Eigenfunction Definition and 121 Threads

I am kind of new to this eigenvalue, eigenfunction and operator things, but i have come across this quote many times: First i need some explanation on how do we know this? All i know about operator ##\hat{H}## so far is this equation where ##\langle W \rangle## is an energy expected value...

In the picture attached I understand everything up to 1.12. I wrote "think of it like a matrix" at the time and that made sense but now I don't really get it. There's obviously an analogy between decomposing a matrix into its eigenvector basis and a function into its eigenfunction basis but I'm...

Homework Statement We are considering the Superposition state: ψ(x,t) = 1/2 u1(x)e(-i/hbar)E1t + √3/2 u4(x) e(-i/hbar)E4t We had to verify that ψ is a solution of the time dependent Schrodinger Equation, which I have done. Next we are asked to show that ψ is NOT an energy eigenfunction...

I am asked to simulate the first two eigenvectors for a periodic potential V(x)=V0cos(x). I have attached those two (blue is first, green is second). And I am then asked to explain why they look the way they do. I wrote: Because a particle needs to tunnel through a lot of potential barriers to...

Hi, I am stuck with a problem which effectively boils down to this: Given the eigenstates of a Hamiltonian with a step potential in the x direction H=-\hbar^2/2m \nabla^2 + V_0 \Theta(x) \psi(q)_{in}=cos(qx)-\frac{\sqrt{K_{V_0}^2-q^2}}{q}sin(qz) \qquad x<0...

I am trying to find out just how to solve for the eigenfunction given a system, namely the parameters of an optical system (say a polarizer) in the form of a 2 by 2 Jones Vector. I know how to derive the eigenvalue, using the the constituent det(λI -A) = 0, 'A' being the system at hand and 'λ'...

$$ \varphi\psi'' = -c^2\varphi^{(4)}\psi\Rightarrow \frac{\psi''}{c^2\psi} = -\frac{\varphi^{(4)}}{\varphi} = -\beta^4. $$ The spatial component is $\varphi^{(4)} - \beta^4\varphi = 0$. It is obvious that $\pm\beta$ are solutions to $m^4 - \beta^4 = 0$. That is, $(m - \beta)(m + \beta)(m^2 +...

The problem is attached. I am not quite sure how to do the 3 parts. for part a: Do I just plug in the f(x) equation into T(f)? If so that gives me k2ekx-2*kekx-3ekx I'm just not sure what I'm supposed to be doing.

I'm having trouble understanding the derivation of the the position operator eigenfunction in Griffiths' book : How is it "nothing but the Dirac delta function"?? (which is not even a function). Couldn't g_{y}(x) simply be a function like (for any constant y) g_{y}(x) = 1 | x=y...

My ODE textbook does not help me much here; and a large number of master's exam practice tests (with worked out solutions) also isn't helping me. I need someone to recommend a (preferably ONLINE) source that clearly states how to solves ODES using eigenfunction expansion. For example...

I have previously taken PDE's and ODE's. I understand obtaining the equation y''+lambda*y=0 (lambda then giving the eigenvalues). But I've encountered now the use of eigenfunction expansion for an ODE; and what I don't understand, is that in solving it they're making some assumption that y''+y=0...

What is the difference between eigenfunction and wave function? I'm always get confused when i am asked to write wave function and eigenfunction..

Homework Statement Solve BVP y''[x]-y[x]=Cos[2x] using eigenfunction expansion. We know y'[0]=y'[Pi]=0 Homework Equations Fourier it up The Attempt at a Solution Is the corresponding Sturm-Liouville problem: f''[x]+/lambda*f=0? All the examples we've done have been of form...

Homework Statement Show that the eigenfunction of Lx can be written as a combination of eigenfunctions from Lz with the same l but different m. Using the eigenfunction Y_{x} = \frac{1}{\sqrt{2}}(Y^{-1}_{1} - Y^{1}_{1}) as the eigenfunction of Lx Homework Equations Y^{m}_{l} =...

Homework Statement "Prove that any two different nondegenerate bound eigenfunctions \psij(x) and \psii(x) that are solutions to the time-independent Schroedinger equation for the same potential V(x) obey the orthogonality relation \int-∞∞ \psi j*\psii(x)dx=0 " Homework Equations I...

Homework Statement I know that,if (operator)(function)=(value)(samefunction) that function is said to be eigenfunction of the operator. in this case i need to show this function to be eigenfunction of the Lz angular momentum:Homework Equations function: ψ=(x+iy)/r operator: Lz= (h bar)/i (x...

