It can be obtained by a Legendre transform of the Lagrangian, and it contains both a kinetic and potential energy term.
Tosio Kato showed that it is self-adjoint, making it a useful tool for its most famous application. If the Poisson bracket is applied to this quantity, the result will be the time derivative for that quantity therefore necessitating that a Poisson bracket of zero is a constant of motion for that system. It is given by the formula h-bar omega times quantity a dagger a plus one half, and when this quantity is applied to the (*) wave function, it gives a multiple of the wave function, which means the time-independent Schrodinger equation is actually an eigenvalue equation in it. It can be obtained by a Legendre transform of the Lagrangian, and it contains both a kinetic and potential energy term. FTP, name this operator from quantum mechanics, named for the Irish inventor of quaternions and symbolized with a capital H.
Tosio Kato showed that it is self-adjoint, making it a useful tool for its most famous application.
{Hamiltonian} (prompt on "H")