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.

View More On Wikipedia.org
Recent contents Top users
  1. gfd43tg

    Eigenfunctions orthogonal in Hilbert space

    Hello, I am having a question regarding how eigenfunctions are orthogonal in Hilbert space, or what does that even mean (other than the inner product is zero). I mean, I know in ##\mathbb {R^{3}}##, vectors are orthogonal when they are right angles to each other. However, how can functions be...
  2. S

    Finding eigenfunctions for given Hamiltonian

    Homework Statement I am having too many troubles finding the eigenfunctions of a given Hamiltonian. I just never seem to know what exactly to do. My idea here is not for you to help me solve each problem below, but I would like to just set the equations. I know you guys don't like it when...
  3. Robsta

    Showing functions are eigenfunctions of angular momentum.

    Homework Statement Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz. Homework Equations I know that Lz = -iћ(∂/∂φ) I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged...
  4. P

    Question about the Hamiltonian Operator

    In my physical chemistry course, we are learning about the Schrödinger Equation and were introduced to the Hamiltonian Operator recently. We started out with the simple scenario of a particle in 1D space. Our professor's slide showed the following "derivation" to arrive at the expression for the...
  5. tasos

    Given HamiltonianFind eigenvalues and eigenfunctions

    Homework Statement We have the hamiltonian H = al^2 +b(l_x +l_y +l_z) where a,b are constants. and we must find the allowed energies and eigenfunctions of the system. Homework EquationsThe Attempt at a Solution [/B] I tried to complete the square on the given hamiltonian and the result is: H =...
  6. V

    Expecting the possible event of zero probability

    Consider a potential well in 1 dimension defined by $$ V(x)= \begin{cases} +\infty &\text{if}& x<0 \text{ and } x>L\\ 0 &\text{if} &0\leq x\leq L \end{cases} $$ The probability to find the particle at any particular point x is zero. $$P(\{x\}) = \int_S \rho(x)\mathrm{d}x=0 ;\forall\; x \in...
  7. kmm

    Commuting operators require simultaneous eigenfunctions?

    Here is what I understand. The generalized uncertainty principle is: \sigma^{2}_{A} \sigma^{2}_{B} \geq ( \frac{1}{2i} \langle [ \hat{A}, \hat{B} ] \rangle )^2 So if \hat{A} and \hat{B} commute, then the commutator [ \hat{A}, \hat{B} ] = 0 and the operators are compatible. What I don't...
  8. tasos

    Finding Energy Eigenvalues and Eigenfunctions for a Particle Well

    Homework Statement (a) Find the energy eigenvalues and eigenfunctions for this well. (b) If the particle at time t = 0 is in state Ψ = constant (0 <x <L)). Normalize this state. Find the state that will be after time t>0 (c) For the previous particle, if we measure the energy at time t = 0...
  9. S

    1d potential V (-x)=-V (x) eigenfunctions.

    Homework Statement Show that for a 1d potential V (-x)=-V (x), the eigen functions of the Schrödinger equation are either symmetric/ anti-symmetric functions of x.Homework EquationsThe Attempt at a Solution I really don't know how to do it for odd potential. Let me show you how I am doing it...
  10. deedsy

    Finding the energy eigenfunctions of infinite well with potential step

    Homework Statement The potential for a particle mass m moving in one dimension is: V(x) = infinity for x < 0 = 0 for 0< x <L = V for L< x <2L = infinity for x > 2L Assume the energy of the particle is in the range 0 < E < V Find the energy eigenfunctions and the equation...
  11. L

    Info on Bessel functions & their use as basis functions.

    Hello all,As an exercise my research mentor assigned me to solve the following set of equations for the constants a, b, and c at the bottom. The function f(r) should be a basis function for a cylindrical geometry with boundary conditions such that the value of J is 0 at the ends of the cylinder...
  12. Whovian

    Eigenfunctions of Laplace Transform

    Firstly, if this is an inappropriate forum for this thread, feel free to move it. This is a calculus-y equation related to differential equations, but I don't believe it's strictly a differential equation. The question I'm asking is which functions...
  13. H

    Eigenfunctions corresponding to a particular energy value

    Suppose we want to get eigenfunctions of a One-Particle Hamiltonian corresponding to one of its eigenvalues, say E, in bases of free particle eigenfunctions (plane waves). Can we use the plane waves corresponding to energies near E to get a reasonable solution?
  14. kq6up

    All Eigenfunctions can be Represented by a Linear Superposition

    Homework Statement The time-independent wave function ##\psi (x)## can always be taken to be real (unlike ##\Psi (x,t)##, which is necessarily complex). This doesn't mean that every solution to the time-independent Schödinger equation is real; what it says is that if you've got one that is...
  15. M

