# Time evolution of a two-state system

1. Oct 26, 2016

### Milsomonk

1. The problem statement, all variables and given/known data
Hey guys,
I have a question that asks;
Assume an Electron Nuetrino (U1) is produced at t = 0. Find the state U(t) for later times t > 0.

To give some context the question is based on a two state system where U1 = Collumn vector (sin(theta) cos(theta)) and U2 = Collumn vector (-sin(theta) cos(theta)).

2. Relevant equations

3. The attempt at a solution

I get that its asking me to essentially write the system with the time dependance, and I cant simply bolt the time dependance exponential on the end because U1/U2 are not generally eigenstates of the hamiltonian. So I need to expand U(t=0) in terms of energy eigenstates and then I can include the time dependant exponential for t>0.
Im just not sure how to do this, since I know its in the state U1 at t=0 whats to expand? any advice would be appreciated :)

2. Oct 26, 2016

### Orodruin

Staff Emeritus
A good start would be to express U1 in the mass eigenstates.

3. Oct 26, 2016

### Milsomonk

Hmmm, we havn't covered that, we havn't looked in any detail at the specific case of nuetrino oscillations. I think he was just hoping to make us think about simple two state systems rather than nuetrino oscillations themselves.

4. Oct 27, 2016

### Orodruin

Staff Emeritus
Well, there is nothing specifically peculiar for the case of neutrino oscillations. The same procedure will be applicable to any two level system.

Are you saying you have not seen how to write an arbitrary vector as a linear combination of a set of basis vectors?

5. Oct 27, 2016

### Milsomonk

No sorry, it was the mass eitrnstates I haven't seen before.