Homework Statement Given f(x)1 = cos(kx) and f(x)2 = sin(kx), find a linear combination which is an eigenfunction of the momentum operator. Homework Equations The momentum operator: -iħ d\dx(Acos(kx)+Bsin(kx) = λ[Acos(kx) + Bsin(kx)] where λ is some constant. The Attempt at a Solution I've...

I'm trying to solve a non-linear time-dependent diffusion equation to find R(x,t). To do so, I'm positing that : R(x,t)=\sum^{J}_{1}X_{i}(x)T_{i}(t) which allows me to arrive at something that looks like : dT_{i}/dt=A_{i}T_{i}(t)-B*T_{i}(t)^{2} The problem I'm having, through sheer...

hi i know how to calculate eigenvalue of given matrix. I want to know if two non homogenous simutaneous equation are given - than can we find its eigenvalue.

Homework Statement Consider the eigenfunction in the top part of the figure. (a) Which of the three potentials illustrated in the bottom part of the figure could lead to such an eigenfunction? Give qualitative arguments to justify your answer. (b) The eigenfunction shown is not the one...

Homework Statement Already it is on title ! Homework Equations attachents provide the most relevant equations i could find. The Attempt at a Solution I have found some old books written in 1920 decade ,dealing with this subject,but the solution they provide seems very hard...

Homework Statement The square of the angular momentum operator is (in SPC); http://img6.imageshack.us/img6/67/54712598.png Show that Y(\theta,\phi) = Csin^{2}\thetae^{2i\phi} Is an eigenfunction. C is a constant. Homework Equations The Attempt at a Solution I am not able to get it in the...

Homework Statement Normalised energy eigenfunction for ground state of a harmonic oscillator in one dimension is: 〈x|n〉=α^(1/2)/π^(1/4) exp(-□(1/2) α^2 x^2) n = 0 α^2=mω/h suppose now that the oscillator is prepared in the state: 〈x|ψ〉=σ^(1/2)/π^(1/4) exp(-(1/2) σ^2 x^2)...

Homework Statement For the following equation y" - λy=0 find the values of λ which produce a non-trivial solution on the interval 0 <= x <= a The given initial conditions are y(0) = 0 y(a) = 0 Homework Equations The Attempt at a Solution see attached pdfs My...

Hello again everyone! I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator...

[b]1. A function "v" where v(x,y,z)≠0 is called an eigenfunction of the Laplacian Δ (on some region Ω - with specified homogenous BC) if v satisifies the BC ad also Δv=-λv on Ω for some number λ. Part A: Give an example of an eigenfunction of Δ when Ω is the cube [0,∏]3 with Dirichlet BCs...

Homework Statement From qualitative arguments, make a sketch of the form of a typical unbound standing wave eigenfunction for a finite square well potential. An unbound particle is one which has total energy E greater than the Potential V of the well Is the amplitude of the oscillation the...

Homework Statement Let the angular part of a wave function be proportional to x2+y2 Show that the wave function is an eigenfunction of Lz and calculate the associated eigenvalue. Homework Equations Lz = xpy-ypx px = -i\hbar\frac{\partial}{\partialx} py =...

Homework Statement Show that the rotational wavefunction 3cos2? -1 is an eigenfunction of the Hamiltonian for a three dimensional rigid rotor. Determine the corresponding eigenvalue. Homework Equations the eigenstates are |l,m> the quantum number of the total angular momentum is l...

What exactly is an eigenfunction, and what are some of the applications of an eigenfunction?

Homework Statement In quantum mechanics a physical observable is represented by an operator A. Define the terms eigenstate, eigenvalue and eigenfunction of a quantum mechanical operator. Homework Equations The Attempt at a Solution I think I know in that eq 'f' is the eigenfunction, and...

I recently had a probelm in QM to find the time evolution of a hydrogen prepared in a state with a wave function that is not an energy eigenfunction: specifically, psi = Y21*R2p where Y is then the D spherical harmonic. Of course, n=2 hydrogen doesn't have d oribtals. So the problem is I...

Homework Statement given the following functions: y(x)= Acos(kx) y(x)=A sin(kx)-Acos(kx) y(x)=Acos(kx)+iAsin(kx) y(x)=A d(x-x0) Which are eigenfunctions of the position, momentum, potential energy,kinetic energy, hamiltonian, and total energy operators Homework Equations y(x) is...

