Recent content by thinkLamp

Oh wow! Nice!

Ah, it should be: $$ \left| \nu_2 \right> = \sin{\theta} \left| \nu_e \right> + \cos{\theta} \left| \nu_{\mu} \right> $$ Thank you so much for your help! You've been very kind and patient.

Ah. Okay, so $$ \left| \nu_2 \right> = \cos{\theta} \left| \nu_e \right> - \sin{\theta} \left| \nu_{\mu} \right> $$ $$ \left| \nu_e \right> = \cos{\theta_M} \left| \nu_1M \right> + \sin{\theta_M} \left| \nu_2 M\right> \\ \left| \nu_{\mu} \right> = - \sin{\theta_M} \left| \nu_1M \right> +...

The mater mixing angle is given as, $$ \Delta_M \sin{2 \theta_M} = \Delta \sin{2 \theta} $$ and $$ \Delta_M \cos{2 \theta_M} = \Delta \cos{2 \theta} - \sqrt{2} G_F N_e $$ So then, I can write, for example, $$ \left| \nu_e \right> = \cos{\theta_M} \left| \nu_1 \right> + \sin{\theta_M} \left|...

That makes sense. So then, how do I go about using this Hamiltonian to propagate my neutrinos? If I diagonalize my Hamiltonian, that would give me my propagation eigenstate, which I can then propagate over distance L and then relate that state back to my flavor eigenstate to find the overlap...

Right. But you said the Hamiltonian I wrote is in flavor basis so wouldn't that imply that its eigenstates are flavor states? Then, it'd be the flavor states that would propagate but that's incorrect like you pointed out in post #2.

Okay, does that mean the eigenstates of this Hamiltonian (which is in flavor basis) would relate to the flavor eigenstates? Or do I have to convert this in the mass basis, find its eigenstates, relate that to mass eigenstates? If my first statement is true, how does the eigenstates of my...

Ah. How do I find by propagation eigenstates then? I'm assuming it'd be just the eigenstates of my Earth-propagation Hamiltonian? So I need to diagonalize my full Hamiltonian. But I'm not sure how to relate that to ##\nu_e## or ##\nu_1##. As in, I took my Hamiltonian (which for this case is...

Thanks for the help! So I should be evolving my mass eigenstate instead. That means, $$ \left| \nu_2, t \right> = \left| \nu_2, 0 \right> e^{-iE_2t} = e^{-iE_2t} (\sin{\theta} \left| \nu_e \right> + \cos{\theta} \left| \nu_{\mu} \right> ) $$ So then when, $$P_{2 \to e} = \left| \left< \nu_2, t...

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...

Homework Statement For a power spectrum density fluctuations ##P(k) \propto k^n##, I need to find the scaling (with respect to ##a##) of the horizon wavenumber ##\frac{2\pi}{\chi_H}## in a matter dominated universe in terms of ##n##. ##\chi_H(a)## is the evolving particle horizon, in a flat...

Okay. I'm not sure how to write ##\left| \psi(x, t) \right>## though. Very generally, my guess is that it would be $$ \left| \psi(x, t) \right> = U_{e1}^* \left| \nu_1 \right> e^{-i \phi_1} + U_{e2}^* \left| \nu_2 \right> e^{-i \phi_2} + U_{e3}^* \left| \nu_3 \right> e^{-i \phi_3} $$ But here...

Does that mean the fraction of mu neutrinos (or tau neutrinos) that arrive at Earth is going to be some constant, that does not depend on the length and neutrino energy? How would I calculate that fraction then? Is it just the square of the corresponding element of the PMNS matrix?

Because since it will oscillate between different flavor states, "how far along" the neutrino has traveled would affect which state it is on?

Homework Statement Consider solar neutrinos of energy 1 MeV (EDIT: 10 MeV not 1 MeV) which are formed at the center of the sun in the ##\nu_2## eigenstate. What fraction of it do you expect to arrive at Earth as ##\nu_\mu## and what fraction as ##\nu_\tau##? Assume that it evolves adiabaticaly...