I'm taking a course in QM at my university, and I'm trying to work out an assignment given to the class by our professor. The setup is as follows:
The problem is about a simplified description of neutrinos. The main simplification is that we only care about the neutrino type. We assume that when measured a neutrino can be found to be one, and only one, of two types: an electron neutrino, denoted $|\nu_e\rangle$, or a muon neutrino $|\nu_\mu\rangle$ (we are not including the tau neutrino).
We represent the two kets as $|\nu_e\rangle = \begin{bmatrix}1\\0\end{bmatrix}$ and $|\nu_\mu\rangle = \begin{bmatrix}0\\1\end{bmatrix}$. Furthermore we have the hamiltonian given in this representation as $$H = \begin{bmatrix} \alpha_e & g_{e\mu}\\ g_{\mu e} & \alpha_\mu \end{bmatrix}$$ where $\alpha_e,\alpha_\mu \in \mathbf{R}$ are constants related to the masses and kinetic energies of the neutrinos, and $g_{e\mu} = g_{\mu e}^*$ are complex numbers related to the interactions between different types of neutrinos.
The last point in this problem prior to where my issues arise asks us to write $H$ as a linear combination of the Pauli matrices and the (2x2) identity matrix. I believe a have this right, and it comes out to the following: $$H = \frac{g_{\mu e} + g_{e \mu}}{2}\sigma_1 + \frac{g_{\mu e} - g_{e \mu}}{2i}\sigma_2 + \frac{\alpha_e - \alpha_\mu}{2}\sigma_3 + \frac{\alpha_e+\alpha_\mu}{2}I_2.$$
Now we assume that a neutrino in the state $|\nu\rangle = a|\nu_e\rangle + b|\nu_\mu\rangle$ (with $|a|^2+|b|^2=1$) has just been measured to be an electron neutrino, so that it is in the state $|\nu\rangle=|\nu_e\rangle$ at time $t=0$. We now want to find the probability of a second measurement on the same neutrino at a later time $t$ giving the result that the neutrino is a muon neutrino.
From here I'm working on the assumption that I need to find $|\nu(t)\rangle$, and use that to find to probability. I know that to do this I can act on $|\nu(t=0)\rangle =|\nu_e\rangle$ with the time evolution operator $U$: $$U = \exp\left(-iHt/\hbar\right) = \sum_{k=0}^\infty \frac{1}{k!}\left(-\frac{iHt}{\hbar}\right)^k.$$
This is where I get stuck. Normally when we have acted with $U$ we have done so on an energy-eigenket so that $H^k|\lambda\rangle = E^k|\lambda\rangle$, and from there things are not that bad. But now, $|\nu_e\rangle$ is not an energy-eigenstate, leading to a very ugly expression.
I have been able to work out a recursive formula (that I think is correct) allowing me to compute $|\nu(t)\rangle$ numerically. Unfortunately, the assignment explicitly asks for an analytical solution for the probability. I'm also thinking that the rewrite of $H$ as a linear combination of Pauli matrices should be used, but I have not been able to figure out how this might help.
I'm also a little unsure as to what "measuring the neutrino to be a muon neutrino" means. What kind of measurement is this referring to? What eigenvalue equation has been solved to give a measurement value that we interpret to give the neutrino type? It seems to me I need to know this in order to find the probability, e.g. what to do if I actually find $|\nu(t)\rangle$?
It has been a full day working with my fellow students on this, so I'm hoping someone here has some helpful insight!
EDIT: Without a doubt the easiest way to do it was to exploit the exponential properties of Pauli matrices, as suggested in comments. Still, I managed to do it using all proposed solutions, although with different algebraic complexities.