Variational principle
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A variational principle is a principle in physics which is expressed in terms of the calculus of variations.
According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation.
Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in physics as Noether's theorem, the Poincaré group of transformations (what is now called a gauge group) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.
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Examples
- Fermat's principle in geometrical optics
- The principle of least action in mechanics, electromagnetic theory, and quantum mechanics, where the dimension is action.
- The Einstein equation also involves a variational principle, according to Stephen Wolfram, (A New Kind of Science, p. 1052.), as a constraint on the Einstein-Hilbert Action.
Variational principle in quantum mechanics
For a hamiltonian H that describes the studied system and any normalizable function Ψ with arguments appropriate for the unknown wave function of the system, we define the functional
- <math> \varepsilon\left[\Psi\right] = \frac{\left\langle\Psi|\hat{H}|\Psi\right\rangle}{\left\langle\Psi|\Psi\right\rangle}.<math>
The variational principle states that
- <math>\varepsilon \geq E_0<math>, where <math>E_0<math> is the lowest energy eigenstate (ground state) of the hamiltonian
- <math>\varepsilon = E_0<math> if and only if <math>\Psi<math> is exactly equal to the wave function of the ground state of the studied system.
The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state.
Further readings
- Epstein S T 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
- Nesbet R K 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
- Adhikari S K 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
- Gray C G, Karl G and Novikov V A 1996 Ann. Phys. 251 1.
See also
External links and references
- Cornelius Lanczos, The Variational Principles of Mechanics
- Stephen Wolfram, A New Kind of Science p. 1052
- Gray, C.G., G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles (http://arxiv.org/abs/physics/0312071)". 11 Dec 2003. physics/0312071 Classical Physics.
- Venables, John, "The Variational Principle and some applications (http://venables.asu.edu/quant/varprin.html)". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
- Williamson, Andrew James, "The Variational Principle (http://www.tcm.phy.cam.ac.uk/~ajw29/thesis/node15.html) -- Quantum monte carlo calculations of electronic excitations". Robinson College, Cambridge, Theory of Condensed Matter Group, Cavendish Laboratory. September 1996. (dissertation of Doctor of Philosophy)
- Tokunaga, Kiyohisa, "Variational Principle for Electromagnetic Field (http://www.d3.dion.ne.jp/~kiyohisa/tieca/26.htm)". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VIru:Вариационные принципы