Eigenfunctions Definition and 173 Threads

What's the proof that eigenfunctions of discrete eigenvalues are in Hilbert Space?

I need to demonstrate the orthonormality of the eigenfunctions for de cero-spin field. I mean, <2k,3k> = 0 for example. nk denotes n particles with wave vector k. I'm trying with the commnutation properties but i´m stuck in the middle of the process. Is there any other thing i need?

Hi everyone! I am answering this problem which is about the eigenvalues and eigenfunctions of the Hamiltonian given as: H = 5/3(a+a) + 2/3(a^2 + a+^2), where a and a+ are the ladder operators. It was given that a = (x + ip)/√2 and a+ = (x - ip)/√2. Furthermore, x and p satisfies the...

Homework Statement How does one find all the permissible values of b for -{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y with boundary conditions y(0)=y(1)=0? Thanks. Homework Equations See aboveThe Attempt at a Solution I assume we have a discrete set of \{b_n\} where they can be regarded as...

In the equation for determining the coefficients of eigenfunctions of a continuous spectrum operator, I have trouble understanding the origin of the Dirac delta. a_f = INTEGRAL a_g ( INTEGRAL F_f F_g ) dq dg a is the coefficient, F = F(q) is an eigenfunction. From this it is shown that...

I've been reading QM by Landau Lifgarbagez, in which I've come across a statement I can't seem to get my head around. It states (just before equation 3.6): a_n = SUM a_m. INTEGRAL f_m. f_n. dq ( a_n is the nth coefficient, f_m is the mth eigenfunction of an operator, dq is the...

Hi, suppose we have an unidimensional finite square well potential and we want to expand an arbitrary wave function in terms of energy eigenfunctions but considering the possibility of bounded (discrete) AND unbounded (continue) states. How do you express the expansion?. The problem is that each...

Hi, I'm a second year physics undergrad currently revising quantum mechanics, and I came across a phrase about angular momentum which has confused me, so I was wondering if anyone could help. We looked at different components of angular momentum (in Cartesian) and decided that they did not...

Homework Statement In this problem all vectors and operators are represented in a system whose basisvectors are the eigenvectors of the operator Lz (the third component of the angular momentum). a) Find the eigenvector |l=1,my=-1> of Ly in terms of the eigenvectors of Lz. b) Go from the...

Homework Statement Homework Equations The Attempt at a Solution Issue is in understanding the content. I am only after a nudge in the right direction. My issue is in getting started as it seems with most of these Quantum Problems.

Homework Statement See figure attached for problem statement. Homework Equations The Attempt at a Solution See figure attached for attempt. I'm confused as to how to do part B? I know that the definition of orthogonal is, \int_\alpha ^{\beta}f(x)g(x) = 0 but how do I...

Homework Statement using X''(x)+ lambda*X(x)=0 find the eigenvalues and eigenfunctions accordingly. Use the case lambda=0, lambda=-k2, lambda=k2 where k>0 Homework Equations X(0)=0, X'(1)+X(1)=0 The Attempt at a Solution I know that for lambda=0 X(x)=C1x+C2 which applying the...

[PLAIN]http://img251.imageshack.us/img251/1050/quantume.png taken from http://quantummechanics.ucsd.edu/ph130a/130_notes/node338.html I see how psi_211 and psi_21-1 are eigenfunctions, because they are just 0. I don't see how they got the other two (+/-). Thanks in advance

I'm doing quantum mechanics with only a little experience in linear algebra. I've been working on eigenstates/values/functions/whatever for a couple days but still having a little trouble. Here's a question I had recently, and if anyone can do a quick check of my work and point me in the right...

My first question is, does any operator commute with itself? If this is the case, is there a simple proof to show so? If not, what would be a counter-example or a "counter-proof"? My second question has to do with the properties of an eigenvalue problem. If you have an operator Q such that...

I have been told that if we have two operators, A and B, such that AB = BA then this is equivalent with that A and B have a common base of eigenfunctions. However, the proof given was made under the assumption that the operators had a non-degenerate spectrum. Now I understand that one rather...

Homework Statement it's already separable, so it's an ODE function. X''+\lambda*X=0 0<x<1 X(0)=-2X(1)+X'(1)=0 Homework Equations The Attempt at a Solution this is a Sturm-Liouville eigenvalue problem. Now, I know how to solve it and everything, but I'm not sure with one...

I've been working my way through some basic quantum mechanics, and have gotten up to perturbation theory. It basically makes sense to me, but there's one thing that bothers me, and I was wondering if somebody could shed some light on it. The essential idea behind perturbation theory is that we...

in this book I have by G.L Squires. One of the questions is: if \phi1 and \phi2 are normalized eigenfunctions corresponding to the same eigenvalue. If: \int\phi1*\phi2 d\tau = d where d is real, find normalized linear combinations of \phi1 and \phi 2 that are orthogonal to a) \phi 1 b)...

I'm enjoying this introductory essay about quantum mechanics found here http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html and I have a question. About five-eighths of the way into it a wave function is given "at time t=0", \psi = \sqrt { \frac {2} {L} }...

