Question

What is the purpose of the Eigen Function ?

Answer 1

Eigenvalues and eigenvectors allow one to “reduce” to different, simpler, problems with a linear operation. For eg, the deformation may be dissected into “plastic” if a stress is applied to a “principal directions strong”, certain directions in which the deformation is greater. For a particular physical structure, the wave function provides observable knowledge about the system.

Complete step by step answer:
In mathematics, an Eigen function of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an Eigenvalue. As an equation, this condition can be written as $Df = \lambda f$ for some scalar Eigen value $\lambda$ . The solutions to this equation may also be subject to boundary conditions that limit the allowable Eigen values and Eigen functions. An Eigenfunction is a type of Eigenvector. Because of the boundary conditions, the possible values of $\lambda$ are generally limited. The set of all possible Eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both. Each value of $\lambda$ corresponds to one or more Eigen functions. If multiple linearly independent Eigen functions have the same Eigenvalue, the Eigenvalue is said to be degenerate and the maximum number of linearly independent Eigen functions associated with the same Eigenvalue is the Eigenvalues degree of degeneracy or geometric multiplicity.
For a particular physical structure, the wave function provides observable knowledge about the system. * “Eigen value” is derived from “Eigen wert” in German, meaning the proper or characteristic value. The presence of a set of orthogonals is fundamental to the technique of Eigenfunction expansion. Eigen functions that can be used for solution construction.
Applications of Eigen Function
Vibrating strings:
Let $h(x,t)$ denote the transverse displacement of a stressed elastic cord, such as the vibrating strings of a string instrument, as a function of the position $x$ along the string and of time $t$. Applying the laws of mechanics to infinitesimal portions of the string, the function h satisfies the partial differential equation
$\dfrac{{{\partial ^2}h}}{{\partial {t^2}}} = {c^2}\dfrac{{{\partial ^2}h}}{{\partial {x^2}}}$
which is called the (one-dimensional) wave equation. Here c is a constant speed that depends on the tension and mass of the string.

Schrodinger equation:
In quantum mechanics , Schrodinger equation
$i\hbar \dfrac{\partial }{{\partial t}}\psi (r,t) = H\psi (r,t)$
With the Hamiltonian operator
$H = - \dfrac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + V(r,t)$
The success of the Schrodinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.

Signals and systems
In the study of signals and systems, an Eigenfunction of a system is a signal $f(t)$ that, when input into the system, produces a response $y(t) = \lambda f(t)$ where $\lambda$ is a complex scalar Eigen value .

Note: Eigenvalues and eigenvectors allow one to “reduce” to different, simpler, problems with a linear operation. For a particular physical structure, the wave function provides observable knowledge about the system. . An Eigenfunction is a type of Eigenvector .