Hamiltonian

The Hamiltonian is the operator that stands for a system's total energy, and in quantum mechanics it is the engine of all time evolution. It is usually built from two pieces: a kinetic-energy term, describing the energy of motion, and a potential-energy term, describing the forces and fields the particle sits in. Add them together and you have the operator written Ĥ.

Its central role is that it sits on the right-hand side of the Schrödinger equation. To know how a quantum system will change, you simply ask what its total energy is, encode that as the Hamiltonian, and let the equation do the rest. Choosing the right Hamiltonian for a problem — a free particle, an atom, a crystal — is most of the work of setting up a quantum calculation.

The Hamiltonian also fixes a system's allowed energies: solving Ĥψ = Eψ yields the energy eigenstates and their eigenvalues. Because energy and time are partners in physics, the operator that represents energy is the very one that generates motion in time. That deep link is why the Hamiltonian, named after a nineteenth-century reformulation of classical mechanics, became the keystone of the quantum theory.