352 CHAPTER 6. QUANTUM PHYSICS
Inserting (6.3.23) into (6.3.20) we deduce that
νμ(t) =cosθ ν 1 (t)+sinθ ν 2 (t) =sinθcosθ(−e−iλ^1 t/ ̄h+e−iλ^2 t/h ̄).Hence, the probability ofνetransforming toνμat timetis
(6.3.24) P(νe→νμ) =|νμ(t)|^2 =
[
sin2θsin(
λ 2 −λ 1
2 h ̄
t)] 2
.
Also, we derive in the same fashion that
νe(t) =cosθ ν 2 −sinθ ν 1 =cos^2 θe−iλ^1 t/ ̄h+sin^2 θe−iλ^2 t/h ̄,and the probability ofνμtoνeis given by
(6.3.25) P(νμ→νe) =|νe(t)|^2 =cos^2
(
λ 2 −λ 1
2 h ̄t)
+cos^22 θsin^2(
λ 2 −λ 1
h ̄t)
From formulas (6.3.24) and (6.3.25), we derive the oscillation betweenνeandνμ, the energy
differenceλ 2 −λ 1 , and the angleθ, if the discrepancy probabilityP(νe→νμ)is measured.
6.3.3 Mixing matrix and neutrino masses
As mentioned in Remark6.16, the current neutrino oscillation requires mass matrixAdefined
in (6.3.19). In this subsection we shall discuss these two topics.
Mixing matrix
The matrixAgiven in (6.3.19) is called the MNS matrix, which is due to Z. Maki, M.
Nakagawa and S. Sakata for their pioneering work in (Maki, Nakagawa and Sakata, 1962 ).
This can be considered as an analog for leptons as the Cabibbo-Kobayashi-Maskawa (CKM)
matrix for quarks. The MNS matrix is written as
(6.3.26) A=
c 12 c 13 s 12 c 13 s 13 e−iδ
−s 12 c 23 −c 12 s 23 s 13 eiδ c 12 c 23 −s 12 s 23 s 13 eiδ s 23 c 13
s 12 s 23 −c 12 c 23 s 13 eiδ −c 12 c 23 −s 12 c 23 s 13 eiδ c 23 c 13
,
whereδis the phase factor, and
cij=cosθij, sij=sinθij,with the valuesθijbeing measured as
θ 12 ≃ 34 ◦± 2 ◦, θ 23 ≃ 45 ◦± 8 ◦, θ 13 ≃ 10 ◦.The matrixAof (6.3.26) is a unitary matrix: A†=A−^1 .Therefore, (6.3.19) can be also
rewritten as
ν 1
ν 2
ν 3
=A†
νe
νμ
ντ