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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]
C.G. Gray, G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles". 11 December 2003. physics/0312071 Classical Physics. Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. John Venables, "The Variational Principle and some applications". Dept of Physics and ...
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 variational theorem states that for a time-independent Hamiltonian operator, any trial wave function will have an energy expectation value that is greater than or equal to the true ground-state wave function corresponding to the given Hamiltonian. Because of this, the Hartree–Fock energy is an upper bound to the true ground-state energy ...
The Hellmann–Feynman theorem is actually a direct, and to some extent trivial, consequence of the variational principle (the Rayleigh–Ritz variational principle) from which the Schrödinger equation may be derived. This is why the Hellmann–Feynman theorem holds for wave-functions (such as the Hartree–Fock wave-function) that, though not ...
In quantum computing, the variational quantum eigensolver (VQE) is a quantum algorithm for quantum chemistry, quantum simulations and optimization problems.It is a hybrid algorithm that uses both classical computers and quantum computers to find the ground state of a given physical system.
[1] [2] [3] In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.
The Schwinger's quantum action principle is a variational approach to quantum mechanics and quantum field theory. [1] [2] This theory was introduced by Julian Schwinger in a series of articles starting 1950. [3]