What is Eigenfunction: Definition and 121 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. I Klein-Gordon Equation with boundary conditions

I am trying to find solutions for the Klien-Gordon equations in 1-d particle in a box. The difference here is the box itself oscillating and has boundary conditions that are time dependent, something like this L(t)=L0+ΔLsin(ωt). My initial approach is to use a homogeneous solution and use...
2. I Completeness of Eigenfunctions of Hermitian Operators

I know that completeness (roughly) implies that (almost) all functions can be decomposed into a sum of eigenfunctions of a Hermitian operator. ##\psi=\sum_n \alpha_n \psi_n##. Clearly, there have to be some restrictions on the function itself, and the operator as well. But what is that? My...
3. I Asymmetric top eigenfunction

Hello! I read that for a symmetric top (oblate or prolate) we can find the exact eigenfunctions (in terms of Winger matrices) and eigenstates, but we can't do it in general for an asymmetric top. I am not sure I understand why. The Hamiltonian for an asymmetric top, for a given J, can be written...
4. Find the eigenfunction and eigenvalues of ##\sin\frac{d}{d\phi}##

Here is what I tried. Suppose ##f(\phi)## and ##\lambda## is the eigenfunction and eigenvalue of the given operator. That is, $$\sin\frac{d f}{d\phi} = \lambda f$$ Differentiating once, $$f'' \cos f' = \lambda f' = f'' \sqrt{1-\sin^2f'}$$ $$f''\sqrt{1-\lambda^2 f^2} = \lambda f'$$ I have no...
5. Eigenfunction proof and eigenvalue

I searched through the courses but I can't find any formula to help me prove that the expression is an eigenfunction. Am I missing something? What are the formulas needed for this problem statement?
6. Normalize the eigenfunction of the momentum operator

I am just solving the equation $$\frac{h}{2\pi i}\frac{\partial F}{\partial x} = pF$$, finding $$F = e^{\frac{ipx2\pi }{h}}C_{1}$$, and$$\int_{-\infty }^{\infty }C_{1}^2 = 1$$, which gives me $$C_{1} = \frac{1}{(2\pi)^{1/2} }$$, so i am getting the answer without the h- in the denominator...
7. Quantum Mechanics hydrogen atom eigenfunction problem

This is a general property of eigenvectors of Hermitian operators. State functions are a particular class of vector, and it is easiest to work in the general formalism (I am hoping to show how ket notation makes qm easier, not just do standard bookwork at this level). Suppose O is a Hermitian...
8. I Differences between the PCA function and Karhunen-Loève expansion

Hello everyone. I am currently using the pca function from MATLAB on a gaussian process. Matlab's pca offers three results. Coeff, Score and Latent. Latent are the eigenvalues of the covariance matrix, Coeff are the eigenvectors of said matrix and Score are the representation of the original...
9. Eigenstates of a free electron in a uniform magnetic field

I started with the first of the relevant equations, replacing the p with the operator -iħ∇ and expanding the squared term to yield: H = (-ħ^2 / 2m)∇^2 + (iqħ/m)A·∇ + (q^2 / 2m)A^2 + qV But since A = (1/2)B x r (iqħ/m)A·∇ = (iqħ / 2m)(r x ∇)·B = -(q / 2m)L·B = -(qB_0 / 2m)L_z and A^2 =...
10. The Eigenfunction of a 2-electron system

Hello! I am stuck at the following question: Show that the wave function is an eigenfunction of the Hamiltonian if the two electrons do not interact, where the Hamiltonian is given as; the wave function and given as; and the energy and Born radius are given as: and I used this for ∇...
11. Eigensolution of the wave function in a potential field.

1. Homework Statement Consider a potential field $$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$ The eigenfunction of the wave function in this field suffices...
12. A Confusing eigensolutions of a wave function

Consider a potential cavity $$V(r)=\begin{cases}\infty, &x\in(-\infty,0]\\\frac{\hslash^2}{m}\Omega\delta(x-a), &x\in(0,\infty)\end{cases}$$ The eigenfunction of the wave function in this field suffices $$-\frac{\hslash^2}{2m}\frac{d^2\psi}{dx^2}+\frac{\hslash^2}{m}\Omega\delta(x-a)\psi=E\psi$$...
13. Eigenfunction of momentum and operators

