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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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} \le \left\langle\phi|H|\phi\right\rangle

This holds for any φ you could have guessed!

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):

\phi = \sum_{n} c_{n}\psi_{n} \,

Then, to find the expectation value of the hamiltonian:

\left\langle\phi|H|\phi\right\rangle \, = \left\langle\sum_{n}c_{n}\psi_{n}|H|\sum_{m}c_{m}\psi_{m}\right\rangle \,
= \sum_{n}\sum_{m}\left\langle c_{n}\psi_{n}|E_{m}|c_{m}\psi_{m}\right\rangle \,
= \sum_{n}\sum_{m}c_{n}^*c_{m}E_{m}\left\langle\psi_{n}|\psi_{m}\right\rangle \,
= \sum_{n} |c_{n}|^2 E_{n} \,

Now, the ground state energy is the lowest energy possible, i.e. E_{n} \ge E_{g}. Therefore, if the guessed wave function φ is normalized:

\left\langle\phi|H|\phi\right\rangle \ge E_{g}\sum_{n} |c_{n}|^2 = E_{g} \,

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

\varepsilon\left[\Psi\right] = \frac{\left\langle\Psi|\hat{H}|\Psi\right\rangle}{\left\langle\Psi|\Psi\right\rangle}.

The variational principle states that

  • \varepsilon \geq E_0, where E0 is the lowest energy eigenstate (ground state) of the hamiltonian
  • \varepsilon = E_0 if and only if Ψ 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.

  • S T Epstein 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.
  • C Lanczos, The Variational Principles of Mechanics (Dover Publications)
  • R K Nesbet 2003 "Variational Principles and Methods In Theoretical Physics and Chemistry". (New York: Cambridge U.P.)
  • S K Adhikari 1998 "Variational Principles for the Numerical Solution of Scattering Problems". (New York: Wiley)
  • C G Gray , G Karl G and V A Novikov 1996 Ann. Phys. 251 1.
  • C.G. Gray, G. Karl, and V. A. Novikov, "Progress in Classical and Quantum Variational Principles". 11 Dec 2003. physics/0312071 Classical Physics.
  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. 
  • Stephen Wolfram, A New Kind of Science p. 1052
  • John Venables, "The Variational Principle and some applications". Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona (Graduate Course: Quantum Physics)
  • Andrew James Williamson, "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)
  • Kiyohisa Tokunaga, "Variational Principle for Electromagnetic Field". Total Integral for Electromagnetic Canonical Action, Part Two, Relativistic Canonical Theory of Electromagnetics, Chapter VI
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