Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
This is the hermitian projection operator associated to the state describing the
system. Note that any arbitrary phase present in the previous formalism does
not appear in the density operator. Its matrix elements are :
ρpn =< up | ˆρ | un >= c ∗ n cp. (C.12)
For a pure state, the density matrix has the following properties:
1. Tr(ˆρ(t)) = 1
2. ˆρ † (t) = ˆρ(t)
3. ˆρ 2 (t) = ˆρ(t);
which express the probability conservation, the fact that ˆρ is hermitian, and a
projection operator. Finally, its time evolution is given by the Liouville-Von
Neumann equation:
i d
ˆρ = ˆH, ˆρ . (C.13)
dt
As an example, let us consider the case of a two-flavour neutrino system in vacuum.
As previously shown one can decompose the flavour state vector on the
basis of the eigenvectors of the Hamiltonian, namely in our case, the mass basis
| νi > with i = 1, 2. For a flavour α, one has | ψνα >= Uα1 | ν1 > +Uα2 | ν2 >
with the lepton mixing matrix U defined by:
cosθV sin θV
U =
, (C.14)
− sin θV cosθV
θV being the vacuum mixing angle. Since the mass basis corresponds to the
eigenvectors, the | νi > just acquires a phase factor | νi(t) >= e −iEit | νi(0) >.
At time t the state for a neutrino created as a να will be:
| ψνα(t) >= Uα1e −iE1t | ν1 > +Uα2e −iE2t | ν2 >
and therefore the corresponding density matrix operator writes in the mass basis:
ρνα = | ψνα(t) >< ψνα(t) |
= ρ11 | ν1 >< ν1 | + ρ22 | ν2 >< ν2 | + ρ12 | ν1 >< ν2 | + ρ21 | ν2 >< ν1 |
In a matrix form one rewrites:
ρνα =
ρ11 ρ12
ρ21 ρ22
170
(C.15)
, (C.16)