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


{\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. cianfa72

    I Quantum particle's state in momentum eigenfunctions basis

    Hi, as discussed in this recent thread, for a particle without spin the quantum state of the particle is described by a "point" in the Hilbert space of the (equivalence classes) of ##L^2## square-integrable functions ##|{\psi} \rangle## defined on ##\mathbb R^3##. The square-integrable...
  2. D

    Determine unit vector n such that (L dot n)psi(r) = (m*hbar)psi(r)

    First I calculated ##(\vec{n} \cdot L) \psi(r) = -i\hbar(n_{x}(3y-z)+n_{y}(z-3x)+n_{z}(x-y))f(r)## and then tried to solve for ##n_{i}## such that I get (x+y+3z)f(r), and then divide ##n_{i}## by the magnitude of ##\vec{n}## to get the unit vector and m, but when I try doing this, I get the...
  3. D

    I Momentum eigenfunctions in an infinite well

    Hi For an infinite well , solving the Schrodinger equation gives wavefunctions of the form sin(nπx/L). These are not eigenfunctions of the momentum operator which means there are no eigenvalues of the momentum operator. Does this mean momentum cannot be measured ? Inside the infinite well the...
  4. H

    A What are the quantum numbers used to label helium atom eigenfunctions?

    Are there any results on the structure of the helium atom eigenfunctions? By this I'm referring to the non-perturbative structure of the eigenfunctions, AKA what are the quantum numbers that one would use to label the eigenfunctions?
  5. P

    I Why the linear combination of eigenfunctions is not a solution of the TISE

    The linear combination of the eigenfunctions gives solution to the Schrodinger equation. For a system with time independent Hamiltonian the Schrodinger Equation reduces to the Time independent Schrodinger equation(TISE), so this linear combination should be a solution of the TISE. It is not...
  6. S

    I Eigenfunctions and wave functions

    I saw this statement from the textbook "Quantum physics of atoms, molecules, solids, nuclei, and particles" second edition pg 166. According to the text, is the author saying the solution to the TISE is the eigenfunction and when you multiply the time dependent part, you get the wave function? I...
  7. Riccardo Marinelli

    A Boundary conditions of eigenfunctions with Yukawa potential

    Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some...
  8. JD_PM

    Finding the eigenfunctions and eigenvalues associated with an operator

    The eigenvalue equation is $$\frac{d^2}{d \phi^2} f(\phi) = q f(\phi)$$ This is a second order linear homogeneous differential equation. The second order polynomial associated to it is $$\lambda ^2 - q = 0 \rightarrow \lambda = \pm \sqrt{q}$$ As both roots are real and distinct, the...
  9. M

    A Eigenfunctions of an Airy Disk

    Does anyone know if the eigenfunctions of the Airy disk function (or Bessel function) \frac{J_1(x)}{x} has a closed form?
  10. medofx

    Help please (concerning eigenfunctions and the Schrödinger equation)

    i have an exam in 2 days and in this question i don't know how should i proceed after that i simplified the wave function but i don't know how to confirm that it's an eigenfunction
  11. Vajhe

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    Hi, I have been reading the Milonni and Eberly book "Laser": in one of the chapters they discuss the Oscillator Model. The treatment is quite straightforward, the Hamiltonian of the process is H=H0+HI where the first term is the "undisturbed" hamiltonian, and the second one is the interaction...
  12. M

    I Sturm-Liouville Eigenfunctions

    Hi PF! Given ##y''+\lambda^2y=0## and BCs ##y'(0)=y'(1) = 0## we know eigenfunctions are ##y=\cos (n\pi x)##, and for ##n=1## this implies there is one zero on the interval ##x\in(0,1)##. However, I read that for SL problems, the ##jth## eigenfunction has exactly ##j-1## zeros on ##x\in(0,1)##...
  13. John Greger

    I Obtain simultaneous eigenfunctions?

    Let's consider two observables, H (hamiltonian) and P (momentum). These operators are compatible since [H,P] = 0. Let's look at the easy to prove rule: 1: "If the observables F and G are compatible, that is, if there exists a simultaneous set of eigenfunctions of the operators F and G, then...
  14. WeiShan Ng

    I Momentum/Position space wave function

    These are from Griffith's: My lecture note says that I am having quite a confusion over here...Does the ##\Psi## in the expression ##\langle f_p|\Psi \rangle## equals to ##\Psi(x,t)##? I understand it as ##\Psi(x,t)## being the component of the position basis to form ##\Psi##, so...
  15. binbagsss

    Hecke Operators and Eigenfunctions, Fourier coefficients

    Homework Statement Consider the action of ##T_2## acting on ##M_k(\Gamma_{0}(N)) ##, and show that ##\theta^4(n)+16F ## and ##F(t)## are both eigenfunctions. Functions are given by: Homework Equations For the Hecke Operators ##T_p## acting on ##M_k(\Gamma_{0}(N)) ##, the Hecke conguence...
  16. binbagsss

    Linear Algebra: 2 eigenfunctions, one with eigenvalue zero

    Homework Statement If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction? I thought for sure yes, but the way I particular question I just...
  17. LarryS

    I QM Orthogonality: Separate & Independent Eigenvalues?

