Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
is represented by
∂tTr(A) = 0 and ∂tA = B × C . (C.33)
Therefore it is straightforward to see that the Liouville Von-Neumann equation
gives in the Pauli matrices representation:
and translates in the polarization form into:
i∂tρ = [H, ρ] (C.34)
∂tTr(ρ) = 0 and ∂tρ = H × ρ , (C.35)
∂tTr(P0) = 0 and ∂tP = B × P , (C.36)
where B corresponds to the Hamiltonian H = 2H expanded on the Pauli matrices
basis. The flavor oscillation which can be calculated using Eq.(C.8) or Eq.(C.13)
are fully analogous to the spin precession in a magnetic field. Here, B plays the
role of a ”magnetic field” and P that of a ”spin vector”. If we define the density
matrix in two flavours (νe and νµ) through the wave functions describing the
neutrino states in the flavour basis by the following equations :
ρνα =
ρνeνe ρνeνµ
ρνµνe ρνµνµ
then the polarization vector writes :
⎛
P = ⎝
=
2Re(ψ ∗ e ψµ)
2Im(ψ ∗ e ψµ)
| ψe | 2 − | ψµ | 2
| ψe | 2 ψeψ ∗ µ
ψ ∗ e ψµ | ψµ | 2
⎞
, (C.37)
⎠ . (C.38)
Consequently, ρνeνe =| ψe | 2 = 1
2 (1 + Pz) and ρνµνµ =| ψµ | 2 = 1
2 (1 − Pz) give the
probability for the neutrino to be measured as νe or νµ respectively.
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