    MHB The eigenvalues are real and that the eigenfunctions are orthogonal

    Hey! :o We have the Sturm-Liouville problem $\displaystyle{Lu=\lambda u}$. I am looking at the following proof that the eigenvalues are real and that the eigenfunctions are orthogonal and I have some questions... $\displaystyle{Lu_i=\lambda_iu_i}$ $\displaystyle{Lu_j=\lambda_ju_j...
  16. N

    Complete set of eigenfunctions

    Hi! So let's say we measured the angular momentum squared of a particle, and got the result ##2 \hbar^2##, so ##l=1##. Now we have the choice of obtaining a sharp value of either ##L_z, L_y## or ##L_x##. Okay, fair enough. But I have two questions: 1) The degeneration degree is ##3## because...
  17. U

    Angular Momentum and Eigenfunctions

    Homework Statement Part (a): What is momentum operator classically and in quantum? Part (b): Show the particle has 0 angular momentum. Part (c): Determine whether angular momentum is present along: (i)z-axis, (ii) x-axis and find expectation values <Lz> and <Lx>. Part (d): Find the result...
  18. M

    Angular momentum operators and eigenfunctions

    Homework Statement Homework Equations The Attempt at a Solution I have tried inserting the first wavefunction into Lz which gets me 0 for the eigenvalue for the first wavefunction. Is this correct? For the second wavefunction, I inserted it into Lz and this gets me -i*hbar*xAe^-r/a which...
  19. M

    Linear combinations of non-eigenfunctions to create eigenfunctions

    Homework Statement Consider the Parity Operator, P', of a single variable function, defined as P'ψ(x)=P'(-x). Let ψ1=(1+x)/(1+x^2) and ψ2=(1+x)/(1+x^2). I have already shown that these are not eigenfunctions of P'. The question asks me to find what linear combinations, Θ=aψ1+bψ2 are...
  20. Runei

    Completeness of the eigenfunctions (Which vectorspace?)

    Completeness of the eigenfunctions (Which vectorspace??) Once again in need of brain power from the interwebz :) So I get that the eigenfunctions to the hamiltonoperator forms a complete set, but I'm unsure now as to which vectorspace it is? And we're talking the one-dimensional case...
  21. U

    Possible measurement, eigenvalues of eigenfunctions and probabilities

    Homework Statement Suppose the angular wavefunction is ##\propto (\sqrt{2} cos(\theta) + sin (\theta) e^{-i\phi} - sin (\theta) e^{i\phi})##, find possible results of measurement of: (a) ##\hat {L^2}## (b)##\hat {L_z}## and their respective probabilities. Homework Equations...
  22. W

    Momentum eigenvalues and eigenfunctions

    Homework Statement For the following wave functions: ψ_{x}=xf(r) ψ_{y}=yf(f) ψ_{z}=zf(f) show, by explicit calculation, that they are eigenfunctions of Lx,Ly,Lz respectively, as well as of L^2, and find their corresponding eigenvalues. Homework Equations I used...
  23. F

    Find eigenfunctions and eigenvalues of an operator

    Homework Statement \hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} Homework Equations Find eigenfunctions and eigenvalues of this operatorThe Attempt at a Solution It leads to the differential eqn - \frac{{{\hbar...
  24. P

    What are the eigenfunctions for the ODE y′′−2xy′+2αy=0?

    Hi, Homework Statement I have the following ODE: y′′−2xy′+2αy=0 I'd like to determine the first three eigenfunctions. Homework Equations The Attempt at a Solution The solution y(x) may be recursively represented as: an+2=an(2n−2α)/[(n+2)(n+1)] I have found the eigenvalues to be...
  25. P

    Eigenfunctions and dirac notation for a quantum mechanical system.

    QUESTION A quantum mechanical system has a complete orthonormal set of energy eigenfunctions, |n> with associate eigenvalues, En. The operator \widehat{A} corresponds to an observable such that Aˆ|1> = |2> Aˆ|2> = |1> Aˆ|n> = |0>, n ≥ 3 where |0> is the null ket. Find a complete...
  26. K

    Eigenfunctions & Potential Diagram

    Homework Statement Consider the stationary state (eigenfunction) Ψ(x) illustrated. Which of the three potentials V(x) illustrated could lead to such an eigenfunction? Homework Equations N/A. The Attempt at a Solution I think it's the second one. Where the wavefunction is...
  27. J

