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.

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, the Einstein-Hilbert action.
• Gauss' principle of least constraint
• Hertz's principle of least curvature

Variational principle in quantum mechanics

Say you have a system for which you know what the energy depends on, or in other words, you know the Hamiltonian H. If you cannot solve the Schrödinger equation to figure out the wavefunction, you can guess any normalized wavefunction whatsoever, say φ, and it turns out that the expectation value of the hamiltonian for your guessed wavefunction will be greater than the actual ground state energy. Or in other words:

$E\_\{ground\}\; \backslash le\; \backslash left\backslash langle\backslash phi|H|\backslash phi\backslash right\backslash rangle$

This holds for any φ you could have guessed!

Proof

Your guessed wavefunction, φ, can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal):

$\backslash phi\; =\; \backslash sum\_\{n\}\; c\_\{n\}\backslash psi\_\{n\}\; \backslash ,$

Then, to find the expectation value of the hamiltonian:

{| \left\langle\phi >H = \left\langle\sum_{n}c_{n}\psi_{n} >H = \sum_{n}\sum_{m}\left\langle c_{n}\psi_{n} >E_{m} = \sum_{n}\sum_{m}c_{n}^*c_{m}E_{m}\left\langle\psi_{n} >\psi_{m}\right\rangle \, = \sum_{n} >c_{n}

If we replace $E\_\{n\}$ with the ground state energy $E\_\{g\}$, then it comes out of the sum, and the relationship changes to greater-than or equal-to. That is,

$\backslash left\backslash langle\backslash phi|H|\backslash phi\backslash right\backslash rangle\; \backslash ge\; E\_\{g\}\; \backslash ,$

In general

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

$\backslash varepsilon\backslash left*=\; \backslash frac\{\backslash left\backslash langle\backslash Psi|\backslash hat\{H\}|\backslash Psi\backslash right\backslash rangle\}\{\backslash left\backslash langle\backslash Psi|\backslash Psi\backslash right\backslash rangle\}.$

The variational principle states that

• $\backslash varepsilon\; \backslash geq\; E\_0$, where $E\_0$ is the lowest energy eigenstate (ground state) of the hamiltonian
• $\backslash varepsilon\; =\; E\_0$ if and only if $\backslash Psi$ 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)
• R.P. Feynman, "The Principle of Least Action", an almost verbatim lecture transcript in Volume 2, Chapter 19 of The Feynman Lectures on Physics, Addison-Wesley, 1965. An introduction in Feynman's inimitable style.
• Lanczos C, The Variational Principles of Mechanics (Dover Publications)
• 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

• Action principle
• History of physics

External links and references

• 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". 11 Dec 2003. physics/0312071 Classical Physics.
• Venables, John, "The Variational Principle and some applications". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
• Williamson, Andrew James, "The Variational Principle -- 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". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI

Theoretical physicsCalculus of variations

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