Physical interpretation of applying a unitary operator to a state When we apply one of the Pauli matrices $\sigma_y$ on one of its eigen-vectors $| \odot \rangle$, what does the eigen-value tell us about $| \odot \rangle$? Is this considered a measurement of $| \odot \rangle$ along the y-axis? What is the physical interpretation of this (e.g. is there an example of an experiment that performs this kind of operation?)
 A: Your question is ambiguous. Applying $\sigma_y$ may mean either applying a particular quantum gate to a qubit, or measuring $\sigma_y$ on that qubit. The gate applying $\sigma_y$ is just represented by $\sigma_y$. The gate that corresponds to measuring $\sigma_y$ from qubit 1 onto qubit 2 is a gate that performs a not on qubit 2 if qubit 1 is in the +1 eigenstate of $\sigma_y$:
$$
U = \tfrac{1}{2}(I-\sigma_{y1})\sigma_{x2}+\tfrac{1}{2}(I+\sigma_{y1}).
$$
It would be very serious error to confuse those two operations because they happen to produce the same result on a particular state.
Now, $\sigma_y = i\sigma_z\sigma_x$, so it is equivalent to a $\sigma_x$ followed by a phase change of $\pi$ on $|1\rangle$ and a phase change of $\pi/2$ on both $|0\rangle$ and $|1\rangle$. You might be able to do this and repeat it by a suitable series of laser pulses on cold atoms or something like that. And there may be other ways it could be instantiated. The physical interpretation of such an experiment will depend on what you actually did in each case, so there won't be a single standard interpretation.
A: Any operator, $A$, acting one of its eigenvector, $| \psi_i \rangle$, will give,
$$ A | \psi_i \rangle = \lambda_i | \psi_i \rangle, $$
where $\lambda_i$ is the corresponding eigenvalue. The eigenvalues of the Pauli matrices are $\pm 1$ corresponding to either spin up or down in the corresponding direction. The eigenvalues are possible results of measuring an observable, which in general is a Hermitian (not unitary) operator. 
The modulus squared of the inner product of the corresponding eigenvector with the state of your system gives the probability of measuring that eigenvalue as the result of measurement. 
We want Hermitian observables because results of measurements should be real numbers and its eigenvectors should be mutually orthogonal (so that different outcomes are mutually exclusive, i.e., you can't measure spin up and down at the same time with the same experiment on the same system) 
A simple Stern–Gerlach experiment is an example of measuring spin: http://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment
A: $\sigma_y$ isn't an observable, but $S_y$ is, so let's focus on that.
If you know that $S_y | \psi \rangle$ can be written $a |\psi\rangle$, where $a$ is a number, then it only tells you that $| \psi \rangle$ a measurement of the $y$-component of spin will definitely yield $a$. That's all. 
I don't think you can consider $S_y|\psi\rangle$ a measurement of $y$-component of spin on the state $| \psi \rangle$. Why? Because if you operate on a general (non-eigen) state, you don't get a nice single number out, whereas physically you will get a single number.
A: A really neat and intuitive way to deal with qubit systems like spin is to use the Bloch sphere. The Bloch sphere represents the two dimensional Hilbert space which the spin $\frac{1}{2}$ state vectors live in by a sphere in $\mathbb{R^3}$. 
Have a look at this wikipedia page for some more info http://en.wikipedia.org/wiki/Bloch_sphere. (Particularly the figure.)
Basically you can represent the spin state vectors as being antipodal points on the Bloch sphere. So if you consider a measurement of the $S_y$ operator on an eigenstate $|\pm y\rangle$
$$S_y |\pm y\rangle=\pm\frac{\hbar}{2}|\pm y\rangle$$ the positive and negative outcomes correspond to real space vectors along the y-axis pointing in either the positive or negative directions. 
A word of caution is that the Bloch sphere is only useful for 2-dimensional Hilbert spaces and so shouldn't be used elsewhere. There is lots of good stuff online about it though so I would have a look around.
A: There are two kinds of operators which have an intuitive meaning when you apply them to a state:


*

*Evolution operators like $e^{-i\hat H\Delta t/\hbar}$ simply take the state at time $t$ and give you the state at $t+\Delta t$

*Projectors tell you what the state will be after you take a measurement. For example, supose you measure an observable $\hat A$ on state $\left|\psi\right>$ and it comes out with the value 5. Then the state after measuring will be $\hat P\left|\psi\right>$ with $\hat P$ the projector for the subspace generated by the eigenvectors of $\hat A$ whose eigenvalue is 5.