    In non-relativistic QM, given a Hilbert Space with a Hermitian operator A and a generic wave function Ψ. The operator A has an orthogonal eigenbasis, {ai}. I have often read that the orthogonality of such eigenfunctions is an indication of the separateness or distinctiveness of the associated...
  18. G

    I Do neutral atom collisions affect the continuous nature of black body radiation?

    I’ve read everything I can here and in the stack exchange on the topic of the continuous nature of black body radiation and it’s been really helpful,but I’m lead now to this question. Do neutral atom collisions shift the eigenfunctions,during the collisions? Do collisions create temporary...
  19. D

    Boundary conditions for eigenfunctions in a potential step

    1. Homework Statement A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##. How is the general form for the eigenfunctions? What is the boundary conditions for this eigenfunctions? Find the degeneracy level for the energy, when it is ##E<V_0## 2...
  20. S

    I Understanding QM Proof: Wavefunction in Orthonormal Eigenfunctions

    Hello! I have a proof in my QM book that: ##\left<r|e^{-iHt}|r'\right> = \sum_j e^{-iHt} u_j(r)u_j^*(r')##, where, for a wavefunction ##\psi(r,t)##, ##u_j## 's are the orthonormal eigenfunctions of the Hamiltonian and ##|r>## is the coordinate space representation of ##\psi##. I am not sure I...
  21. D

    I Do operators A and Hamiltonian share a set of eigenfunctions if they commute?

    If time evolution of a general ket is given by | Ψ > = e-iHt/ħ | Ψ (0) > where H is the Hamiltonian. If i have a eigenbasis consisting of 2 bases |a> and |b> of a general Hermitian operator A and i write e-iHt/ ħ |a> = e-iEat/ ħ |a> and e-iHt/ħ |b> = e-iEbt/ ħ |b> ; does this mean...
  22. stevendaryl

    I What is the Role of Eigenfunctions in Understanding Quantum Mechanics?

    <Moderator's note: this thread was split off from https://www.physicsforums.com/threads/eigenstates-eigenvalues.904774/> Well, the unfortunate thing, pedagogically, is that in teaching about eigenfunctions and eigenvalues, the most obvious operators to use for examples are the position...
  23. F

    I Particle in the box eigenfunctions

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  24. S

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    Homework Statement This is problem 17 from Chapter 3 of Quantum Physics by S. Gasiorowicz "Consider the eigenfunctions for a box with sides at x = +/- a. Without working out the integral, prove that the expectation value of the quantity x^2 p^3 + 3 x p^3 x + p^3 x^2 vanishes for all the...
  25. Kara386

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    Homework Statement I've been given the spherical harmonics ##Y_{l,m}## for the orbital quantum number ##l=1##. Then told to calcute the sum of their squares over all values of m and explain the significance of the result. Homework Equations ##Y_{1,1} =...
  26. H

    Eigenfunctions, eigenstates and eigenvalues

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  27. S

    A Imaginary time propagation to find eigenfunctions

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  28. I

    Eigenfunction of a spin-orbit coupling Hamiltonian

    Dear all, The Hamiltonian for a spin-orbit coupling is given by: \mathcal{H}_1 = -\frac{\hbar^2\nabla^2}{2m}+\frac{\alpha}{2i}(\boldsymbol \sigma \cdot \nabla + \nabla \cdot \boldsymbol \sigma) Where \boldsymbol \sigma = (\sigma_x, \sigma_y, \sigma_z) are the Pauli-matrices. I have to...
  29. A

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    Hi all, I asked for help with one part of this question here. But after thinking about another part of the question, I realized I didn't understand it as well as I'd thought. Homework Statement Ψ(x,0)=A(iexp(ikx)+2exp(−ikx)) is a wave function. A is a constant. Can Ψ be normalised? Homework...
  30. A

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    Hi all, This is from a past exam paper: At t=0 the state of a particle is described by the wavefunction $$ \Psi (x,0) =A(iexp(ikx)+2exp(-ikx)) $$ This is between positive and negative infinity - not in a potential well. What values of momentum are allowed, and with what probability in each...
  31. S

    Engineering Eigenfunctions of the vector Helmholtz equation

    Hi everyone, I'm looking for a reference book that treats the theory behind the eigenfunctions solution of the so called vector Helmholtz equation and its Neumann and Dirichlet problems. I've already found a theory inside the last chapter of Morse & Feshbach's Methods of theoretical physics...
  32. Amy B

    I Eigenfunctions and eigenvalues

    is exp (-kx) an eigenfunction?
  33. T

    I Non-commuting operators on the same eigenfunctions

    In Griffiths chapter 4 (pg. 179-180) there is an example (Ex. 4.3) that details the expectation value of ## S_x ##, ##S_y##, and ##S_z## of a spin 1/2 particle in a magnetic field. In this example, they find an eigenvector of ## H## (which commutes with ## S_z##) but then use this same...
  34. A

    I Are the derivatives of eigenfunctions orthogonal?

