- #1

- 16

- 0

## 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?