Fundamental Understanding of Hamiltonians First of all, my major is CS for several months I have been exploring the area Quantum Computing, therefore my background in Quantum Mechanics is a bit lacking.
I know that a Hamiltonian is a self-adjoint operator which describes the total energy of a system, and its eigenvalues refer to the possible energy levels of that system. Also, it governs the time evolution of the system as the Schrödinger Equation shows.
But I have a basic conceptual question about the Hamiltonian itself, and what does it represent.
To give an example, Suppose there is a 2 level quantum system,
$$
| \psi (t_1) \rangle = \left[ \begin{array}{cc}
 1 \\
 0 \\
\end{array} \right] $$
and we apply the Pauli-X gate,
$$
X | \psi (t_1) \rangle = | \psi (t_2) \rangle = \left[ \begin{array}{cc}
 0 \\
 1 \\
\end{array} \right] $$
Is there a unique Hamiltonian that characterizes the state $| \psi (t_1) \rangle$, or the operation applied? If not, what exactly does the Hamiltonian represent? And are we able to find this Hamiltonian by a series of linear differential equations?
Any help regarding my lack of fundamental knowledge is appreciated.
 A: Evolution of a quantum state is unitary, that is it can be wrtitten as
$$ |\psi(t)\rangle = U(t)|\psi(0)\rangle $$
where $U(t)$ is an unitary operator. Hamiltonian is the generator of this evolution, that is the operator $U(t)$ satisfies the equation
$$ \frac{dU(t)}{dt} = -\frac{i}{\hbar} H(t) U(t)$$
from which we can calculate
$$ H(t) = i\hbar \frac{dU(t)}{dt} U(t)^{-1}$$
Note that you need continuous evolution to be able to define the hamiltonian. The quantum gate is an unitary transformation, but generally you only know its final effect, and you don't know the full function $U(t)$, and because of that you can't say what the hamiltonian is.
Only in some cases, when you know how the gate is actually built, you can tell what the hamiltonian is.
A: 
I know that a Hamiltonian is a self-adjoint operator which describes the total energy of a system, and its eigenvalues refer to the possible energy levels of that system.

If you knew that, you wouldn't be asking the following question:

Is there a unique Hamiltonian that characterizes the state $ |ψ(t_1)⟩ $, or the operation applied? If not, what exactly does the Hamiltonian represent?

The Hamiltonian is not a property of a specific state or transformation. In the usual formalism of QM, your states live in a (rigged?) Hilbert space $ V $, and there's a canonical self-adjoint operator $ H $ on this space called the Hamiltonian. The Hamiltonian generates the time evolution of the system via the Schrodinger equation and it's the unique operator such that for any state $ \psi \in V $, the value $ \langle \psi, H \psi \rangle $ is the expected value of the energy upon doing a measurement on the state $ \psi $. Crucially, the operator $ H $ does not depend on the state $ \psi $.
In the context of quantum computing what matters is that by changing the Hamiltonian of a quantum system, you can in principle perform arbitrary unitary transformations on the state space.
