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In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. [1] The basis for this method is the variational principle. [2] [3]
The variational method of Ritz would found his use quantum mechanics with the development of Hellmann–Feynman theorem. The theorem was first discussed by Schrödinger in 1926, the first proof was given by Paul Güttinger in 1932, and later rediscovered independently by Wolfgang Pauli and Hans Hellmann in 1933, and by Feynman in 1939.
The variational method in quantum mechanics; Hellmann–Feynman theorem; Gauss's principle of least constraint and Hertz's principle of least curvature; Hilbert's action principle in general relativity, leading to the Einstein field equations. Palatini variation; Hartree–Fock method; Density functional theory; Gibbons–Hawking–York ...
Variational method (quantum mechanics), one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states; Variational Bayesian methods , a family of techniques for approximating intractable integrals arising in Bayesian inference and machine learning;
The algorithm is based on the variational method of quantum mechanics. It was originally proposed in 2014, with corresponding authors Alberto Peruzzo, Alán Aspuru-Guzik and Jeremy O'Brien . [ a ] [ 1 ] [ 2 ] The algorithm has also found applications in quantum machine learning and has been further substantiated by general hybrid algorithms ...
In quantum mechanics, where a system of particles is described using a Hamiltonian, the Ritz method uses trial wave functions to approximate the ground state eigenfunction with the lowest energy. In the finite element method context, mathematically the same algorithm is commonly called the Ritz-Galerkin method .
Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system.
In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system. [ 1 ] The basic building block is a generic wave function | Ψ ( a ) {\displaystyle |\Psi (a)\rangle } depending on some parameters a {\displaystyle a} .