“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

Two-Level Systems by Hand — A neutrino changes flavor over 500 km

Exercises

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

Practice Problem 3 of 9

A neutrino changes flavor over 500 km

Problem

Two-flavor neutrino oscillations obey

P_{μ \to τ} = sin^{2} (2 θ) sin^{2} (1.267 Δ m^{2} [eV^{2}] L [km] / E [GeV])

. For

Δ m^{2} = 2.5 \times 1 0^{- 3}

^{2}

sin^{2} 2 θ = 0.99

L = 500

km,

E = 1

GeV: compute

P_{μ \to τ}

Solution

Predict before reading on. Map it onto the worked example: flavor states are $∣ L ⟩, ∣ R ⟩$ ; mass eigenstates are the field-free eigenstates; the oscillation phase is the splitting times time. What plays the role of $2 A$ ?

phase = 1.267 \times \frac{2.5 \times 1 0 ^{- 3} \times 500}{1.0} = 1.584 rad, sin^{2} (1.584) = 1.000

P_{μ \to τ} = 0.99 \times 1.000 = 0.99

This baseline and energy sit almost exactly at the first oscillation maximum — nearly every muon neutrino has become a tau neutrino. The mass-squared splitting $Δ m^{2}$ IS the level splitting of the two-level system; the beam’s flight is the free evolution between “measurements” at production and detection.

Result

P_{μ \to τ} \approx 0.99 (first oscillation maximum)