A Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system (this addition is the total energy of the system in most of the cases under analysis).

The expression of the Hamiltonian can take different forms and simplifications taken into account the concrete characteristics of the system under analysis: single or several particles in the system; interaction between particles; kind of potential energy; time varying potential or time independent one; etc.

Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.