has wave functions around the electron.)

With this, there is a possibility that the two particles are in the same space at a point in time. So... annihilation!

 

[Bill_K]

The 'reason' electrons do not collapse into the nucleus of atom is due to the standing wave nature of the electron. This is all fine and dandy.

And wrong. S-wave orbitals in atoms overlap the nucleus, just as the electron and positron overlap in positronium. The reason that atoms do not 'collapse' is that in most cases it is not energetically possible. For example a hydrogen atom does not collapse because the result would be a neutron, and since a neutron is more massive than proton there is not enough energy available for it to happen. But in many other cases the absorption of an electron into the nucleus is energetically possible, and the process does take place, and is known as K-capture. See the Wikipedia article on electron capture, which mentions Be⁷ as an example.

 

[TomCurious]

in many other cases the absorption of an electron into the nucleus is energetically possible, and the process does take place, and is known as K-capture.

I am glad you raised this, because it was another question I was hoping to ask. So, in effect, you answered two of them! (thank you)

Unfortunately, the wikipedia article you referred me to is rather heuristic. Could you direct me to a more detailed treatment of K-capture (i.e. more rigorous), please? Specifically, I am interested in decay rate calculations.