| Hamiltonian (quantum mechanics) | |
|---|---|
| Name | Hamiltonian |
| Field | Quantum physics |
| Introduced | 1920s |
| Notable examples | Schrödinger Hamiltonian, Pauli Hamiltonian, Dirac Hamiltonian |
Hamiltonian (quantum mechanics)
The Hamiltonian in quantum mechanics is the operator corresponding to the total energy of a quantum system and generates the time evolution of quantum states via the Schrödinger equation. It is central to predicting dynamics, determining spectra, and formulating conservation laws in quantum theory, and underpins practical applications in quantum chemistry, condensed matter physics, and quantum computing.
In the canonical formulation of non-relativistic quantum theory the Hamiltonian operator H acts on a system's state vector in a Hilbert space and represents observables associated with energy. The Schrödinger picture uses H to define unitary time evolution through the time-dependent Schrödinger equation iħ ∂/∂t |ψ(t)⟩ = H |ψ(t)⟩, while in the Heisenberg picture H governs operator evolution via commutators. The Hamiltonian encodes interactions, external fields, and kinetic contributions; symmetry properties of H lead to conserved quantities through Noether's theorem in quantum contexts. First quantization procedures map classical Hamiltonians to H, and second quantization represents H in terms of creation and annihilation operators for many-body systems.
Mathematically H is typically an unbounded, self-adjoint operator on a separable Hilbert space to ensure real spectra and unitary time evolution via Stone's theorem. For a particle in coordinate representation H often takes the form T + V where T is the kinetic term (e.g., −(ħ^2/2m)∇^2) and V is a multiplication operator by a potential function. Domain issues, self-adjoint extensions, and essential self-adjointness are central in rigorous treatments developed by functional analysts; these topics connect to the work of John von Neumann and Reed and Simon. Commutation relations [H, A] determine time dependence of observables A and feature prominently in perturbation theory and linear response.
A time-independent Hamiltonian H (no explicit t-dependence) allows separation of variables and leads to stationary states satisfying the time-independent Schrödinger equation H |ψ⟩ = E |ψ⟩. Time evolution is given by the unitary operator e^(−iHt/ħ). Time-dependent Hamiltonians H(t) describe driven systems, time-dependent potentials, and interaction pictures; they require time-ordered exponentials (Dyson series) to construct evolution operators, and are fundamental in treatments of quantum control, time-dependent perturbation theory, and phenomena like quantum quenches in many-body physics.
Canonical examples include the single-particle Schrödinger Hamiltonian for a particle in a potential, the Pauli Hamiltonian incorporating spin and magnetic interactions, and the relativistic Dirac Hamiltonian for spin‑1/2 fermions. Many-body models include the Heisenberg model and Hubbard model in condensed matter, the BCS Hamiltonian for superconductivity, and the Anderson impurity model. Quantum field theories are defined by Hamiltonian densities in canonical quantization and lattice regularizations such as lattice gauge theory. In quantum optics the Jaynes–Cummings model and cavity QED Hamiltonians describe light–matter coupling; in quantum information, model Hamiltonians underpin adiabatic quantum computing and quantum annealing.
Spectral properties of H—discrete, continuous, or mixed spectra—determine bound states, scattering states, and resonances. Eigenvalues correspond to allowed energy levels measurable in spectroscopy; eigenstates form bases (or generalized eigenbases) for state decomposition. Techniques from spectral theory and scattering theory classify spectra and resonances; examples include the hydrogen atom spectrum solved via separation of variables leading to Rydberg series and the continuum of free-particle Hamiltonians. Degeneracies relate to symmetry groups (e.g., SU(2 for spin) and lead to multiplets; perturbations lift degeneracy described by degenerate perturbation theory.
The quantum Hamiltonian arises from quantization prescriptions applied to a classical Hamiltonian function H_class(q,p). Canonical quantization replaces phase-space coordinates by operators satisfying canonical commutation relations [q̂, p̂] = iħ, guided historically by pioneers such as Paul Dirac and Werner Heisenberg. Alternative schemes include path integral formulation (action principle), Weyl quantization, and geometric quantization. The correspondence principle ensures classical dynamics emerge in the ħ → 0 limit or for coherent states; features like Poisson brackets correspond to quantum commutators up to factors of iħ.
Exact solutions exist for select Hamiltonians (harmonic oscillator, hydrogen atom, free particle) often via symmetry and special functions. General systems employ approximation methods: time-independent perturbation theory, time-dependent perturbation theory, variational methods (e.g., Rayleigh–Ritz), mean-field theories (Hartree–Fock), density functional theory (DFT) in quantum chemistry, and numerical diagonalization techniques including exact diagonalization, Lanczos algorithm, and matrix product states/DMRG for many-body problems. Semiclassical methods (WKB), Monte Carlo sampling (quantum Monte Carlo), and renormalization group methods analyze interacting Hamiltonians and critical phenomena. Experimental control and spectroscopy probe Hamiltonians in platforms like ion trap quantum computers, superconducting qubits (e.g., devices by IBM and Google), and ultracold atoms in optical lattices.