Eigenfunction

This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function λ. The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.

An eigenfunction is a type of eigenvector.

Df=λf, 1

where λ is a scalar. The solutions to Equation ddtf(t)=λf(t).

This differential equation can be solved by multiplying both sides by dfdt|t=0=2. We then find that

⟨ui,uj⟩=∫Ω ui∗(t)uj(t)dt=δij={1i=j0i≠j,

where δij is the Kronecker delta and can be thought of as the elements of the identity matrix.

Functions can be written as a linear combination of the basis functions,

Aij=⟨ui,Duj⟩=∫Ω ui∗(t)Duj(t)dt.

We can write the function Df(t) either as a linear combination of the basis functions or as D acting upon the expansion of f(t),

Aij=⟨ui,Duj⟩=∫Ωdt ui∗(t)Duj(t).

integrated over some range of interest for t denoted Ω.

By analogy with Hermitian matrices, D is a Hermitian operator if Aij = Aji*, or:

x along the string and of time ∂2h∂t2=c2∂2h∂x2,

which is called the (one-dimensional) wave equation. Here X(x)T(t), we can form a pair of ordinary differential equations:

−ω, respectively. For any values of X(x)=sin⁡(ωxc+φ),T(t)=sin⁡(ωt+ψ), where the phase angles x = 0 and T(0) = 0, we constrain the eigenvalues. For these boundary conditions, φ = ψ = 0, and

ωn = ⁠ncπ/L⁠, where ωn is the frequency of the iℏ∂∂tΨ(𝐫,t)=HΨ(𝐫,t)

with the Hamiltonian operator

E. As shown in an earlier example, the solution of Equation (2) is the time-independent Schrödinger equation. The eigenfunctions k</sub>"}},"i":0}}]}'/>φk of the Hamiltonian operator are stationary states of the quantum mechanical system, each with a corresponding energy k</sub>"}},"i":0}}]}'/>Ek. They represent allowable energy states of the system and may be constrained by boundary conditions.

The Hamiltonian operator Ψ(𝐫,t)=∑kckφk(𝐫)e−iEkt/ℏ or, for a system with a continuous spectrum,

y(t) = λf(t), where