Definition 0.1 The Hamiltonian operator H introduced in quantum mechanics (QM) by Schrödinger (and thus sometimes also called the Schrödinger operator) on the Hilbert space $L^2(\Rset^n)$ is given by the action: $$ \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), $$

The operator defined above $[-\nabla^2 +V(x)]$ , for a potential function $V(x)$ specified as the real-valued function $V\colon \Rset^n \to \Rset$ is called the Hamiltonian operator, H, and only very rarely the Schrödinger operator.

Schrödinger formulation of QM

Heisenberg formulation of QM

Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is ``more natural and fundamental than the Schrödinger'' formulation because the Lorentz invariance from General Relativity is also encountered in the Heisenberg picture, and also because there is a `correspondence' between the commutator of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of classical mechanics. If the state vector $\psi$ , or $\left| \psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of motion' of a (quantum) observable operator is : $$ \frac{d}{dt} A_{quantum} = (i\hbar)^{-1}[A,H] + \left(\frac {\partial A}{\partial t}\right)_{classical} $$