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Question
Positronium is just like a H-atom with the proton replaced by the positively charged anti-particle of the electron (called the positron which is as massive as the electron). What would be the ground state energy of positronium?
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Solution
Positronium (Ps) is a system consisting of an electron and its anti-particle a positron, bound together into an exotic atom, specifically anonium. The system is unstable: the two particles annihilate each other to predominantly produce two or three gamma-rays, depending on the relative spin states. The orbit and energy levels of the two particles arc similar to that of the hydrogen atom (which is a bound slate of a proton and an electron). However, because of the reduced mass, the frequencies of the spectral lines are less than half of the corresponding hydrogen lines.
As in the new H-atom (positronium), the proton is replaced by the position of mass m = me/2 as under
Mass of positronium = m = `m_e^- + m_e^+`
`m_e^+ = m_e^(-1) = m_e/2`
As En = – 13.6 and so the energy of positron
`E_n = (-m_e^+e^4)/(8ε_0n^2h^2) = (-[m_e/2]e^4)/(8ε_0n^2h^2) = (-13.6)/2`
So `E_n = 13.6/2` ......`(∵ m_e = m/2)`
En = – 6.8 eV
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