eigenfunktion

An eigenfunction is a specific type of function that, when acted upon by a linear operator, returns a scaled version of itself. This scaling factor is known as an eigenvalue. Mathematically, if $L$ is a linear operator and $f$ is an eigenfunction, then the relationship is expressed as $Lf = \lambda f$, where $\lambda$ is the eigenvalue. This means that the operator $L$ does not change the fundamental form of the eigenfunction $f$, but merely stretches or shrinks it by the factor $\lambda$.

Eigenfunctions and eigenvalues are fundamental concepts in many areas of mathematics, physics, and engineering. In quantum mechanics, for instance, the eigenfunctions of an observable operator represent the possible states of a system, and the corresponding eigenvalues represent the possible values that can be measured for that observable. For example, the eigenfunctions of the momentum operator are plane waves, and their eigenvalues are the possible momenta.

The set of all eigenfunctions of a given operator often forms a basis for the space of functions it operates on. This means that any arbitrary function can be expressed as a linear combination of these eigenfunctions, making them incredibly useful for solving problems. Finding eigenfunctions and eigenvalues is a core task in solving differential equations, analyzing vibrations, and understanding wave phenomena. The properties of these functions and their associated values provide deep insights into the underlying structure of the systems they describe.