Hamiltonianens

Hamiltonians are mathematical constructs used to describe the total energy of a physical system. They appear in classical mechanics, quantum mechanics, and related fields as a way to encode kinetic and potential energy and, more generally, the dynamics of a system.

In classical mechanics, the Hamiltonian H(q, p, t) is a function of generalized coordinates q, conjugate momenta p, and time. The evolution of the system is governed by Hamilton's equations: dq_i/dt = ∂H/∂p_i and dp_i/dt = -∂H/∂q_i. When the Hamiltonian has no explicit time dependence, its value is conserved along trajectories. The Hamiltonian formalism emphasizes symplectic structure and canonical transformations that preserve the form of the equations of motion.

In quantum mechanics, the Hamiltonian Ĥ is an operator corresponding to the total energy. It generates time evolution through the Schrödinger equation iħ ∂ψ/∂t = Ĥ ψ. For time-independent systems, energy eigenstates satisfy Ĥ|E> = E|E>. The Hamiltonian framework underpins many approximation methods, spectra calculations, and dynamics in quantum systems.

Hamiltonians are central to many-body physics and statistical mechanics, where H models interactions among particles and degrees of freedom. Examples include the quantum harmonic oscillator, the rigid rotor, and lattice models such as tight-binding and Hubbard models. The structure and symmetries of H determine conserved quantities and emergent phenomena.

Beyond physics, energy-like functionals bearing the same name appear in other disciplines to generate or describe dynamical evolution, reflecting the broad utility of the Hamiltonian concept.