[urgent] Eigenfunction again! Find the eigenvalue and eigenfunction of the following \intcos(2X)+cox(2t)y(t) dt = ky , k=eigenvalue , intergrate from 0 to Pi thx, urgent this is the original question

if y''-2y'+y=ky k=eigenvalue, y(0,pi)=0, 0<x<pi find corresponding eigenvalues and eigenfunctions. thx a lot!

Homework Statement at t=0 a particle is described by the eigenfunction Ψ(x) given by: Ψ(x)= iAe^(-x/2) given that x≥0 and 0 given that x<0 where A is a real number. assuming the system is in a well defined eigenstate with total energy E, write an expression for the corresponding...

Homework Statement Determing the constant c such that \psi_{c}(x,y,z) = x^{2}+cy^{2} is an eigenfunction of \hat{L_{z}} Homework Equations \hat{L_{z}} = -i \hbar (x\frac{\partial \psi}{\partial y} - y\frac{\partial \psi}{\partial x} The Attempt at a Solution x\frac{\partial...

Homework Statement Homework Equations Stated in the question. The Attempt at a Solution It is a eigenfunction of L_z as it has no dependence on Z? Not sure if I can just state this, I do need to actually prove it but I can't get the calculations to work. I managed a similar...

Homework Statement Consider a free electron in a constant magnetic field \vec{B}=B\hat{z} and a perpendicular electric field \vec{E}=\varepsilon\hat{y}. Find the energy eigenvalues and eigenfunctions in terms of harmonic oscillator eigenfunctions Hint: Use Landau gauge \vec{A}=-By\hat{x}...

If one measures a system for momentum, it will collapse into a delta in momentum space. (right?) How would that look in position space, and how would the function evolve in time, if no subsequent measurements are made, in both position and momentum space? A mathematical answer would be appreciated.

Homework Statement T(f(x)) = 5 f(x) T is defined on C. Find all real eigenvalues and real eigenfunction. V:R -> R Homework Equations Not sure. The Attempt at a Solution No, clue. I can find eigenvalues for matrices, that's not a problem. I'm having problem that its a T(function) =...

Homework Statement Consider a particle that moves in one dimension. Two of its normalized energy eigenfunctions are \varphi_1(x) and \varphi_2(x), with energy eigenvalues E_1 and E_2. At time t=0 the wave function for the particle is \phi= c_1*\varphi_1+c_2*\varphi_2 and c_1 and...

Please: I need another Definition of the eigenfunction and the book the include this Definition I found in SCHAUM’S DIFFERENTIAL EQUATIONS that: Those values of Lamda for which nontrivial solutions do exist are called eigenvalues; the corresponding nontrivial solutions are called...

what the differ between eigenstate and eigenfunction ?

Homework Statement find eigenfunction of the the given operator. compute the vaule of px and x for this eigenfunction Homework Equations O= p + x O is made up by adding the momentum operator(p) and the position operator(x); it is one dimensional The Attempt at a Solution i tried...

Hi there, I don't have Mathematica and the standard books didn't work. I am looking for the eigenfunctions to this operator: \hat{O} = (\sin \alpha) \, \partial_x +(\cos \alpha) \, x \qquad \alpha \in \mathbb{R}

Homework Statement Suppose that two operators P and Q satisfy the commutation relation: [P,Q]=P. Suppose that psi is an eigenfunction of the operator P with eigenvalue p. Show that Qpsi is also an eigenfunction of P, and find its eigenvalue. Homework Equations The Attempt at a...

I am a seond year Quantum Chemistry student. I am having a hard time understanding these concepts. I was wondering I can get help in this concept. How can it be demonstrate mathematically in the Hamiltonian operator that the function φ(x) = A sin(2x) + B cos(2x) is an eigenfunction of...

Homework Statement Find all eigenfunction of momentum operator in x(px=h/i*d/dx) and their eigenvalues. Homework Equations operator*eigenfunction=eigenvalue*eigenfunction Operator=px The Attempt at a Solution I really don't have any clues Thank you

Homework Statement How to use eigenfunction expansion in Legendre polynomials to find the bounded solution of (1-x^2)f'' - 2xf' + f = 6 - x - 15x^2 on -1<= x <= 1 Homework Equations eigenfunction expansion The Attempt at a Solution [r(x)y']' + [ q(x) + λ p(x) ]...