Positronium: Para, Ortho, and the Zeeman Effect

Positronium is hydrogen with the proton replaced by a positron. The Coulomb problem is identical; only the reduced mass changes, from $\mu \approx m_e$ to exactly $\mu =m_e/2$ (two equal partners). Every hydrogen energy is therefore halved and every length doubled: binding energy $6.803\mathrm{eV}$ instead of 13.606, mean electron–positron separation $2a_0=1.058$ Å, and a Lyman-α analog at 5.102 eV (2430 Å, in the UV). One knob — the reduced mass — regenerates the whole spectrum.

The difference is that this atom annihilates. The electron and positron overlap at the origin ($|\psi (0){|}^2\ne 0$ for S states), and where matter meets antimatter with nonzero density, photons come out. The ground state of positronium is not a stable floor; it is a starting line.

Two spin-$\frac{1}{2}$ particles couple to total spin 0 or 1: a singlet — para-positronium — and a triplet — ortho-positronium. Charge conjugation sorts their deaths. A state with quantum numbers $L,S$ has C-parity $(-1{)}^{L+S}$, and a system of $n$ photons has $(-1{)}^n$. So the ground-state singlet ($C=+1$) must decay to an even number of photons — two — while the triplet ($C=-1$) is forced to three.

Three photons cost an extra factor of $\alpha$ and a three-body phase space, and the price is enormous:

a factor of 1114 at leading order (measured: 125.14 ps and 142.05 ns — Ore & Powell's three-photon rate, 1949). Same atom, same orbital, three decades of lifetime apart, decided entirely by spin. Everything else on this page exploits that gap.

Para and ortho are not degenerate. The spin–spin interaction splits them — and positronium adds a contribution no ordinary atom has: the pair can annihilate into a virtual photon and reform, a process available only to the $C=-1$ triplet. At leading order,

That is 204.39 GHz computed, 203.3887 GHz measured — leading order lands within 0.49%, and the remainder is the $O({\alpha }^3)$ QED corrections that make positronium a precision test bench: a bound state built from two point leptons, with no nuclear structure to smear the comparison. Note the proportions: 43% of the splitting comes from the annihilation channel. Nearly half of this "hyperfine" interval exists because the atom can momentarily stop existing.

Here is the odd part. In hydrogen, a magnetic field shifts levels — the ordinary Zeeman fan. In ground-state positronium the first-order shifts cancel: the positron's magnetic moment is equal and opposite to the electron's, so the total moment of any definite-spin state is zero. What survives is the operator ${\mu }_{B}Bg({\sigma }_z^{e^{-}}-{\sigma }_z^{e^{+}})/2$, which is antisymmetric under exchanging the two spins — and an antisymmetric operator has no diagonal elements in states of definite exchange symmetry. Its only nonzero matrix element in the ground state connects $|1,0⟩$ to $|0,0⟩$.

So the field does not shift positronium's levels; it mixes its identities. The problem collapses to a two-level system: singlet and triplet $m=0$ states separated by the hyperfine gap, coupled by $x=g{\mu }_{B}B$, while $|1,\pm 1⟩$ — with no $m=0$ partner to talk to — sail through untouched.

You cannot see the 203 GHz transition directly with 1951 technology. But you do not need to: the singlet and triplet die differently, so any admixture of para character into an ortho state shows up as a shortened lifetime and a shifted photon budget (2γ events where 3γ belonged). The lifetime is a meter that reads out the wavefunction's composition.

That is how it was actually done. Deutsch discovered positronium (1951), and Deutsch & Brown (1952) applied a magnetic field, watched the $m=0$ triplet quench, and fit the curve — extracting the hyperfine interval from lifetimes years before direct microwave spectroscopy could touch it. The same logic runs today in reverse: in materials, the environment steals lifetime from ortho-positronium, and reading the stolen amount measures the size of the holes the atom sits in.