- #1
thinkLamp
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Homework Statement
I'm trying to solve problem 3 from http://www.ictp-saifr.org/wp-content/uploads/2018/07/hw_ICTP-SAIFR-2.pdf The problem is as follows:
Assume that solar neutrinos arrive at the surface of the Earth in the ##\left| \nu_2 \right>## state (a mass eigenstate). Assume this neutrino propagates a distance L through the Earth before reaching the detector. Assume that the the electron number density in the neutrino's path is constant which means the oscillation frequency is
$$
\Delta_M = \sqrt{ \Delta^2 \sin^2{2 \theta} + (\Delta \cos{2 \theta} - \sqrt{2} G_F N_e )^2 }
$$
Compute the probability that this neutrino is detected as an electron-type neutrino.
Homework Equations
$$
\Delta_M = \sqrt{ \Delta^2 \sin^2{2 \theta} + (\Delta \cos{2 \theta} - \sqrt{2} G_F N_e )^2 }
$$
$$
\left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right>
$$
The Attempt at a Solution
I started by writing
$$
\left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right>
$$
Then, because it's the flavor states that's going to be oscillating, I wrote
$$
\left| \nu_e (t) \right> = \left| \nu_e \right> e^{-i \Delta_M t} = \left| \nu_e \right> e^{-i \Delta_M L}
$$
But then, that would mean, the probability that the ##\left| \nu_2 \right>## neutrino is detected as a ##\left| \nu_e \right>## neutrino is just
$$
\left| \left< \nu_2 (t) | \nu_e \right> \right|^2 = \cos^2{\theta} \left| e^{-i \Delta_M L} \right|^2 = \cos^2{\theta}
$$
But, this does not depend on ##L## even though in the next sub-problem, I need to use a numeric value of ##L## to compute the probability so it seems wrong.
Where does this dependency on ##L## come from?