What is Eigenfunctions: Definition and 181 Discussions

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. T

    Finding eigenfunctions of the linear momentum operator

    Homework Statement http://img508.imageshack.us/img508/7199/46168034nt3.jpg The Attempt at a Solution I'm just totally lost with this question. The theory just eludes me totally. Just how do you determine whether it is/isn't an eighenfunction of the linear momentum operator?
  2. C

    Physical meaning of the 2 eigenfunctions of Free Particle

    For Schrodinger's equation \frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi Solving to find that \psi = Aexp(ikx)+Bexp(-ikx) I am curious about the physical meanings of the two terms of the solutions. In solving a free particle encountering a potential barrier, In the...
  3. M

    How to prove the orthogonality of eigenfunctions in Sturm-Louisville problems?

    [SOLVED] orthogonal eigenfunctions From Sturm-Louisville eigenvalue theory we know that eigenfunctions corresponding to different eigenvalues are orthogonal. For example, \Phi_{xx} + \lambda \Phi = 0 would be of Sturm-Louisville form (note: \Phi_{xx} represents the second derivative of...
  4. L

    How to find eigenfunctions given the wave function?

    Hi all, How do we find the eigenfunctions if we are given the wavefunction? I have a wave function at time = 0 and it is of a *free* particle and I need to find the wave function at a later time t. I did : \Psi(x,t)=\Psi(x,0)*e^{-iHt/hbar} then \Psi(x,t)=\sum_{n}(<\phi_{n}|\Psi(x,0)>...
  5. R

    Linear Combinations of Eigenfunctions

    Homework Statement Suppose that f(x) and g(x) are two eigenfunctions of an operator Q^{\wedge}, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q^{\wedge}, with eigenvalue q. Homework Equations I know that Q^{\wedge}f(x) = qf(x) shows...
  6. T

    Exponentials as eigenfunctions in LTI Systems

    I hope I catched the correct forum. Under http://en.wikipedia.org/wiki/LTI_system_theory e^{s t} is an eigenvalue. I don't really understand that the following is an eigenvalue equation: \quad = e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau \quad = e^{s t}...
  7. S

    Eigenvalues & Eigenfunctions: Exploring Physical Significance

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  8. B

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

    How can i prove that u(x)=exp(-x^{2}/2) is the eigenfunction of \hat{A} = \frac{d^{2}}{dx^{2}}-x^2
  9. G

    Eigenfunctions of Rigid Rotator

    Homework Statement Two particles of mass m are attached to the ends of a massless rigid rod of length a. The system is free to rotate in three dimensions about the centre (but the centre point itself is fixed). Homework Equations (a) Show that the allowed energies of this rigid rotator...
  10. P

    Solving the Mystery of Energy Eigenfunctions

    It seems the Schrodinger equation is written so that psi is an energy eigenfunction. So all psi are energy eigenfunctions? But how can it turn into other eigenfunctions like momentum? Or is it already a momentum eigenfunction as welll as the energy eigenfunction and so also position and so on...
  11. U

    Energy and momentum eigenfunctions

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  12. quasar987

    Explicit formula the nth eigenfunctions of the quantum harmonic oscillator?

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  13. N

    Eigenfunctions in Hilbert Space, Infinite Square Wells and Uncertainty

    Hi I'm kinda stuck with a couple quantum HW questions and I was wondering if you guys could help. First, Is the ground state of the infinite square well an eigenfunction of momentum?? If so, why. If not, why not?? Second, Prove the uncertainty principle, relating the uncertainty in...
  14. E

    Eigenvalues and eigenfunctions of the lowering operator

    Homework Statement Consider lowering and rising operators that we encountered in the harmonic oscillator problem. 1. Find the eigenvalues and eigenfunctions of the lowering operator. 2. Does the rising operator have normalizable eigenfunctions?Homework Equations a-= 1/sqrt(2hmw) (ip + mwx) a+...
  15. N

    Eigenfunctions (orthogonality & expansion)

    1) If you have a particle in 1D bound within range "-a" and "a". You come up with one eigenfunction that is sinusoidal (since it satisfies the problem). Now, you get all the necessary constants through the usual way... I want to know whether more than one eigenfunction can be produced and...
  16. B

    Eigenfunctions problem - i have the answer, explanation required

    Homework Statement Now in a revision lecture given a few weeks ago, the lecturer gave this as the answer. The Attempt at a Solution No I think generally I'm fine with it (apart from it doesn't seem very obvious that this is what you should do with the maths!). BUT 1)...
  17. C

    Calculating <E> & <E^2> with Eigenfunctions of Parity Operator

    Q1 energy no. of times measured a1 n1 a2 n2 a3 n3 a4 n4 expectation value <E> = (a1n1+a2n2+a3n3+a4n4) / (n1+n2+n3+n4) is this correct? Also, how do you caluculate expectation...
  18. B

