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

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5. ### 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...
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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...
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8. ### Finding the eigenfunctions and eigenvalues associated with an operator

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12. M

### I Sturm-Liouville Eigenfunctions

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15. ### Hecke Operators and Eigenfunctions, Fourier coefficients

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17. ### 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. ### 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

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21. ### 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. ### 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...
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25. ### What does the sum of eigenfunctions represent?

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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...
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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...
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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. ### I Eigenfunctions and eigenvalues

is exp (-kx) an eigenfunction?
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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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### Trial function and Eigenfunction....

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41. ### 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. ### 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. ### Expand function as series of eigenfunctions

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44. ### 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. ### Momentum eigenfunctions proof and Fourier Transform question

I have the following problem:
46. ### 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...
47. ### KE operator and eigenfunctions

I have just done a question and then looked at the solution which I don't get. The question gives a wavefunction as u = x - iy. It then asks if this function is an eigenfunction of the kinetic energy operator in 3-D. Applying this operator to u gives zero. I took this to mean that u is an...
48. ### 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...
49. ### How do you plot eigenfunctions of perturbed HO?

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50. ### Calculate the eigenfunctions for a spin half particle

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