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

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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:

<math>E_{ground} \le \left\langle\phi|H|\phi\right\rangle </math>

This holds for any φ you could have guessed!

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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):

<math>\phi = \sum_{n} c_{n}\psi_{n} \,</math>

Then, to find the expectation value of the hamiltonian:

H|\phi\right\rangle \, </math> H|\sum_{m}c_{m}\psi_{m}\right\rangle \,</math>
E_{m}|c_{m}\psi_{m}\right\rangle \,</math>
\psi_{m}\right\rangle \,</math>
c_{n}|^2 E_{n} \,</math>

If we replace <math>E_{n}</math> with the ground state energy <math>E_{g}</math>, then it comes out of the sum, and the relationship changes to greater-than or equal-to. That is,

<math>\left\langle\phi|H|\phi\right\rangle \ge E_{g} \,</math>
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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

<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.

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Further readings

  • Epstein S T 1974 "The Variation Method in Quantum Chemistry". (New York: Academic)
  • 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.
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See also

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External links and references

ja:変分原理

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