Homework Statement Homework Equations ##\hat{P}= -ih d/dx## The Attempt at a Solution To actually obtain ##\psi_{p_0}## I guess one can apply the momentum operator on the spatial wavefunction. If we consider a free particle (V=0) we can easily get obtain ##\psi = e^{\pm i kx}##, where ##k=...
14. How to find eigenvalues and eigenfunction

defind ## \hat{A}f(x)=f(-x) ## find eigenfunction and eigenvalue I think ## \frac{d}{dx} ( \hat{A}f(x) ) = \frac{d}{dx} f(-x) ## ## \hat{A} \frac{d}{dx}f(x) + f(x) \frac{d}{dx} \hat{A} = -\frac{d}{dx} f(x)## ## \hat{A} \frac{d}{dx}f(x) + \frac{d}{dx} f(x) = -f(x) \frac{d}{dx} \hat{A}## ##...
15. Schrodinger equation and boundary conditions

Hi at all, I'm tring to solve Schrodinger equation in spherically symmetry with these bondary conditions: ##\lim_{r \rightarrow 0} u(r)\ltimes r^{l+1}## ##\lim_{r \rightarrow 0} u'(r)\ltimes (l+1)r^{l}## For eigenvalues, the text I'm following says that I have to consider that the...
16. Eigenfunction energy levels in a harmonic well

Homework Statement If the first two energy eigenfunctions are ## \psi _0(x) = (\frac {1}{\sqrt \pi a})^ \frac{1}{2} e^\frac{-x^2}{2a^2} ##, ## \psi _1(x) = (\frac {1}{2\sqrt \pi a})^ \frac{1}{2}\frac{2x}{a} e^\frac{-x^2}{2a^2} ## Homework EquationsThe Attempt at a Solution Would it then be...
17. Why are we only considering the first eigenfrequencies?

Hey, I have a question concerning eigenfrequencies: Let us assume we examine a beam that is fixed at one end and free at the other end. It is possible to get an analytical solution in form of a unlimtied series: sum_i=1..infinity eigenfunction(i)*exp(i*eigenfrequencie(i)*t). (something...
18. A Can the convolution operator be diagonalized using the Fourier transform?

Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
19. Show that the total eigenfunction must be antisymmetric

Homework Statement [/B] By considering the eigenfunctions for 2 noninteracting particles at distances r1 and r2, show that their total eigenfunction must be antisymmetric. . Homework Equations Spatial wavefunctions: Ψ(x1,x2) = 1/√2 [ ψA(x1)ψB(x2) ± ψA(x2)ψB(x1)] Where + gives a symmetric...
20. Applying particle in a box boundaries to an eigenfunction

Homework Statement Use the eigenfunction Ψ(x) =A'eikx + B'e-ikx rather than Ψ(x)=Asinkx + Bcoskx to apply the boundary conditions for the particle in a box. A. How do the boundary conditions restrict the acceptable choices for A’ and B’ and for k? B. Do these two functions give different...
21. Physical chemistry: Energy operator and eigenfunction

Homework Statement The energy operator for a time-dependent system is iħ d/dt. A possible eigenfunction for the system is Ψ(x,y,z,t)=ψ(x,y,z)e-2πiEt/h Show that the probability density is independent of time Homework Equations ĤΨn(x) = EnΨn The Attempt at a Solution [/B] I understand the...
22. 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...
23. I Is Ψ2Px an Eigenfunction of L2 or Lz in Quantum Mechanics?

hi, am major new on quantum mechanics. please help me understand. is the real wave function Ψ2Px= [Ψ2p+1 +Ψ2p-1]1/2 an eigen function of L2 or Lz? if so, how is it? and if so kindly explain the values of l and m thanks
24. Prove the nth energy eigenfunction has n-1 zeros

How do we show that ##\psi_n(x)## has ##(n-1)## zeros for all ##n\in Z^+##? Assuming ##\psi_k(x)## has ##(k-1)## zeros for some ##k\in Z^+##, by oscillation theorem, we can only get ##\psi_{k+1}(x)## has ##\geq k## zeros. Also, how do we show that ##\psi_1(x)##, the eigenfunction corresponding...
25. Show that the next eigenfunction has a zero between 2 zeros

Homework Statement Homework Equations Wronskian theorem: The Attempt at a Solution I've gotten the relationship given by the question but I do not know how to continue. Since ##\psi_n(a)=\psi_n(b)=0##, LHS ##=\psi_n'(b)\,\psi_{n+1}(b)-\psi_n'(a)\,\psi_{n+1}(a)## If LHS ##=0##, RHS...
26. Eigenfunction of multiple operators simultaneously?

