I have a question about difference between Hamiltonian function (the description of system in classical physics) and the Hamiltonian operator (quantum mechanics).
I think that there two different points of view: a physical one and mathematical (more technical) one.
In classical Hamiltonian mechanics state of the system (just for sake of simplicity let's consider one dimensional case) is determined by the variables $p, q$. It actually means that if one has defined the initial values of $p$ and $ q$ in arbitrary point of time $t$ then one can find their values in subsequent moment of time $t + \Delta t$ $$ q(t + \Delta t) = q(t) + \dot q(t) \Delta t $$ $$ p(t + \Delta t) = p(t) + \dot p(t) \Delta t $$ using canonical equations: $$\dot q = \partial H / \partial p$$ $$\dot p = - \partial H / \partial q$$ where $H$ classical Hamiltonian function.
In case of quantum mechanics. The state of the system is defined by $\Psi(q, t)$. And if we know $\Psi$ at given moment of time $t$ we can calculate it at subsequent moment $t + \Delta t$: $$\Psi(q, t + \Delta t) = \Psi(q, t) + \dot \Psi(q, t) \Delta t$$ where $i \hbar \dot \Psi = \hat H \Psi$ and $\hat H$ is the Hamiltonian operator.
As for me this leads to the following consequences
- In classical physics Hamiltonian defines canonical variables, but in QM Hamiltonian operator defines only one quantity (psi function)
- Classical motion is defined by canonical equation (principle of the least action), QM Hamiltonian constructed in such a way to satisfy Schrodinger equation (it is not derived from principle of least action)
- Mathematically Hamiltonian in CM is just a function of $q,p$ variables but in quantum mechanics it is a Hermitian operator
Please could you tell me if I am right or if I have something missed here? I am actually interested in what is the difference between quantum and classical Hamiltonian? I will be very pleased because it is very interesting topic for me.