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.
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...
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...
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...
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?
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...
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...
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...
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...
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
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...
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)##...
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...
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...
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...
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...
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...
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...
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...
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...
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...
<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...
Hello Everyone,
For the particle in the 1D box of width ##L##, the time invariant Schrodinger equation is cast in the form of the Hamiltonian operator and automatically leads to the energy eigenfunctions
$$\Psi(x) = \sqrt{\frac 2L} sin(n\pi x/L)$$
I know that these energy eigenfunctions...
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...
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} =...
Homework Statement
The problem states consider A_hat=exp(b*(d/dx)). Then says ψ(x) is an eigenstate of A_hat with eigenvalue λ, then what kind of x dependence does the function ψ(x) have as x increases by b,2b,...?
Homework EquationsThe Attempt at a Solution
Started out by doing...
Hi, I have been trying to use imaginary time propagation to get the ground state and excited states eigen function but the results I got is different from the analytical solution. I know that to get excited states, I should propagate 2 or more orthogonal functions depending on the number of...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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 =...