Quantum Mechanincs - Dirac notation and solving time dependant schrodinger The $\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}$ obviously correlate to $x,y,z$ components of the operators.
Consider the Hamiltonian:
                 $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$
where $C$ is a constant and the magnetic field is given by 
$$\vec{B} = (0,B,0) $$
and the spin is 
$$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$ 
with
$$\hat{S}_{x} =\frac{\hbar}{2}∣↑⟩⟨↓∣+ \frac{\hbar}{2}∣↓⟩⟨↑∣ $$
$$\hat{S}_{y} =−\frac{i\hbar}{2}∣↑⟩⟨↓∣+ \frac{i\hbar}{2}∣↓⟩⟨↑∣ $$
$$\hat{S}_{z} =\frac{\hbar}{2}∣↑⟩⟨↑∣− \frac{\hbar}{2}∣↓⟩⟨↓∣,$$
where the basis vectors are assumed orthogonal and normalised. Knowing that at $t=0$
$$ ∣ψ(0)⟩= \frac{1}{2}∣ψ_1⟩+ \frac{\sqrt{3}}{2}∣ψ_2⟩$$
where the ${∣ψ_i⟩}$ are the eigenvectors of the Hamiltonian, solve the time-dependent Schrodinger equation. Write the solution in terms of the basis vectors $∣↑⟩$ and $∣↓⟩$.
From the question I understand that the Hamiltonian will cancel out the $x$ and $y$ spin operators through the dot product. I have tried following this forward and using a presumed general solution to the time dependent Schrodinger equation but I got no where. Could someone just explain to me how to look at attempting this question?
 A: I will get you started with the first bit and finding the eigenvectors to the Hamiltonian, so you are right in that this Hamiltonian is proportional to the $\hat{S}_y$ operator so you need to find the eigenvectors of the $\hat{S}_y$ operator and they will also be eigenvectors of the full Hamiltonian. So we want some states $\vert y_+\rangle$ and $\vert y_-\rangle$ which are eigenvectors, which means they satisfy $$\hat{S}_y\vert y_{\pm}\rangle=\pm\frac{\hbar}{2}\vert y_{\pm}\rangle$$.
Once you have had a bit of experience you will just be able to "see" what the eigenvectors must be for these matrices. Or knowing what an eigendecomposition is helps for this (google it). However, if you are stuck you can always go the old fashioned way by writing out the matrix in the computational basis such that $\hat{S}_y=\frac{\hbar}{2}\begin{pmatrix}0 & -i\\i & 0\end{pmatrix}$ and finding the eigenvectors and eigenvalues by solving the charactersitic equation. 
HINT the eigenvectors should be $\pm\frac{\hbar}{2}$. This should really help if you know what an eigendecomposition is (if not I will explain if google isn't helpful).
Also for this bit it is worth noting that the state $\vert \uparrow\rangle=\begin{pmatrix}1\\ 0\end{pmatrix}$ and $\vert \downarrow\rangle=\begin{pmatrix}0\\ 1\end{pmatrix}$. 
I will put the solution at the end of my answer so you don't have to read it if you don't want to.
It is true that an eigenstate of a system with eigenenergy $E$ evolves in time as $\vert\psi\left(t\right)\rangle=\exp\left(\frac{-iEt}{\hbar}\right)\vert\psi\left(0\right)\rangle$ which is just a special case of general unitary time evolution of a state.
Since the Schrodinger equation is linear we can use the superposition of two solutions as a new solution. I think in the question they want $\vert\psi_1\rangle=\vert y_+\rangle$ and $\vert\psi_2\rangle=\vert y_-\rangle$. From what you have written you can't tell which way round they are but that isn't really important. (I also think my notation is better).
So if you know how each of the eigenstates evolve individually and you know that we can use linear superposition to get a new solution, you should be able to get the form for $\vert\psi\left(t\right)\rangle$. If not comment and I will show some more steps in a few days.
The Eigenvectors (spoilers ahead)

The eigenvectors are $\vert y_{\pm}\rangle=\frac{1}{\sqrt{2}}\left(\vert\uparrow\rangle\pm i \vert\downarrow\rangle\right)$
A: I'll tell you a very powerful way to think about this problem. I myself would not find the eigenvectors like Chris2807 did. I'd proceed as such:
The general solution you refer to is $\exp(-\frac{i t}{\hbar}\hat{H})|\varphi_0\rangle$. So, you seem to be aware that this is $\exp(-\frac{i t}{\hbar}C B \hat{S}_y)|\varphi_0\rangle$. It is useful to think of $S_y$ in this way: $\hat{S}_y=\frac{\hbar}{2}\hat{\sigma}_y$, where $\hat{\sigma}_y=-i∣↑⟩⟨↓∣+i∣↓⟩⟨↑∣$  is a Pauli matrix with the property that $\hat{\sigma}_y^2=\mathbf{1}$. So we have our solution as $\exp(-i\left( \frac{t C B}{2}\right) \hat{\sigma}_y)|\varphi_0\rangle$. The solution is in your hands: Expand the operator exponential in terms of the series $e^x=1+x+\frac{1}{2}x^2+\frac{1}{6}x^3+\cdots$. Each "$\sigma^2$" in the even terms cancels out to the identity operator and in the infinite sum this gives you a cosine term. Each "$\sigma^n$" for odd $n$ term is just a $\pm i \sigma$ term and you wind up with an i times sine term. You're left with something of the form $(\cos(\cdot)+i\sin(\cdot)\hat{\sigma}_y)|\psi_0\rangle$, which can be solved simply by applying the operators correctly.
The Pauli matrices are very cool and if you learn their properties well you can solve lots of problems like these quickly.
A: Look again carefully at the way you have written  your spin operator components (up and down together?)and compare them  to any q .m textbook spin components. Maybe it's just a typo in the way you have written the question, 
New to this myself but if u and d are orthogonal this would make each spin component equal zero, right? 
So I think question is written wrong, you can switch the bra and ket around and get zero
Actually thinking about it, if the question is written correctly, imho they are asking you to consider spin components as operators, i just never saw them written that way before. 
It's x and z not x and y that dot product will cause to disappear. 
I think I have as much to learn from this problem as you do, good while since I have covered this. Hope some one gives us both complete picture
Ok Ben, that's a great set of answers to read through and my apologies for my mistakes above, but you absolutely do remember and learn from the mistakes much more than anything else. I always get the feeling I should take up knitting insead of physics when the right way of doing the problem is shown to me but that feeling does pass. 
I got to learn so much from your question (and my mistakes trying to answer it) that I intend writing it up as a pdf within the next week, so if you are interested in my answers let me know and i can post on my own hosting site, if is this allowed on this forum.   
