Are the classical hamiltonian and quantum hamiltonian different types of objects? Context: I'm not a physicist.
I've come across the Hamiltonian in classical physics and in quantum physics, and I can't recognise why they have the same name. They seem very different. So I probably am missing something.
Classical. My understanding of the classical Hamiltonian is: it is a function $H(p,q,t)$ of the (generalized) coordinates of the system, letting $q$ be the generalized "position" vector, and $p$ be the generalized momentum vector. The time evolution of the system is obtained as:
$$\dot q=\frac  {\partial H} {\partial p}\quad\quad\quad \dot p=-\frac  {\partial H} {\partial q}$$
Where the dots denote time derivatives.
Quantum. My understanding of the quantum Hamiltonian, is that it is a Hermitian operator $H$ on the quantum state $|\Psi \rangle $, that is used to calculate the time derivative (generalized Schrodinger equation) :
$$|\dot \Psi\rangle = \alpha H|\Psi \rangle$$
(Where $\alpha=-\frac i {2\pi h}$, which is not important for now).
How are these related? They seem totally different objects:


*

*Type signature. The classical hamiltonian is a function from  state-space to $\mathbb R$. The quantum Hamiltonian is an operator from state space to state space. (both possibly time dependent, in the classical case I mean by state space, $(q,p)$ space, in the quantum case I mean the quantum state space).

*Derivatives. In the classical case, we take the derivative of the Hamiltonian. In the quantum case, we don't, we just use the hamiltonian itself, and use it to operate on the state space.

*Analog of Newton's law?. In classical physics, the Hamiltonian is based on a variational principle (the Hamiltonian is derived from minimizing the Action functional). The "local" laws of motion in classical physics are Newton's laws: $\dot p = F(q,p)$. It seems to me that the Schrodinger equation is the quantum analog of Newton's law (insofar as there is one), not the analog of Hamilton's equations in classical mechanics: The "force operator" also is an operator from state space to state space. Using nonstandard notation by letting the classical state $s=(q,p)$ to emphasize the analogy with the Schrodinger equation, Newton's laws can be written as:
$$\dot s = F s$$
Where $F_q(q,p) = \frac 1 m p$ and $F_p(q,p)$ is the force. It is surprising to me that the Schrodinger equation, which has the form of a local law of motion, contains the energy of the system (the Hamiltonian), whereas the classical law of motion contain forces rather than energy.
Am I wrong that the Schrodinger equation is more analogous to Newton's laws than to the classical Hamiltonian equations? Why do they have the same name, if they seem like very different types of objects? 
 A: The relationship between classical and quantum mechanics is expressed the clearest in the process of Dirac's canonical quantisation. This consists of promoting replacing all Poisson brackets
$$ \{A, B\} = \frac{\partial A}{\partial q} \frac{\partial B}{\partial p} - \frac{\partial B}{\partial q} \frac{\partial A}{\partial p} $$
by the corresponding commutators
$$ -\frac{i}{\hbar} [\hat{A}, \hat{B}] = -\frac{i}{\hbar} (\hat{A} \hat{B} - \hat{B} \hat{A}). $$
In, particular, a classical observable $A$ evolves according to
$$ \dot{A} = \{A, H\} $$
and a quantum observable $\hat{A}$ in the Heisenberg picture evolves according to
$$ \dot{\hat{A}} = -\frac{i}{\hbar} [\hat{A}, \hat{H}] $$
(an alternative formulation to the Schrödinger equation).
I don't claim to have a good intuition for what this correspondence "means", but at least I can comment on your point on the type signature. As QM observables are Hermitian, they are completely characterised by their eigenvalues and eigenvectors. So you could see it as a mapping assigning each eigenvector a value, the eigenvalue. Representing this function as an operator is nice as it allows you to apply the function to all parts of a superposition simultaneously.
For the mathematics of how canonical quantisation works (and why the naive version doesn't work), you should probably look into deformation quantisation or the like. Seems interesting but I don't know any details. It might illuminate your question of where the derivatives come from. (Let me know if you figure it out! I am also puzzled.)
A: As far as I know, the classical and quantum Hamiltonian are related because of the Hamilton-Jacobi equation. The Hamilton Jacobi equation is a classical equation of motion that uses the Hamiltonian itself, and it reads $$\frac{\partial S}{\partial t}+H=0$$, and $S$ is termed Hamilton's principal function, together with the equation $p_j=\frac{\partial S}{\partial q_j}$, and there is even an analogue of the time independent Schödinger equation, by inserting $S=-Et+W$, where $W$ is called Hamilton's characteristic function, one gets $H=E$. And in the Schrödinger equation of the form, $$i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m} \nabla \cdot \nabla \psi +U\psi$$, if one inserts $\psi=\sqrt{\rho}e^{\frac{iS}{\hbar}}$, and substitutes $\hbar \rightarrow 0$ it becomes the following form of the Hamilton-Jacobi equation: 
$$\frac{\partial S}{\partial t}+\frac{|\nabla S|^2}{2m}+U=0
$$
Another reason why they have the same name is because of their relation with energy. The QM Hamiltonian is of the form $$\hat{H}=-\frac{\hbar^2}{2m}\nabla\cdot\nabla+U$$, while the CM Hamiltonian is of the form $$H=\frac{|\mathbf{p}|^2}{2m}+U$$
Also, it is worth mentioning that one can derive the QM Hamiltonian from CM Hamiltonian by substituting dynamical variables with operators.
In short, their name is because of analogy between them. Hope this helps.
