Rayleigh-Ritz Principle

The Rayleigh-Ritz principle (also Ritz method or Rayleigh- Ritz 's variation method ) is a variation principle for determining the smallest eigenvalue of an eigenvalue problem . It goes back to The Theory of Sound by John William Strutt, 3rd Baron Rayleigh (1877) and was published in 1908 by the mathematician Walter Ritz as a mathematical method.

Let it be a self-adjoint operator with domain in a Hilbert space . Then the infimum of the spectrum is given by ${\displaystyle H}$ ${\displaystyle D(H)}$ ${\displaystyle \sigma (H)}$

${\displaystyle E_{0}=\inf \sigma (H)=\inf _{\psi \in D(H)\setminus \{0\}}{\frac {\langle \psi ,H\psi \rangle }{\langle \psi ,\psi \rangle }}}$.

If the infimum is an eigenvalue, the inequality is obtained ${\displaystyle E_{0}}$

${\displaystyle E_{0}\leq {\frac {\langle \psi ,H\psi \rangle }{\langle \psi ,\psi \rangle }}}$

with equality if and only if one eigenvector to be. The quotient on the right is known as the Rayleigh quotient . ${\displaystyle \psi }$ ${\displaystyle E_{0}}$

In practice, it is also suitable as an approximation method by making an approach for with indefinite parameters and optimizing the parameters in such a way that the Rayleigh quotient is minimal. Instead of using vectors in the domain of definition , one can also optimize using vectors in the square shape domain , which then corresponds to a weak formulation of the eigenvalue problem. ${\displaystyle \psi }$ ${\displaystyle D(H)}$ ${\displaystyle Q(H)}$

Applications

The principle is used, for example, when calculating parameters of the vibration behavior of elastic plates , but also other elastic bodies (such as beams ), when exact solutions can no longer be achieved with elementary calculation methods.

The basic idea is the balance of the potential forces of external, imprinted and internal forces. These potentials are expressed in terms of deformation values ​​(e.g. deflection). The stresses are expressed by stretching or shearing according to Hooke's law .

In quantum mechanics , the principle says that for the total energy of the system in the ground state (i.e. for the associated expectation value of the Hamilton operator ) and for any wave functions or states, the expectation value is greater than or equal to (the same in the case of the exact ground state wave function ) the ground state energy of the system is: ${\displaystyle {\mathcal {}}E_{0}}$ ${\displaystyle |\psi _{0}\rangle }$ ${\displaystyle {\hat {H}}}$ ${\displaystyle |\psi \rangle }$ ${\displaystyle \langle \psi |{\hat {H}}|\psi \rangle }$

${\displaystyle E_{0}\leq \langle {\hat {H}}\rangle [\psi ]\,:=\,\langle \psi |{\hat {H}}|\psi \rangle ,\qquad \|\psi \|=1.}$

As a rule, the Hamilton operator is restricted downwards and has a (non-degenerate) eigenvalue (“ground state”) at the lower limit of the spectrum. The sample wave function can deviate considerably from the exact ground state function, but it becomes more similar the closer the calculated total energy is to the ground state energy.

Ritz process

The Ritz variation method applies the Rayleigh-Ritz principle directly. For this purpose, a family of test vectors, which are varied over a set of parameters β , is used. A (not necessarily finite) set of vectors can be selected and the test vector can be represented as a linear combination: ${\displaystyle \psi _{n}}$

${\displaystyle \psi _{\vec {\beta }}=\sum _{n}\beta _{n}\psi _{n}}$

Or you can choose a family of functions that can be varied via a parameter, such as Gaussian curves with different widths : ${\displaystyle \beta }$

${\displaystyle \psi _{\beta }(x)={\frac {1}{\beta {\sqrt {2\pi }}}}\cdot \exp \left[-{\frac {x^{2}}{2\beta ^{2}}}\right]}$

Now insert these functions in the above expression and look for the minimum value of . In the simplest case this can be done by differentiation according to the parameter : ${\displaystyle \langle {\hat {H}}\rangle [\psi _{\beta }]}$ ${\displaystyle \beta }$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \beta }}\langle {\hat {H}}\rangle [\psi _{\beta }]=0}$

