Etudes des proprietes des neutrinos dans les contextes ...

tel-00450051, version 1 - 25 Jan 2010
where the double subscripts corresponds to the initial state of the considered
system. Therefore, |ψe,e(t)| 2 = |Uee| 2 = P(νe → νe), which is the probability to go
to the state νe starting with the initial state where a νe has been created. Finally,
the Schrödinger equation that contains all three initial states of a supernova
environment where all the neutrinos are created is:
i ∂
⎛
⎞ ⎛
⎞
Uee Ueµ Ueτ Uee Ueµ Ueτ
⎝ Uµe Uµµ Uµτ ⎠ = H ⎝ Uµe Uµµ Uµτ ⎠ , (C.8)
∂t
Uτe Uτµ Uττ
where H is the hamiltonian of the system.
Uτe Uτµ Uττ
C.3 The Liouville-Von Neumann equation: the
density matrix formalism
The Liouville-Von Neumann equation describes the evolution of the density matrices
The density matrix can be used in two cases : like wave functions one can
consider pure quantum states like Eq.(C.6), or one can consider superimposed
states or statistical mixture of states. This formalism is useful for the description
of oscillations of neutrino ensembles with more than one initial flavour with
possible loss of coherence, like in the environment of the early Universe.
The pure states
By definition, in a pure state, the description of the system is given by a state
vector that can be expanded on a basis of a finite dimensional Hilbert space
| un >, and
| ψ(t) > =
cn(t) | un > with
| cn(t) | 2 = 1 (C.9)
n
The coefficients cn are probability amplitudes, and such amplitudes can interfere
just as for neutrinos. The temporal evolution of the state vector is given by the
Schrödinger equation:
n
i d
dt | ψ(t) >= ˆ H | ψ(t) > (C.10)
where ˆ H is the hamiltonian of the system. We can then define the density operator
by:
ˆρ(t) =| ψ(t) >< ψ(t) |=
c ∗ ncp | up >< un | . (C.11)
n,p
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