    Figuring symmetries of a differential operator from its eigenfunctions

    So, I understand that the derivative operator, D=\frac{d}{dx} has translational invariance, that is: x \rightarrow x - x_0, and its eigenfunctions are e^{\lambda t}. Analogously, the theta operator \theta=x\frac{d}{dx} is invariant under scalings, that is x \rightarrow \alpha x, and its...
  28. D

    Finding normalized eigenfunctions of a linear operator in Matrix QM

    Homework Statement Hey everyone! The question is this: Consider a two-state system with normalized energy eigenstates \psi_{1}(x) and \psi_{2}(x), and corresponding energy eigenvalues E_{1} and E_{2} = E_{1}+\Delta E; \Delta E>0 (a) There is another linear operator \hat{S} that acts by...
  29. B

    Expansion in terns of Laplacian eigenfunctions

    Hi ! I am currently studying String theory in Polchinski's book. In section 6.2, eq. 6.2.2, he takes an arbitrary function X(\sigma) defined on a Riemann Surface M. Then he expands it on a complete set of eigenfunctions of the laplacian, X(\sigma)=\sum_I x_I X_I(\sigma) with \Delta X_I =...
  30. B

    Confirmation concept questions eigenfunctions and operators

    Homework Statement Are the momentum eigenfunctions also eigenfunctions of e free particle energy. Operator? Are momentum eigenfunctions also eigenfunctions of the harmonic oscillator energy operator? An misplayed system evolves with time according to the shrodinger equation with potential...
  31. T

    Rewriting an Initial State with Normalized Eigenfunctions

    Homework Statement Consider the Hamiltonian H=0.5p^2+ 0.5x^2, which at t=0 is described by: ψ(x,0)= 1/sqrt(8*pi) θ1(x) + 1/sqrt(18pi) θ2(x), where: θ1= exp(-x^2/2); θ2=(1-2x^2)*exp(-x^2/2) a) Normalize the eigenfunctions and rewrite the initial state in terms of normalized...
  32. B

    Formal properties of eigenfunctions

    Homework Statement Give a physicist's proof of the following statements regarding energy eigenfunctions: (a) We can always choose the energy eigenstates E(x) we work with to be purely real functions (unlike the physical wavefunction, which is necessarily complex). Note: This does not mean...
  33. F

    Eigenfunctions of spin operator

    What are the eigenfunctions of the spin operators? I know the spin operators are given by Pauli matricies (https://en.wikipedia.org/wiki/Spin_operator#Mathematical_formulation_of_spin), and I know what the eigenvalues are (and the eigenvectors), but I have no idea what the eigenfunctions of the...
  34. J

    The HUP, simultaneous measurements, and eigenfunctions

    Hi everyone, I know this topic has been discussed quite a bit -- and in particular it's been done in this thread and this thread. But there are still some things I want to talk about in order to (hopefully) clarify my own thoughts. One of the threads discusses this Ballentine article in which...
  35. R

    Show the functions are eigenfunctions of the hamiltonian

    Given the hamiltonian in this form: H=\hbar\omega(b^{+}b+.5) b\Psi_{n}=\sqrt{n}\Psi_{n-1} b^{+}\Psi_{n}=\sqrt{n+1}\Psi_{n+1} Attempt: H\Psi_{n}=\hbar\omega(b^{+}b+.5)\Psi_{n} I get to H\Psi_{n}=\hbar\omega\sqrt{n}(b^{+}\Psi_{n-1}+.5\Psi_{n-1}) But now I'm stuck. Where can I...
  36. J

    Momentum Eigenfunctions and Definite Momentum

    Hi everyone, On page 138 of my Shankar text, Shankar states: "...since the plane waves are eigenfunctions of P, does it mean that states of well-defined momentum do not exist? Yes, in the strict sense. However, there do exist states that are both normalizable to unity (i.e. correspond to...
  37. D

    MHB Orthogonality of Eigenfunctions of Mixed Boundary Conditions

    $$ \left.(\phi_n\phi_m' - \phi_m\phi_n')\right|_0^L + (\lambda_m^2 - \lambda_n^2)\int_0^L\phi_n\phi_m dx = 0 $$ where $\phi_{n,m}$ and $\lambda_{n,m}$ represent distinct modal eigenfunctions which satisfy mixed boundary conditions at $x = 0,L$ of the form \begin{alignat*}{3} a\phi(0) + b\phi'(0)...
  38. T

    Eigenvalues and eigenfunctions

    Homework Statement Hello! I don't know how to solve this problem: find eigenvalues and eigenfunctions of quadratic membrane which is fixed in three edges. Fourth edge is flexible bended in the middle (at this edge membrane is in the shape of triangular). Surface tension of membrane is γ...
  39. S