Hello, I am taking an introductory course in quantum mechanics. One thing I am confused about is, Schrodinger's equation seems to be regarded as the "ultimate" formula which determines a particle's possible wavefunctions and energies, given a certain potential (Hamiltonian of psi = Energy...
27. 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...
28. MHB Eigenfunction boundary conditions order

Hi, just want to confirm that with the eigenfunction boundary condition $p(x) v^*(x)u'(x)|_{x=a} = 0$, the order of (solutions) v, u doesn't matter? I ask because a problem like this had one solution = a constant, so making that the u solution makes $p(x) v^*(x)u'(x) = 0$ no matter the...

Homework Statement ## \psi_1 ## and ## \psi_2 ## are momentum eigenfunctions corresponding to different momentum eigenvalues ## p_1 \not= p_2 ##. Is ## \psi_1 ## + ## \psi_2 ## also momentum eigenfunction ? Homework Equations Is the right answer[/B] Yes No It Depends ?The Attempt at a...

Homework Statement Show that the radial eigenfunction unr,l is a solution of the differential equation: ħ2/2me×d2unr,l/dr2+[l(l+1)ħ2/2mer2 - e2/4πε0r]unr,l=Enr,lunr,lHomework Equations The radial function is R(r)=u(r)/r, so that the expression on the RHS is E×u. The Attempt at a Solution I know...
31. Momentum operator eigenfunction

This might be trivial for some people but this has been bothering lately. If P is momentum operator and p its eigenvalue then the eigenfunction is up(x) = exp(ipx/h). where h is the reduced Planck constant (sorry can't find a way to make the proper notation). While it can also be proved that...
32. Finding a constant from eigenfunction

Homework Statement Since Hamiltonian operator is: Ĥ = - (ħ2/(2m))(delta)2 - A/r where r = (x2+y2+z2) (delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 A = a constant from Ĥg(r) = Eg(r) form, where: g(r) = D e-r/b(1-r/b) with b, D as constants, is an EIGENFUNCTION of Ĥ, find the correct b and give the...
33. What is the difference between H and E in the equation Hψ = Eψ?

Could someone please explain Hψ = Eψ? I understand that H = Hamiltonian operator and ψ is a wavefunction, but how is H different from E? I am confused. I am trying to understand "Hψ = Eψ" approach
34. MHB Eigenfunction expansion with chebyshev

I am absolutely dying with this question. Ok so referring to the attached image; we can re-iterate the given equation in SL-Form. so $(1-x^2)u'' - xu' + 2u =0$ divide everything by $\sqrt{1-x^2}$ so we get $\sqrt{1-x^2}u'' - \frac{x}{\sqrt{1-x^2}} u' + \frac{2}{\sqrt{1-x^2}}u=0$ which...
35. Eigenfunction of a system of three fermions

I have to find the eigenfunction of the ground state \Psi_0 of a three independent s=1/2 particle system. The eigenfunctions \phi_{n,s}(x) = \varphi_n(x) \ \chi_s and eigenvalues E_n of the single particle Hamiltonian are known. Becuse of the Pauli exclusion principle, there must be...
36. The Simplest Eigenfunction for Schrödinger's Wave EQ?

I am just getting into quantum after a long absence of working on modern physics. I am having a go at "Introduction to Modern Physics" by Griffiths What is the simplest equation that satisfies Schrödinger's Wave Equation. It looks like $$Ae^{x}$$ would do the trick, but it would not solve...
37. Struggling with (I think pretty trivial) eigenfunction work for QM.

