Positronium is the bound state of an electron and a positron

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · Radiative transitions, external fields and collisions

Deriving the Spectrum, Ch Radiative transitions and lifetimes

Problem

Positronium is the bound state of an electron and a positron, with reduced mass

\mu = m_e/2

. Its energy levels and Bohr radius follow the hydrogenic formulas with

\mu

in place of

m_e

(Chapter 1), so its term energies, and hence its

2p\to1s

transition frequency, are half those of hydrogen (treating hydrogen's reduced mass as

\approx m_e

), while its characteristic radius is twice hydrogen's Bohr radius

a_0

. Hydrogen's

2p\to1s

transition has rate

A_{\rm H} = 6.27\times10^8\,\mathrm{s^{-1}}

and lifetime

\tau_{\rm H} = 1.60\,\mathrm{ns}

, with

A \propto \omega^3|\vec d_{21}|^2

and the dipole matrix element proportional to the atom's characteristic radius. ((a)) Find the spontaneous emission rate

A_{\rm Ps}

and lifetime

\tau_{\rm Ps}

of the positronium

2p\to1s

transition. ((b)) Find the natural linewidth (FWHM in frequency) of this positronium line, in MHz.

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