Hamiltonian
The Hamiltonian is the operator that represents the total energy of a quantum system, combining its kinetic and potential energy. Its eigenvalues are the allowed energy levels of the system, and its eigenstates are the states of definite energy. The Hamiltonian is named after the mathematician William Rowan Hamilton, whose formulation of classical mechanics it generalizes.
Time evolution
The Hamiltonian governs how a quantum state changes in time through the Schrodinger equation, in which the rate of change of the state is set by the Hamiltonian acting on it. Finding the energy levels of a system therefore reduces to finding the eigenvalues of its Hamiltonian, which makes it the starting point for describing atoms, molecules, and engineered devices. It is also the object programmed in Quantum annealing, where a problem is encoded so that its solution is the lowest-energy state of a final Hamiltonian, all within the framework of quantum mechanics.
Sources
- Quantum Mechanics (Stanford Encyclopedia of Philosophy) (Stanford Encyclopedia of Philosophy, 2021)
- Quantum annealing in the transverse Ising model (arXiv, 1998)
Cite this entry
"Hamiltonian." postquantum.wiki. Updated July 11, 2026. https://postquantum.wiki/hamiltonian@misc{pqwiki-hamiltonian,
title = {Hamiltonian},
howpublished = {\url{https://postquantum.wiki/hamiltonian}},
year = {2026},
note = {postquantum.wiki, updated 2026-07-11}
}