This is the sort of thing that should be easy but my brain has blanked due to be overloaded in other areas of Physics. I've tried a great deal of different approaches but I'll walk though the one that led closest. I assumed that \hat{A} and \check{B} where both hermitian so there...
38. Show that the two wave functions are eigenfunction

Homework Statement Consider the dimensionless harmonic oscillator Hamiltonian H=½ P2+½ X2, P=-i d/dx. Show that the two wave functions ψ0(x)=e-x2/2 and ψ1(x)=xe-x2/2 are eigenfunction of H with eigenvalues ½ and 3/2, respectively. Find the value of the coefficient a such that...
39. QM 1-D Harmonic Oscillator Eigenfunction Problem

Homework Statement A particle of mass m moves in a 1-D Harmonic oscillator potential with frequency \omega. The second excited state is \psi_{2}(x) = C(2 \alpha^{2} x^{2} + \lambda) e^{-\frac{1}{2} a^{2} x^{2}} with energy eigenvalue E_{2} = \frac{5}{2} \hbar \omega. C and \lambda are...
40. Finding the eigenfunction of momentum

the questions is: is the ground state of an infinite square well an eigenfunction of momentum, if so what is the momentum? solution: i was working it out and i got something different from the solutions, and i don't understand where they're getting the cotangent term from.. and...
41. Collapsing the wavefuntion to an Energy Eigenfunction?

Is there an experiment that can measure the energy of a single particle so immediately after it has collapsed to one of the energy eigenfunctions? The problem is that all experiments i can think of are about measuring the position of a the particle so we collapse it to its delta function...
42. Finding the probability density function given the eigenfunction

Homework Statement I need to find the probability density function given the eigenfunction Homework Equations \psi=C\exp^({\frac{ipx}{\hbar}-\frac{x^2}{2a^2}}) The Attempt at a Solution I tried to square the function but that gave me a nasty integral that I could not solve. I...
43. Finding Hydrogen eigenfunction u(2,0)

Taken from Physics of Quantum Mechanics, by James Binney. I try to calculate ##u_{n}^{l=n-2}##, something goes wrong: Starting, we define operator A by: A_{n-2} = \frac{a_0}{\sqrt 2}\left(\frac{i}{\hbar}p_r + \frac{1-n}{r} + \frac{Z}{(n-1)a_0}\right) Substituting ##p_r = -i\hbar...
44. Quantum Chemistry Eigenfunction

1. Consider a particle of mass m in a cubic (3-dimensional) box with V(x,y,z) = 0 for 0 < x < L, 0 < y < L, and 0 < z < L and V(x,y,z) = ∞elsewhere. Is 1/\sqrt{2} * (ψ(1,1,5)+ψ(3,3,3)) an eigenfunction of the Hamiltonian for this system? If so, what is the eigenvalue? Explain your reasoning 2...
45. MHB Eigenvalue and eigenfunction for Fredholm method

Given $f(x) = \lambda\int_0^1xy^2f(y)dy$ At order $$\lambda^2$$ and $$\lambda^3$$, we have repeated zeros so $D(\lambda) = 1 - \frac{\lambda}{4}.$ Then we have $\mathcal{D}(x, y;\lambda) = xy^2$ so $f(x) = \frac{\lambda}{D(\lambda)}\int_0^1\mathcal{D}(x, y;\lambda)dy.$ How do I...
46. Checking that a coherent state is an eigenfunction of an operator

Homework Statement Hey guys, I'll type this thing up in Word. http://imageshack.com/a/img716/8219/wycz.jpg
47. MHB Eigenfunction of \frac{d^2}{dx}-x^2: e^{-0.5x^2} with eigenvalue x^2-1

show that e^{-0.5x^2} is an eigenfunction of the operator \frac{d^2}{dx}-x^2 and finds it's eigenvalue. I get e^{-0.5x^2}(x^2-1)-x^2 so it doesn't seem like its an eigenfunction.
48. Eigenfunction of all shift operators

Prove that if a continuous function e\left( x \right) on \mathbb{R} is eigenfunction of all shift operators, i.e. e\left( x+t \right) = \lambda_t e\left( x \right) for all x and t and some constants \lambda_t , then it is an exponential function, i.e. e\left( x \right)= Ce^{ax} for some...
49. Wave function is a combination of eigenfunction?

hi, i read in quantum mechanics wave function is a combination of eigenfunctions and according to Orthodox interpretation measurement causes the wave function to collapse into one of the eigenfunction of the quantity being measured. Is this explanation still valid?
50. Eigenfunction & Potential Barrier

Homework Statement A particle of total energy E is incident on a potential barrier V0 (E<V0) between x=0 and x=a. Write down the allowed eigenfunctions in the regions x<0, 0<x<a and x>a in terms of five unknown constants A, B, C, D and F where A and F are the amplitudes of the incident and...