    Eigenstates and eigenfunctions and Schrodinger - HELP PLEASE

    A couple of things first - Whats an eigenfunction? Whats an eigenvalue? I've been doing a course on quantum mechanics for nearly 2 months and whilst these words have popped up in both notes and lectures no-one has actually bothered to explain what they are or what they mean! Next thing. Can...
  19. G

    Eigenfunctions and their Eigenvalues

    If I have two eigenfunctions of an operator with the same eigenvalue how do I construct linear combinations of my eigenfunctions so that they are orhtogonal? My eigenfunctions are: f=e^(x) and g=e^(-x) and the operator is (d)^2/(dx)^2
  20. A

    Spherical symmetri of eigenfunctions?

    I have been given these 4 eigenfunctions of the hydrogen atoms first 2 n-shells. \psi_{100}(r, \theta, \phi )=\frac{1}{\sqrt{\pi a^3_0}}e^{-r/a_0} \psi_{200}(r, \theta, \phi )=\frac{1}{\sqrt{8\pi a^3_0}}(1-\frac{r}{2a_0})e^{-r/2a_0} \psi_{210}(r, \theta, \phi )=\frac{1}{4\sqrt{2\pi...
  21. B

    Finding a General Solution for Eigenfunctions in Quantum Mechanics

    First it asks a few questions about what if it were a classical particle approaching the barrier. Much of this I understand and am OK with. Then we start treating the particle as a quantum thing so its governed by the TI Schrodinger EQ. So, what it wants me to do which I am a bit unsure about...
  22. A

    Angular Momentum, L_x eigenvalues and eigenfunctions

    This is a very simple question, but I can't seem to get it right, there's probably something silly that I'm missing here. Here's the question: I have A system in the l=1 state, and I have L_z|\ket{lm} = \hbar m\ket{lm}and L^2 \ket{lm} = \hbar^2 l(l+1)\ket{lm} I need to find the eigenvalues...
  23. E

    Eigenfunctions and eigenvalues of Fourier Transform?

    :rolleyes: :cool: I have a question..yesterday at Wikipedia i heard about the "Hermite Polynomials2 as Eigenfunctions of Fourier (complex?) transform with Eigenvalues i^{n} and i^{-n}...could someone explain what it refers with that?...when it says "Eigenfunctions-values" it refers to the...
  24. resurgance2001

    Eigenfunctions versus wavefunctions

    Hi - hope that someone can help me with this. I am new to quantum mechanics - trying to answer a question about eigenfunctions and don't have a decent textbook at the moment. Can someone tell me please, what is the difference between a wavefunction and an eigenfunction for a particle in an...
  25. S

    Serious conceptual problem with QM (eigenfunctions)

    The wavefunction psi is often separated into two parts, the time dependent part and the part which has only spatial dependence (phi), and this I think can only be done if we assume that the potential is not a function of time. I often see proofs where we have H acting on phi (not psi) and we get...
  26. S

    Is exp (- mod(x)/a) an eigenfunction of momentum?

    Is exp (- mod(x)/a) an eigenfunction of momentum. I know that this is not differentiable at x = 0, but does this completely disqualify it from being a momentum eigenfunction?
  27. C

    Eigenfunctions and eigenvalues

    This is probably a straight forward question, but can someone show me how to solve this problem: \frac {d^2} {d \phi^2} f(\phi) = q f(\phi) I need to solve for f, and the solution indicates the answer is: f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi} I know...
  28. C

    Help Needed With Sketching Eigenfunctions

    Hi there, I was hoping someone could maybe give me a hand with sketching eigenfunctions (although I reliase it can be difficult over the net!). I we have a particle at n=2 inside a finite potential well but we have a potential barrier in the middle of it, how do we draw this (the particle has...
  29. F

    Calculating Energy Eigenvalues & Eigenfunctions for a 2D Particle

    let's say.. there is a particle, with mass m, in a 2-dimensions x-y plane. in a region 0 < x < 3L ; 0 < y < 2L how to calculate the energy eigenvalues and eigenfunctions of the particle? thx :smile: and.. 2nd question.. there is a particle of kinetic energy E is incident from...
  30. homology

    Solving Kevin's QM Mystery: Sketchy Eigenfunctions

    So What's the dealy-do with the eigenfunctions of the position operator x and the momentum operator p? As a blossoming mathematician the thought of using functions that don't even reside in the space as a basis gives me the chills. Moreover, not only using functions that are not normalizable...
  31. C

    What is the physical interpretation of orthogonal eigenfunctions?

    Can anyone give me a physical interpretation of what orthogonal eigenfunctions are please? I understand the mathematical idea, the overlap integral, but I'm not clear about what it implies for the different states. At the moment the way I'm thinking of it is that the energy eigenfunctions of an...
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