Ritz method

The Rayleigh - Ritz principle ( also method of Ritz or Rayleigh Ritz cal variation procedure) is a variational principle for the determination of the smallest eigenvalue of an eigenvalue problem. It goes on The Theory of Sounds by John William Strutt, 3rd Baron Rayleigh (1877 ) back in 1908 and published by the mathematician Walter Ritz as a mathematical method.

It is a self-adjoint operator with domain in a Hilbert space. Then the infimum of the spectrum is given by

If the infimum is an eigenvalue, we obtain the inequality

With equality if and only if an associated eigenvector. The ratio on the right side known as the Rayleigh quotient.

In practice, it is also suitable as an approximation method by making an approach with undetermined parameters and the parameters optimized so that the Rayleigh quotient is minimal. Instead of vectors in the domain can be optimized, which then corresponds to a weak formulation of the eigenvalue problem via vectors in quadratic form field.

Applications

The principle is used for example in the calculation of parameters of the vibration characteristics of elastic plates, but also other elastic body ( such as bars), used when exact solutions are not accessible by elementary calculation methods.

Basic idea is to balance the potential of external forces, embossed and internal forces. These potentials are expressed by deformation parameters (eg deflection ). The voltages are thereby expressed by stretching or shearing according to Hooke's law.

In quantum mechanics states the principle that the total energy of the system in the ground state (ie, for the relevant expectation value of the Hamiltonian ) and for arbitrary wave functions or states the expectation value greater than or equal ( equal in the case of the exact ground state wave function) of the ground state energy of the system is:

In general, the Hamiltonian is bounded from below and has at the lower limit of the spectrum a ( non-degenerate ) eigenvalue ( " ground state "). Although the trial wave function may significantly differ from the exact ground state function, but it is more similar, the closer is the calculated total energy of the ground state energy.

Ritz method

The Ritz variational method uses the Rayleigh - Ritz principle directly. This is a family of test vectors, which are varied over a set of parameters β, is used. Thus, a (not necessarily finite) set of vectors are selected and the test vector can be represented as a linear combination:

Or choosing a family of functions that can be varied via a parameter, such as Gaussian curves with different widths:

Now one uses these functions in the above expression, and seeks the minimum value of. In the simplest case, this can be accomplished by differentiation with respect to the parameters:

Solving this equation, we obtain a value for which the ground state energy is minimized. With this value, then an approximate solution, but do not know how well the approach really is (eg Gaussian functions are for the hydrogen atom very bad ), so that one speaks of " uncontrolled process ". At least you can use the minimum value as the " best approximation " to the actual ground state energy.

For the proof

The principle is immediately apparent if we assume that an orthonormal basis of eigenvectors has with associated eigenvalues. These eigenvalues ​​are sorted, then is obtained by developing

An arbitrary vector by this orthonormal basis

In the general case of an arbitrary range can be made to prove an analogous argument by replacing according to the spectral theorem, the sum of an integral of the spectral function.

Extensions

An extension is the min-max theorem which is a variation principle for all eigenvalues ​​below the essential spectrum. An accurate estimate of intrinsic value above and below provides the Temple 's inequality.