Eigenfunction
in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
[1][2][3] The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.
In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue.
In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions.
That is, a function f is an eigenfunction of D if it satisfies the equation where λ is a scalar.
The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.
[4][5] A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator
is the eigenfunction of the derivative operator, where f0 is a parameter that depends on the boundary conditions.
Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value.
where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.
As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.
where δij is the Kronecker delta and can be thought of as the elements of the identity matrix.
The coefficients bj can be stacked into an n by 1 column vector b = [b1 b2 … bn]T. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite dimension.
Additionally, define a matrix representation of the linear operator D with elements
Taking the inner product of each side of this equation with an arbitrary basis function ui(t),
This is the matrix multiplication Ab = c written in summation notation and is a matrix equivalent of the operator D acting upon the function f(t) expressed in the orthonormal basis.
Consider the Hermitian operator D with eigenvalues λ1, λ2, … and corresponding eigenfunctions f1(t), f2(t), ….
This Hermitian operator has the following properties: The second condition always holds for λi ≠ λj.
[5] Depending on whether the spectrum is discrete or continuous, the eigenfunctions can be normalized by setting the inner product of the eigenfunctions equal to either a Kronecker delta or a Dirac delta function, respectively.
In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
Here c is a constant speed that depends on the tension and mass of the string.
where the phase angles φ and ψ are arbitrary real constants.
If we impose boundary conditions, for example that the ends of the string are fixed at x = 0 and x = L, namely X(0) = X(L) = 0, and that T(0) = 0, we constrain the eigenvalues.
This last boundary condition constrains ω to take a value ωn = ncπ/L, where n is any integer.
Thus, the clamped string supports a family of standing waves of the form
can be solved by separation of variables if the Hamiltonian does not depend explicitly on time.
The eigenfunctions φk of the Hamiltonian operator are stationary states of the quantum mechanical system, each with a corresponding energy Ek.
They represent allowable energy states of the system and may be constrained by boundary conditions.
When the Hamiltonian does not depend explicitly on time, general solutions of the Schrödinger equation are linear combinations of the stationary states multiplied by the oscillatory T(t),[11]
The success of the Schrödinger equation in explaining the spectral characteristics of hydrogen is considered one of the greatest triumphs of 20th century physics.
