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In Griffith book on elementary particles, page-162, it is written:

In positronium, the positron and electron can annihilate temporarily giving rise to a virtual photon only when it is in a triplet state with $\ell =0$. And this raises the energy of the triplet $S$ state.

Here I am attaching a screenshot of that page. Please explain how this raises the energy of that state.

enter image description here

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  • $\begingroup$ you should give a link. the edition I found on the net does not have the phrase $\endgroup$
    – anna v
    Commented May 7, 2020 at 14:33
  • $\begingroup$ Thank you for checking, although I have not written the exact phrase. I have uploaded a screenshot of that page (check the first paragraph). Let me know if any other information is needed. Thank you again. $\endgroup$
    – Sujit
    Commented May 8, 2020 at 2:31
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    $\begingroup$ This link has the same page zamalik.weebly.com/uploads/5/6/1/9/56198443/… $\endgroup$
    – anna v
    Commented May 8, 2020 at 3:50

1 Answer 1

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The way I understand it is the following:

There are three quantum numbers in the hydrogen solution, n,l,m.

l can go from 0 to to n. The triplets discussed are the ones with l=1,for each n, so as to have the angular momentum of a virtual photon in the diagram shown. There are many triplets , up to the continuum of very large n. The states of l=1 that can contribute to an annihilation diagram are the S states, where there is a probability for the electron positron orbitals to overlap ( in hydrogen for the electron to exist in the proton).

In the calculations he plans to make later many virtual photon diagrams will contribute to the first order diagram of annihilation into two photons, from all n energy levels, thus the virtual photon will have a range of energy limits in the integration. He states what he will show later, that this increases the energy available for the annihilation at each n.

In a sense , the potential V entering in the equation is modified by corrections coming from higher order terms resulting in the next figure, to be calculated later.

So it has to be taken as given ,i.e. that is what the calculations will give.

Now to intuitively interpret the

it raises the energy of the triplet S states by an:

The energy levels are negative ones, if it is the binding energy in the potential well, when it is raised, it means the particle is less bound. So the virtual photon contributions reduce the binding of the electron and positron in the positronium.

If I am wrong I hope someone will correct me.

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  • $\begingroup$ Here triplet state means both positron and electron have the same spin, i.e. either both up or down. So that triplet state will be S=1. And overlapping of wavefunctions is more probable for l=0. $\endgroup$
    – Sujit
    Commented May 8, 2020 at 22:03
  • $\begingroup$ @Sujit in my vocabulary S is the 0 state of the qunatum number l l=1,0,-1,l=2,1,0.-1,-2 etc( S,P,D,F) $\endgroup$
    – anna v
    Commented May 9, 2020 at 3:27

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