decays or in reactions within the sun and other stars, is associated to
an electron (e) and is called electron neutrino ( 
). The others are the muon neutrino 
and the tau neutrino 
. "Neutrino oscillation" is based on what is known as "the wave-particle duality", that is, according to quantum mechanics laws, a particle, when it is propagated, behaves like a wave.
The fundamental "weak" interaction not only can generate neutrinos
(as is the case in nuclear reactions within the sun), but it can also make
them observable when they interact in an experimental apparatus, producing
various electrically charged particles that can be detected. The analysis
of weak interaction has shown the existence of three different kinds, or
"flavours" as physicists say, of neutrinos. In previous studies
it has been observed that the neutrino emitted in nuclei
decays or in reactions within the sun and other stars, is associated to
an electron (e) and is called electron neutrino (
). The others are the muon neutrino
and the tau neutrino
.
We could now have a look at the solar neutrinos to see how the assumed
"oscillation" takes place. We have to consider the propagation
of neutrinos (generated as
) towards the Earth, where our detector is situated. We shall see that,
because of neutrino oscillation, some of our
are "magically" transformed into different neutrinos (but the
secret lies in the laws of quantum mechanics).
To be simple, let us consider just two neutrino families,
and . This is the
typical case in which the prevailing "mixing" for the
is with the , closer
to it in our elementary particles classification: at a mass scale for electrically
charged particles associated to neutrinos, the muon is indeed the closest
to the electron. What if the property of being propagated without changes
does not apply to
and (called flavour
eigenstates), but just to some of their "mixings"[1],
the so-called 
and
, mass eigenstates?
Clearly, in the case of neutrino propagation we have to argue in terms
of mass eigenstate and not flavour eigenstates.
To clarify the notion of mixing, let us consider (Figure
5) a system of Cartesian coordinates where x and y are the flavour
eigenstates and another system of Cartesian coordinates x'-y' (where x'
and y' are the mass eigenstates) rotated by a small angle with respect
to the system x-y. In the second system, a point P(x,O) on x (standing
for a pure flavour eigenstate) is no longer represented by a single
coordinate (x') , but takes a small component also in the other coordinate
(y'). In other words, it is represented by a combination or a "mixing"
of two components x' and y'. The extent of the mixing is uniquely determined
by the rotational angle of one Cartesian system to the other. This angle
is used as a quantitative parameter to describe the situation[2]
and is called "mixing angle". If the mixing angle is small, the
eigenstates are almost pure flavour eigenstates and vice versa. The flavour
eigenstate generated
in the Sun thus breaks up into its two mass eigenstates
and ; we then follow
the latter on their way towards the Earth. We see what role their possible
masses[3], or rather the mass difference,
do play during propagation. It should be noted that if a neutrino mass
is not zero, different neutrino types will possibly have different masses
and on an increasing scale, as is the case for the corresponding charged
particles e, 
and 
. For this reason, any mass difference will be close to the mass of the
heaviest neutrino, since the other masses are presumably much smaller,
hence negligible.
For a given momentum[4] the energy associated to mass eigenstates is all the more higher as is their "rest" mass; indeed, according to the mass-energy equation, a particle "rest" mass contributes to its total energy, together with kinetic energy (associated to the momentum). Like all elementary particles, neutrino eigenstates in propagation are represented by waves, and their rate grows with energy. Thus, if neutrinos have a mass, and each neutrino type a different one, their corresponding rates (so called phases) are also different. This is not without consequences.
Even without resorting to the mathematical instruments used by physicists,
necessary to reach a thorough understanding of the phenomenon, we can follow
the neutrinos on their way towards the Earth and regard their mass eigenstates
as waves propagated with a different rate according to the neutrino mass.
If the eigenstates have the same mass, the corresponding waves reach the
Earth with the same time relation (or with the same "phase").
These waves can then be combined again, resulting exactly into a
, as flavour eigenstate, the one detected by experiments thanks to the
weak interaction. If the masses of the eigenstates are different, the corresponding
waves are propagated with a different rate, thus they would not reach the
Earth with the same phase as at the start. When they are combined, the
waves are no longer the pure flavour state
they had when they started. There is rather a 
mixing. As quantum mechanics prescribes, we see a
at times and a ,
at other times, and the probability is given by the size of the mixing.
These are neutrino oscillations, whereby an observer on the Earth could
see a that was never
generated !
An analogy (Figure 6) can help us to better understand how a change of phase in the waves can change the flavour eigenstates. In the theory of colours we can distinguish "basic colours" (red, blue and green) and "compound colours", such as violet, a mixture of red and blue. We can use colour mixing as an analogy for neutrino oscillation.
Imagine that a given source generated a "violet" wave. Violet (analogy for a flavour eigenstate) is actually a compound colour, made up by mixing basic colours (an analogy for mass eigenstates) red and blue. The emitted wave is then made by a "red" wave (dotted curve in the figure) and by a "blue" one (continuons curve), whose initial values give the right shade of violet when they are mixed. Let us now take basic colours red and blue[5] for propagation. If the red and blue waves are propagated the same way, their overlap at every distance from the source results in the same shade of violet everywhere. If they are propagated differently, the proportion of each colour is different at each point, and so is the resulting colour seen by an observer, whose eye is globally sensitive rather than to isolated basic colours, to their mixing or overlap. Since the starting colour is actually a compound of two different basic colours (mass eigenstates) and since they are propagated differently, a compound colour (flavour self state) can be observed that is different from the starting one and can change in all points! So the word "oscillation" is not due to the wave representation of particles, but rather to the fact that the colour observed (flavour eigenstate) changes at each point as it leaves its source, following the oscillatory law. In some points, the wave could even appear completely red or completely blue to some observer.