Eigenfunction

In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has

${\displaystyle {\mathcal {A}}f=\lambda f}$

for some scalar, λ, the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends upon any boundary conditions required of ${\displaystyle f}$. In each case there are only certain eigenvalues ${\displaystyle \lambda =\lambda _{n}}$ (${\displaystyle n=1,2,3,...}$) that admit a corresponding solution for ${\displaystyle f=f_{n}}$ (with each ${\displaystyle f_{n}}$ belonging to the eigenvalue ${\displaystyle \lambda _{n}}$) when combined with the boundary conditions. The existence of eigenvectors is typically the most insightful way to analyze ${\displaystyle A}$.

For example, ${\displaystyle f_{k}(x)=e^{kx}}$ is an eigenfunction for the differential operator

${\displaystyle {\mathcal {A}}={\frac {d^{2}}{dx^{2}}}-{\frac {d}{dx}}}$

for any value of ${\displaystyle k}$, with a corresponding eigenvalue ${\displaystyle \lambda =k^{2}-k}$. If boundary conditions are applied to this system (e.g., ${\displaystyle f=0}$ at two physical locations in space), then only certain values of ${\displaystyle k=k_{n}}$ satisfy the boundary conditions, generating corresponding discrete eigenvalues ${\displaystyle \lambda _{n}=k_{n}^{2}-k_{n}}$.

Applications

Eigenfunctions play an important role in many branches of physics. An important example is quantum mechanics, where the Schrödinger equation

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\psi ={\mathcal {H}}\psi }$

has solutions of the form

${\displaystyle \psi (t)=\sum _{k}e^{-iE_{k}t/\hbar }\phi _{k},}$

where ${\displaystyle \phi _{k}}$ are eigenfunctions of the operator ${\displaystyle {\mathcal {H}}}$ with eigenvalues ${\displaystyle E_{k}}$. The fact that only certain eigenvalues ${\displaystyle E_{k}}$ with associated eigenfunctions ${\displaystyle \phi _{k}}$ satisfy Schrödinger's equation leads to a natural basis for quantum mechanics and the periodic table of the elements, with each ${\displaystyle E_{k}}$ an allowable energy state of the system. The success of this equation in explaining the spectral characteristics of hydrogen is considered one of the great triumphs of 20th century physics.

Due to the nature of the Hamiltonian operator ${\displaystyle {\mathcal {H}}}$, its eigenfunctions are orthogonal functions. This is not necessarily the case for eigenfunctions of other operators (such as the example ${\displaystyle A}$ mentioned above). Orthogonal functions ${\displaystyle f_{i}}$, ${\displaystyle i=1,2,\dots ,}$ have the property that

${\displaystyle 0=\int f_{i}^{*}f_{j}}$

where ${\displaystyle f_{i}^{*}}$ is the complex conjugate of ${\displaystyle f_{i}}$

whenever ${\displaystyle i\neq j}$, in which case the set ${\displaystyle \{f_{i}\,|\,i\in I\}}$ is said to be linearly independent.