Ad
related to: eigenvalues and eigenfunctions diff eq formulaeducator.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
As shown in an earlier example, the solution of Equation is the exponential = /. Equation is the time-independent Schrödinger equation. The eigenfunctions φ k of the Hamiltonian operator are stationary states of the quantum mechanical system, each with a corresponding energy E k. They represent allowable energy states of the system and may be ...
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.
The eigenvalue equation for D is the differential equation = () The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions .
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon ...
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation : ∇ 2 f = − k 2 f , {\displaystyle \nabla ^{2}f=-k^{2}f,} where ∇ 2 is the Laplace operator, k 2 is the eigenvalue, and f is the (eigen)function.
Weyl applied his theory to Carl Friedrich Gauss's hypergeometric differential equation, thus obtaining a far-reaching generalisation of the transform formula of Gustav Ferdinand Mehler (1881) for the Legendre differential equation, rediscovered by the Russian physicist Vladimir Fock in 1943, and usually called the Mehler–Fock transform.
This proof of the Hellmann–Feynman theorem requires that the wave function be an eigenfunction of the Hamiltonian under consideration; however, it is also possible to prove more generally that the theorem holds for non-eigenfunction wave functions which are stationary (partial derivative is zero) for all relevant variables (such as orbital rotations).
Ad
related to: eigenvalues and eigenfunctions diff eq formulaeducator.com has been visited by 10K+ users in the past month