Eigenfunction Definition and 121 Threads

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as




D
f
=
λ
f


{\displaystyle Df=\lambda f}
for some scalar eigenvalue 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.

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  1. P

    Eigenfunction for Adjoint Operstor

    Hi, I am having a question again (I should not work on so many things at the same time). I came across an operator A acting on C^2 - functions f:R^2->R: (Af)(x,y)=(\Delta f)(x,y)-f(x,y)+f(0,y) My goal is to find the adjoint operator (acting on measures) with respect to the inner...
  2. B

    Eigenfunction, Eigenvalue, Wave Function and collapse

    Reading Sam Treiman's http://books.google.de/books?id=e7fmufgvE-kC" he nicely explains the dependencies between the Schrödinger wave equation, eigenvalues and eigenfunctions (page 86 onwards). In his notation, eigenfunctions are u:R^3\to R and the wavefunction is \Psi:R^4\to R, i.e. in contrast...
  3. L

    Linear Combos of Y; eigenfunction of L_x

    Homework Statement I have to find the linear combinations of Y_10, Y_11, and Y_1-1 that are eigenfunctions of L_y. There are three such combinations... Homework Equations The Attempt at a Solution Starting from (L_y)(psi_y)=(alpha)(psi_y), Using the relationshiP: psi_y= aY_11 +...
  4. S

    Functions, operator => eigenfunction, eigenvalue

    [SOLVED] Functions, operator => eigenfunction, eigenvalue Homework Statement Show, that functions f1 = A*sin(\theta)exp[i\phi] and f2 = B(3cos^{2}(\theta) - 1) A,B - constants are eigenfunctions of an operator http://img358.imageshack.us/img358/3406/98211270ob1.jpg and find...
  5. B

    Is u(x) = exp(-x²/2) an Eigenfunction of \hat{A} = d²/dx² - x²?

    How can i prove that u(x)=exp(-x^{2}/2) is the eigenfunction of \hat{A} = \frac{d^{2}}{dx^{2}}-x^2 .(if i don't know the eigenfunction how can i find it from expression of A operator)
  6. D

    QM - Eigenfunction / Eigenvalue Problem

    Homework Statement Find the eigenfunctions and eigenvalues for the operator: a = x + \frac{d}{dx} 2. The attempt at a solution a = x + \frac{d}{dx} a\Psi = \lambda\Psi x\Psi + \frac{d\Psi}{dx} = \lambda\Psi x + \frac{1}{\Psi} \frac{d\Psi}{dx} = \lambda x + \frac{d}{dx}...
  7. P

    Eigenfunction is made entirely of sin functions

    Homework Statement The text claims that any function can be constructed from eigenfunctions. BUt if the eigenfunction is made entirely of sin functions than it cannot construct even functions? So it cannot construct any function? That is why the Fourier series has both sin and cos functions.
  8. P

    Can Zero Be Considered an Eigenfunction?

    Homework Statement If y(x)=0 satisfies the ode and all the boundary conditions than does it count as the first eigenfunction? The Attempt at a Solution It wouldn't satisfy the orthogonality relation though? In that the integral of 0 and 0 is 0 even though the integral is over two...
  9. I

    Eigenfunction Equation with Boundary Conditions

    I've been going round in circles with this problem for days: Find the eigenvalues and associated normalised eigenfunctions of the operator L: L_y = x^2 y'' + 2 xy' + \frac{y}{4} Boundary conditions y(1)=y(e)=0 So what I've done: substitute x = \exp(t) Then L_y = \frac{d^2y}{dt^2} +...
  10. S

    Proving the Potential of Ground State Eigenfunction

    A particle in a potnetial V(x) has a definite energy E = - \frac{\hbar^2 \alpha^2}{2m} and its eigenfunction is given by \psi(x) = Nx \exp(-alpha x) if 0 <= x < infinity zero elsewhere Prove that V(x) = -alpha \hbar^2/mx 0 <= x < infinity infinity elsewhere Given that the...
  11. J

    Schrodinger equation Eigenfunction problem

    So, here's the question: \psi(x) = A*(\frac{x}{x_{0}})^n*e^(\frac{-x}{x_{0}}) Where A, n, and X0 are constants. Using Schrodinger's equation, find the potential U(x) and energy E such that the given wave function is an eigenfunction (we can assume that at x = infinity there is 0...
  12. P

    Eigenfunction expansion method in PDE solutions

    How does this method work? What are the mathematical ideas behind this method? Unlike separation of variables techniques, where things can be worked out from first principles, this method of solving ODE seems to find the right formulas and apply which I feel uncomfortable about.
  13. U

    Sin(kx) is an eigenfunction of the KE operator?

    The function sin(kx) is an eigenfunction of the KE operator? my work: KE=\frac{- \hbar^2}{2m}\frac{\partial^2}{\partial x^2} sin(kx) \frac{\hbar^2}{2m}\frac{sin(kx)}{k^2} i'm not sure how to show that a function is an eigenfuntion. what other work do I need to do?
  14. U

    Eigenfunction of the momentum "operator"

    which of the following functions is an eigenfunction of the momentum "operator" -i \hbar \frac{\partial}{\partial x}: f_1 =cos(kx- \omega t) f_2 =e^{a^2x} f_3 =e^{-(\omega t+kx)} for this question, I'm not sure what they are looking for... for f1 i \hbar k sin(k x -\omega t)...
  15. Reshma

    Verify whether eigenfunction or not?

    The wavefunction of a particle moving inside a one dimensional box of length L is non-zero only for 0<x<L. The normalised wavefunction is given by: \psi (x) = \sqrt{\frac{2}{L}}\sin \frac{n\pi x}{L} Is this wavefunction an eigenfunction of the x-component of the momentum operator \vec p =...
  16. J

    Mathematica Fourier transform with Mathematica (Dirac mean position eigenfunction)

    Mathematica can't calculate Fourier transform (Dirac mean position eigenfunction) Hi, I'm attempting to use Mathematica to calculate a mean-position eigenfunction of the Dirac equation. To do so I need to evaluate Fourier transforms (from p-space to r-space) of wavefunctions dependent on...
  17. K

    Green's function expansion in a set of eigenfunction

    Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|} and I need to express it into series of...
  18. homology

    And while I'm at it: eigenfunction of position

    I've yet to see a decent argument as to why the eigenfunctions of the position operator are delta functions. (Griffith's argues this, but oh so weakly). Could someone provide one or a couple dozen Kevin
  19. P

    Hey,guys is ther a thing name after the eigenfunction of the time operator

    we know that there's position eigenket,P eigenket and energy eigenket but is there something called the time operator,and the time eigenket or this is only a terminology that doesn't even exist,it's just my illusion! thanks gratefully
  20. S

    Is the Momentum of a Particle in a 1D Box Truly Known?

    According to my book, uncertainty Q = 0 (where Q is an observable) is true when the state function is an eigenfunction. The energy eigenfunction for a particle in a 1-D box with infinitely high walls is sin(n*pi*x/a). This implies that the linear momentum, p, is known with zero uncertainty...
  21. A

    Help with Eigenfunction Questions

    I am a little stuck understanding and answering the following questions. Can anyone help me with them? "A system has four eigenstates of an observable, with corresponding eigenvalues 3/2, 1/2, -1/2 and -3/2, and normalized eigenfunctions Psi_{3/2}, Psi_{1/2}, Psi_{-1/2} and Psi_{-3/2}...
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