    We know that modes of vibration of an Euler-Bernoulli beam are given by eigenfunctions, with the natural frequency of each mode being given by its eigenvalue. Thus these modes are all mutually orthogonal.Can anything be said of the derivatives of these eigenfunctions? For example, I have the...
  35. thegirl

    I Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

    Hi, why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity. I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and...
  36. P

    3D Harmonic Oscillator - Eigenfunctions and Eigenvalues

    Homework Statement Due to the radial symmetry of the Hamiltonian, H=-(ħ2/2m)∇2+k(x^2+y^2+z^2)/2 it should be possible to express stationary solutions to schrodinger's wave equation as eigenfunctions of the angular momentum operators L2 and Lz, where...
  37. L

    Eigenstates of Orbital Angular Momentum

    Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators L^2 and L_z. We know that...
  38. Amcote

    Sturm-Liouville Orthogonality of Eigenfunctions

    Homework Statement Consider the following Sturm-Liouville Problem: \dfrac{d^2y(x)}{dx^2} + {\lambda}y(x)=0, \ (a{\geq}x{\leq}b) with boundary conditions a_1y(a)+a_2y{\prime}(a)=0, \ b_1y(b)+b_2y{\prime}(b)=0 and distinguish three cases: a_1=b_1, a_2{\neq}0, b_2{\neq}0a_2=b_2=0, a_1{\neq}0...
  39. S

    MHB Software for calculating eigenvalues and eigenfunctions of an integral operator

    Hi can someone direct me to a free software to calculate eigenvalues and normalized eigenfunctions of a linear integral operator. I am trying to solve a fredholm integral equation with degenerate kernel using it instead of linear equations thanks sarrah
  40. Ronf

    Trial function and Eigenfunction....

    Homework Statement Hello, I just started to study QM, I just have a general question, how to know if a trial function is not an eigenfunction of a hamiltonian (that has the lowest value in a graph) ? - Thanks and sorry for the stupid question. Homework EquationsThe Attempt at a Solution I have...
  41. I

    Self adjoint operators, eigenfunctions & eigenvalues

    Homework Statement Consider the space ##P_n = \text{Span}\{ e^{ik\theta};k=0,\pm 1, \dots , \pm n\}##, with the hermitian ##L^2##-inner product ##\langle f,g\rangle = \int_{-\pi}^\pi f(\theta) \overline{g(\theta)}d\theta##. Define operators ##A,B,C,D## as ##A = \frac{d}{d\theta}, \; \; B=...
  42. K

    Eigenfunctions of the angular momentum operator

    Hi everyone, I tried to find the Eigenstate of the angular momentum operator myself, more specifically I tried to find a Function Y_{lm}(\theta,\phi) with L_zY_{lm}=mħY_{lm} and L^2Y_{lm}=l(l+1)ħ^2Y_{lm} where L_z=-iħ\frac{\partial}{\partial \phi} and...
  43. I

    Expand function as series of eigenfunctions

    Homework Statement Determine all eigenvalues and eigenfunctions for the Sturm-Liouville problem \begin{cases} -e^{-4x}\frac{d}{dx} \left(e^{4x}\frac{d}{dx}\right) = \lambda u, \; \; 0 < x <1\\ u(0)=0, \; \; u'(1)=0 \end{cases} Expand the function ##e^{-2x}## as a series of...
  44. AwesomeTrains

    Eigenvalues of disturbed Hamiltonian

    Hello everyone! I'm trying to follow a solution to a problem from the book "Problems and Solutions on Quantum Mechanics", it's problem 1017. There's a step where they go on too fast, and I can't follow. I've posted the solution and where my problem is down below. Homework Statement The dynamics...
  45. D

    Understanding Eigenfunctions and Operators in Quantum Mechanics

    Hello, so I have a couple of related questions. 1) If you have a wavefuction Ψ, and act on it with some operator, does it have to give you the same wavefunction back (ie. does the wavefunction have to be an eigenfunction of the operator)? Could you have a wavefunction like e-iħtSin(x)? Since...
  46. D

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  47. A

    Proving Coherent States are Eigenfunctions of Annihilation Operators

    Look at the following attached picture, where they prove the coherent states are eigenfunctions of the annihiliation operators by simply proving aexp(φa†)l0> = φexp(φa†)l0>. I understand the proof but does that also prove that: aiexp(Σφiai†)l0> = φiexp(Σφiai†)l0> ? I can see that it would if you...
  48. A

    How do you plot eigenfunctions of perturbed HO?

    Homework Statement Find eigenvalues and eigenvectors of a perturbed harmonic oscillator (H=H0+lambda*q4 numerically using different numerical methods and plot perturbed eigenfunctions. I wrote a code in c++ which returns a row of eigenvalues of the perturbed matrix H and a matrix of...
  49. B

    Calculate the eigenfunctions for a spin half particle

    Homework Statement Spin can be represented by matrices. For example, a spin half particle can be described by the following Pauli spin matrices s_x = \frac{\hbar} {2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} , s_y = \frac{\hbar} {2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} , s_z =...