| variational method | |
|---|---|
| Name | Variational method |
| Field | Quantum mechanics |
| Introduced | 1920s |
| Founder | Paul Dirac; foundations in Variational principle |
| Notable uses | Hartree–Fock method, Density functional theory, Quantum Monte Carlo |
variational method
The variational method is a computational and theoretical technique in quantum mechanics that estimates eigenvalues and eigenstates of a Hamiltonian by optimizing a trial state according to a variational principle. It matters because it provides rigorous upper bounds for ground-state energies, informs approximate methods like Hartree–Fock method and Density functional theory, and underpins modern computational approaches in both atomic and condensed-matter physics.
The variational approach traces its roots to classical variational principles such as the principle of least action and was adapted to quantum problems during the early development of quantum theory in the 1920s and 1930s by figures including Paul Dirac and John von Neumann. The method gained practical prominence through atomic and molecular calculations by researchers like Douglas Hartree and later refinements by V. Fock leading to the Hartree–Fock formalism. Development accelerated with numerical computing advances at institutions such as Los Alamos National Laboratory and Bell Labs, and with algorithmic innovations from the field of computational physics and numerical analysis.
The central statement is the Rayleigh–Ritz principle: for a normalized trial wavefunction |ψ_trial⟩, the expectation value ⟨ψ_trial|Ĥ|ψ_trial⟩ is an upper bound to the true ground-state energy of the Hamiltonian Ĥ. This result connects to operator theory as developed by John von Neumann and spectral theory in Hilbert space. The method formalizes energy minimization and justifies variational bounds used in atomic physics, molecular physics, and solid-state physics. Extensions include constrained variational principles for excited states (e.g., orthogonality constraints) and time-dependent variational principles in the style of Dirac–Frenkel variational principle applied to dynamical problems.
Choosing an effective trial wavefunction (ansatz) is central. Common ansätze include Slater determinants underpinning Hartree–Fock method and configuration interaction (CI), multi-configurational self-consistent field (MCSCF), geminal and pairing wavefunctions for superconductivity, and Jastrow factors used in Quantum Monte Carlo to capture correlation. In modern work, tensor network states such as matrix product states and projected entangled pair states are variational classes for one- and two-dimensional lattice models. Machine-learning inspired ansätze, e.g., neural-network quantum states like the restricted Boltzmann machine approach proposed by researchers connected to institutes such as Caltech and Google DeepMind, represent a recent direction.
Practical implementations use analytic variation where integrals are tractable, and numerical optimization when parameters are many. Deterministic solvers include self-consistent field iterations (SCF) for Hartree–Fock and gradient-based minimizers in quantum chemistry packages such as Gaussian and NWChem. Stochastic approaches employ variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC) as implemented in codes like QMCPACK and CASINO. For lattice models, density matrix renormalization group (DMRG) algorithms perform variational optimization over matrix product states. Optimization techniques draw from linear algebra, nonlinear conjugate gradient, and modern machine-learning optimizers like Adam and stochastic reconfiguration.
The variational method is widely used to compute ground-state energies of atoms, molecules, and nuclei, and to study phase diagrams in condensed-matter systems such as the Hubbard model and Heisenberg model. In quantum chemistry, variational calculations underpin benchmarks for chemical bonding and reaction energies in work by communities around IUPAC standards and computational chemistry groups. Excited states are accessed via state-specific variational functionals or linear-response variants such as time-dependent density functional theory (TDDFT). In nuclear physics, variational Monte Carlo and Green’s function Monte Carlo techniques from groups at Argonne National Laboratory have produced accurate nuclear spectra.
While the Rayleigh–Ritz bound guarantees an upper bound for ground states, the quality depends critically on the expressiveness of the ansatz and numerical optimization. Systematic error estimation involves basis-set extrapolation in quantum chemistry, variance extrapolation in Monte Carlo, and finite-size scaling in condensed-matter simulations. Convergence issues—local minima, variational bias, and sign problems in fermionic Monte Carlo—are persistent challenges. Cross-validation against perturbative expansions (e.g., Møller–Plesset perturbation theory) and experimental data from facilities like synchrotrons or cold-atom experiments at institutions such as MIT and Max Planck Institute help assess reliability.
The variational method complements perturbation theory by providing nonperturbative bounds and initial approximations for resummation techniques. In quantum field theory, variational techniques appear in the Gaussian effective potential and variational approximations to path integrals; influential work has come from researchers at Princeton University and Imperial College London in the context of symmetry breaking and renormalization. Lattice gauge theory employs variational bases for low-lying spectra, and continuum approaches use variationally optimized trial actions to study strong-coupling phenomena. These connections bridge traditional theoretical physics with computational practice, supporting stable and conservative progress in understanding quantum many-body systems.
Category:Quantum mechanics Category:Computational physics