Search results
Results from the WOW.Com Content Network
Its eigenfunctions form a basis of the function space on which the operator is defined [5] As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy. However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for ψ E {\displaystyle \psi _{E}} within the ...
These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
The eigenfunctions of the position operator (on the space of tempered distributions), represented in position space, are Dirac delta functions. Informal proof. To show that possible eigenvectors of the position operator should necessarily be Dirac delta distributions, suppose that ψ {\displaystyle \psi } is an eigenstate of the position ...
For a free particle, the Hamiltonian = is invariant under translations. Translation commutes with the Hamiltonian: [ H , T ^ ] = 0 {\displaystyle [H,\mathbf {\hat {T}} ]=0} . However, if we express the Hamiltonian in the basis of the translation operator, we will find that H {\displaystyle H} has doubly degenerate eigenvalues.
The covariance function K X satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λ k, e k (t) of eigenvalues and eigenfunctions of T K X forming an orthonormal basis of L 2 ([a,b]), and K X can be expressed as (,) = = ()
Definition: An eigenvalue is said to have algebraic multiplicity if is the smallest integer such that the th derivative of (()) with respect to , in is nonzero.