    Eigenfunctions of Operators with Continuous Sprectra

    I'm self-studying Griffith's Intro to Quantum Mechanics, and on page 100 he makes the claim that the eigenfunctions of operators with continuous spectra are not normalizable. I can't see why this is necessarily true. Hopefully I am not missing something basic. Thanks in advance.
  40. P

    Determining stability about a critical point using eigenfunctions

    I admit I am a bit out of practice when it comes to DiffEq. I think I am either forgetting a simple step or getting my methods mixed up. Homework Statement The problem concerns a pendulum defined by d2θ/dt2 + (c/mL)(dθ/dt) + (g/L)sinθ = 0 where m=1, L=1, c=0.5, and of course g=9.8 After...
  41. O

    Sturm-Liouville Problem. Two eigenfunctions have the same number of zeros

    Hello, As I know, the "standard" Sturm-Liouville problem \frac{d}{dx}p\frac{d}{dx}y + \lambda\rho y = 0 with boundary conditions \begin{align}y^{\prime}(a)&=0 \\ y^{\prime}(b)&=F(\lambda)\end{align} has sequence of eigenvalues \lambda < \lambda_1 < \lambda_2 < ... ... with corresponding...
  42. R

    Corresponding Energy Levels for Normalised Eigenfunctions

    Homework Statement Particle of mass m in a 1D infinite square well is confined between 0 ≤ x ≤ a Given that the normalised energy eigenfunction of the system is: Un(x) = (\frac{2}{a})\frac{1}{2} sin (\frac{nx\pi}{a}) where n = 1, 2, 3... what are the corresponding energy levels...
  43. S

    Differential Equations - Eigenvalues and Eigenfunctions

    Homework Statement Find the eigenvalues and the eigenfunctions for x^2y"+2xy'+λy = 0 y(1) = 0, y(e^2) = 0 Homework Equations See problem The Attempt at a Solution My book has one paragraph on this that does not help me. I tried using an auxiliary equation and solving for lambda...
  44. P

    Boundary Value Problem; Eigenvalues and Eigenfunctions

    Homework Statement Find the eigenvalues and eigenfunction for the BVP: y'''+\lambda^2y'=0 y(0)=0, y'(0)=0, y'(L)=0 Homework Equations m^3+\lambdam=0, auxiliary equation The Attempt at a Solution 3 cases \lambda=0, \lambda<0, \lambda>0 this first 2 give y=0 always, as the only...
  45. P

    MHB Eigenfunctions and Boundary Conditions for $$\frac{d^2}{dx^2}(xy) - λxy=0$$

    consider $$\frac{d^2}{dx^2}(xy) - λxy=0$$. Show eigenfunctions are $$y_{n}=\frac{\sin(n\pi x)}{x}$$. Boundary conditions are y(1)=0 and y regular at x=0 I integrated twice to obtain $$6xy=λx^3y+6Ax+6B$$ where A,B constants. I can't apply the condition y is regular because I don't know what it...
  46. C

    Help with Eigenfunctions: Find Eigenvalue-Eigenfunction Pair

    eigenfunctions. help! I would be very grateful for any help on the following question: Find any single eigenvalue-eigenfunction pair, with a real eigenvalue, for the following operator: \textit{L} = (\partial^2/\partialx) + (\partial/\partialx) + 2Id subject to the initial boundary...
  47. M

    Spin state function of a beam of particles in terms of eigenfunctions.

    Hi there, I apologise that I should probably know this/its a stupid question but I seem to have forgotten all physics over the holiday and so any help would be great! I have been told that there is a beam of atoms with spin quantum number 1/2 and zero orbital angular momentum, with spin +1/2...
  48. T

    Angular Momentum Eigenfunctions for Bead on a Wire

    Homework Statement A bead of mass m on a circular ring has the wave function Acos\stackrel{2}{}θ. Find expectation value, eigenfunctions & eigenvalues. Homework Equations The differential operator for the angular momentum is L = \hbar/i (\partial/\partialθ). The Attempt at a...
  49. R

    Do the eigenfunctions for the position operator form an orthogonal set?

    Starting with, \hat{X}\psi = x\psi then, x\psi = x\psi \psi = \psi So the eigenfunctions for this operator can equal anything (as long as they keep \hat{X} linear and Hermitian), right? Well, McQuarrie says that "the eigenfunctions of a Hermitian operator are orthogonal", which can be...
  50. J

    Determining eigenfunctions + arbritary value constant

    When we determine an eigenfunction of a given differential equation, is it necessary to include the arbirtary value in front of the solution ? If not, is it because of the term's arbritary nature which means we can choose to include/reject it from the determined eigenfunction ?
Back
Top