Hamiltonians

A Hamiltonian is a mathematical construct used to describe the total energy of a physical system, serving as a central element in both classical and quantum mechanics. In classical mechanics, the Hamiltonian H(q,p,t) is a function of generalized coordinates q=(q1,…,qn) and conjugate momenta p=(p1,…,pn). The equations of motion are given by Hamilton's equations: dq_i/dt = ∂H/∂p_i and dp_i/dt = −∂H/∂q_i. If H does not depend explicitly on time, its value is conserved along the system’s trajectories.

In quantum mechanics, the Hamiltonian becomes an operator Ĥ acting on states in a Hilbert space. The dynamics are governed by the Schrödinger equation iħ ∂ψ/∂t = Ĥ ψ. The eigenvalues of Ĥ are the possible energy measurements, and for a time-independent Ĥ the time evolution is governed by the unitary operator U(t) = exp(−iĤt/ħ).

Hamiltonians are model-dependent and reflect the physical interactions present. Classic examples include the simple harmonic oscillator H = p^2/2m + (1/2)mω^2 q^2 and the hydrogen atom with a Coulomb potential. In many-body and condensed‑matter physics, Hamiltonians describe interactions among many degrees of freedom, such as the tight-binding model, the Hubbard model, or spin-chain models.

Generalizations include time-dependent Hamiltonians Ĥ(t), which allow explicit time variation in energy, and the role of symmetries in identifying conserved quantities (Noether’s theorem). In statistical mechanics, the Hamiltonian underpins the partition function Z = Tr[exp(−βĤ)], linking microscopic dynamics to macroscopic thermodynamics.