The operator in quantum mechanics that corresponds to the total energy of a system is the Hamiltonian.
Usually written as Ĥ, the Hamiltonian combines the system’s kinetic-energy and potential-energy operators. For a simple particle in a potential, it is commonly expressed as Ĥ = T̂ + V̂. Measuring the Hamiltonian can yield one of the system’s allowed energy eigenvalues, so its spectrum represents the possible total-energy results.
The Hamiltonian also controls time evolution through the time-dependent Schrödinger equation: the Hamiltonian acting on a state determines how that state changes with time. For a time-independent Hamiltonian, energy eigenstates evolve in a particularly simple way, acquiring a phase while preserving their measurable energy distribution.
The name comes from William Rowan Hamilton, whose nineteenth-century reformulation of classical mechanics influenced quantum theory. The Hamiltonian is not the same as the Lagrangian: both describe dynamics, but they use different mathematical formulations. The position and momentum operators represent specific observables, whereas the Hamiltonian represents the system’s complete energy under the chosen model.