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Two-Level Systems by Hand — Magnetic quenching of positronium

Exercises

Introductory Quantum Mechanics › Unit 3 · Two-level systems & spin-1/2 › Two-Level Systems by Hand › all problems

Practice Problem 5 of 9

Magnetic quenching of positronium

Problem

In positronium the

m = 0

triplet and the singlet are split by

Δ E_{hfs} = h \times 203.39

GHz and mixed by a magnetic field with matrix element

2 μ_{B} B

(both particles have

g \approx 2

). At

B = 0.5

T, find the singlet admixture

sin^{2} θ

in the mostly-triplet state, and estimate its lifetime given

τ_{para} = 125

ps and

τ_{ortho} = 142

ns.

Solution

Predict before reading on. Identical structure to ammonia with the roles swapped: here the splitting is intrinsic (hyperfine) and the field provides the off-diagonal coupling. Write $x = 4 μ_{B} B /Δ E_{hfs}$ and find where $x$ sits relative to the crossover $x = 1$ .

x = \frac{4 μ _{B} B}{Δ E _{hfs}} = \frac{4 \times ( 13.996 GHz/T ) \times 0.5}{203.39 GHz} = 0.138

sin^{2} θ = sin^{2} (\frac{1}{2} arctan x) = 0.467%

Small mixing — but the singlet decays 1100× faster, so even 0.47% of it dominates the decay rate:

\frac{1}{τ} = \frac{sin ^{2} θ}{τ _{p}} + \frac{cos ^{2} θ}{τ _{o}} \Rightarrow τ \approx 22.5 ns (vs 142 ns unperturbed)

The lifetime of one Zeeman sublevel collapses sixfold at half a tesla while $m = \pm 1$ stay untouched — magnetic quenching, and a lifetime measurement that doubles as a hyperfine-splitting measurement. The full story lives on the positronium page.

Result

sin^{2} θ = 0.47%, τ_{m = 0} \approx 22.5 ns