If you solve this equation, you get a value for which the ground state energy is minimized. With this value one has an approximate solution, but does not know how good the approach really is, which is why one speaks of "uncontrolled procedures". After all, the minimum value can be used as the “best approximation” of the actual ground state energy. ${\displaystyle \beta }$

For proof

The principle is immediately understandable if one assumes that there is an orthonormal basis consisting of eigenvectors of with associated eigenvalues . Let these eigenvalues be ordered, then one gets through development ${\displaystyle \psi _{n}}$ ${\displaystyle H}$ ${\displaystyle E_{n}}$ ${\displaystyle E_{0}\leq E_{1}\leq \cdots }$

${\displaystyle \psi =\sum _{n}\langle \psi _{n},\psi \rangle \psi _{n}}$

of any vector according to this orthonormal basis ${\displaystyle \psi }$

${\displaystyle {\begin{aligned}\langle \psi ,H\psi \rangle &=\sum _{n}\langle \psi ,H\psi _{n}\rangle \langle \psi _{n},\psi \rangle \\&=\sum _{n}E_{n}\langle \psi ,\psi _{n}\rangle \langle \psi _{n},\psi \rangle \\&\geq E_{0}\sum _{n}\langle \psi ,\psi _{n}\rangle \langle \psi _{n},\psi \rangle \\&=E_{0}\langle \psi ,\psi \rangle \,.\end{aligned}}}$

In the general case of an arbitrary spectrum, an analogous argument can be made as proof by replacing the sum with an integral over the spectral family according to the spectral theorem.

Extensions

An extension is the Min-Max Theorem , which represents a variation principle for all eigenvalues ​​below the essential spectrum . The Temple inequality provides an exact estimate of an eigenvalue upwards and downwards .

literature

• Hans Cycon, Richard G. Froese, Werner Kirsch , Barry Simon: Schrödinger Operators, Springer 1987
• Michael Reed, Barry Simon : Methods of Modern Mathematical Physics, 4 volumes, Academic Press 1978, 1980
• John William Strutt, 3rd Baron Rayleigh, The Theory of Sound, 1877
• W. Ritz: About a new method for solving certain variation problems in mathematical physics. In: Journal for pure and applied mathematics ISSN  0075-4102 , Bd. 135, 1908, pp. 1-61
• W. Ritz: Theory of the transverse oscillations of a square plate with free edges. In: Annalen der Physik ISSN  0003-3804 , (4th episode) Vol. 28, 1909, pp. 737-786
• GM Vainikko: Ritz method . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Gerald Teschl : Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, American Mathematical Society, 2009 ( Free online version )
• Karl-Eugen Kurrer : History of Structural Analysis. In search of balance , Ernst and Son, Berlin 2016, p. 519ff, ISBN 978-3-433-03134-6 .

Individual evidence

1. ↑ WB Krätzig et al .: Structures 2 . Theory and calculation methods of statically indeterminate rod structures. Gabler Wissenschaftsverlage, 2004, ISBN 978-3-540-67636-2 , p. 232 ( online [accessed April 7, 2012]).
2. ↑ JK MacDonald, Successive Approximations by the Rayleigh-Ritz Variation Method , Physical Review ISSN  0031-899X , Vol. 43, (1933), pp. 830-833.
3. ^ Gerald Teschl: Mathematical Methods in Quantum Mechanics . With Applications to Schrödinger Operators. American Mathematical Society, 2009, ISBN 978-0-8218-4660-5 , pp. 119 ( online [accessed April 7, 2012] Theorem 4.10).
4. ↑ George Temple: The theory of Rayleigh's principle as applied to continuous systems . In: Proc. Roy. Soc. London . Ser. A 119, 1928, p. 276-293 .
5. ^ Gerald Teschl: Mathematical Methods in Quantum Mechanics . With Applications to Schrödinger Operators. American Mathematical Society, 2009, ISBN 978-0-8218-4660-5 , pp. 120 ( online [accessed April 7, 2012] Theorem 4.13).