Hi, I am having a lot of difficulty conceptually understanding what eigenfunctions and eigenvalues actually are, their physical meaning, i.e. what they represent, and how they interact. Would anybody happen to be able to explain them in relatively simple terms? I didn't know whether to put...

hi one of my past papers needs me to show that if 2 eigenfunctions, A and B, of an operator O possesses different eigenvalues, a and b, they must be orthogonal. assume eigenvalues are real. we are given \int A*OB dx = \int(OA)*B dx * indicates conjugate

Homework Statement This is problem 18.1 from Merzbacher. "The hamiltonian of a rigid rotator in a magnetic field perpendicular to the x-axis is of the form H=AL^2+BL_z+CL_y, if the term \that is quadratic in the field is neglected. Obtain the exact energy eigenvalues and eigenfunctions of the...

Homework Statement I'm given a standard form of Bessel's equation, namely x^2y\prime\prime + xy\prime + (\lambda x^2-\nu^2)y = 0 with \nu = \frac{1}{3} and \lambda some unknown constant, and asked to find its eigenvalues and eigenfunctions. The initial conditions are y(0)=0 and...

Use separation of variables/Fourier method to solve ut - 4uxx = 0, -pi<x<pi, t>0 u(-pi,t) = -u(pi,t), ux(-pi,t) = -ux(pi,t), t>0. ============================= What I got is that (n+1/2)2 are eigenvalues (n=0,1,2,3,...) and cos[(n+1/2)x)] is an eigenfunction. Instead of two sets of...

Homework Statement If there are any eigenvalues for the following integral operator, calculate them Kf(t) = \int_0^1 (1+st) f(s) \ ds The Attempt at a Solution I've tried making this into a differential equation, to no avail. I've also just tried solving the equation Kf(t) =...

Homework Statement Find the eigenfunctions and eigenvalues of the translation operator \widehat{T_{a}} Translation operator is defined as \widehat{T_{a}}\psi(x)=\psi(x+a) (you all know that, probably you just call it differently) Homework Equations The eigenvalue/eigenfunction equation is...

Homework Statement a) Show that the functions f=sin(ax) and g=cos(ax) are eigenfunctions of the operator \hat{A}=\frac{d^2}{dx^2}. b) What are their corresponding eigenvalues? c)For what values of a are these two eigenfunctions orthogonal? d) For a=\frac{1}{3} construct a linear...

My book on quantum physics says that if two Hermitian operators commute then it emerges that they have common eigenfunctions. Is that true? If A,B hermitian commuting operators and Ψ a random wavefunction then: [A,B]Ψ=0 => ABΨ=BAΨ If we assume that Ψ is B`s eigenfunction: b*AΨ=BAΨ...

I understand what is meant by the orthogonalilty of eigenfunctions... ...but what is measnt by the completeness of eigenfunctions?

Homework Statement What is meant by the completeness of eigenfunctions? The Attempt at a Solution I understand the AX(x)=BX(x) where A is the operator, B is the eigenvalue and X(x) the eigenfunction. I cannot find anywhere anything on what is meant by the completeness of...

Homework Statement Given X''(x) + lambda*X(x) = 0 X(0) = X'(0), X(pi) = X'(pi) Find all eigenvalues and eigenfunctions. Homework Equations Case lambda = 0 Case lambda > 0 Case lambda < 0 The Attempt at a Solution First case, X(x) = Ax + B but the function doesn't satisfy...

"Time" operator, or "Time" eigenfunctions We seem to define hermitian operators for momentum, position, energy ect., but we don't really talk about a "Time" operator, or "Time" eigenfunctions. What does time mean in standard quantum mechanics, and why is it different than the above dynamical...

Homework Statement For each of the following wave functions check whether they are eigenfunctions of the momentum operator, ie whether they satisfy the eigenvalue equation: \hat{p} \psi(x) = p\psi(x) with \hat{p} = i \hbar \frac{\partial}{\partial x} and p is a real number. For those...

can somebody help me with the solution of the following problems? Ques. Find the eigenfunctions and eigenvalues for the operators 1. sin d/d psi 2. cos(i d/d psi) 3. exp(i a d/d psi) 4. (d)square/d (x)square+z/x * d/dx

Hi people, I have this problem to do, and its only worth one mark which makes me think it must be easy, but our lecturer has not taught us very well at all, never explains anything. Anyway, there's a particle confined in an infinite potential well within the region -L/2 < x < L/2, where the...

Homework Statement With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1. 2. The attempt at a solution The first...

I know this is a common and important fact, so I've been willing to accept it, but this has always been something that has been "outside the scope" of my quantum lectures. Does anyone have reference for a proof?

Homework Statement Consider two identical particles of mass m and spin 1/2. They interact via a potential given by V=\frac{g}{r}\hat{\sigma}_1\cdot\hat{\sigma}_2 where g>0 and \hat{sigma}_j are Pauli spin matrices which operate on the spin of particle j. a) Construct the spin...

Homework Statement This is the original question: \frac{d^{2}y}{dx^{2}}-\frac{6x}{3x^{2}+1}\frac{dy}{dx}+\lambda(3x^{2}+1)^{2}y=0 (Hint: Let t=x^{3}+x) y(0)=0 y(\pi)=02. The attempt at a solution This might be all wrong, but this is all I can think of \frac{dt}{dx}=3x^{2}+1 so...

I know that the momentum eigenfunctions are of the form \phi = Ce^{ikx}, but how would we normalize them? We just get \int_{-\infty}^{\infty} C^2 dx = 1 which means that C is infintesimally small...

a) Consider a linear operator L with 2 different eigenvalues a1 and a2, with their corresponding eigenfunction f1 and f2. Is f1 + f2 also an eigenfunction of L? If so, what eigenvalue of L does it correspond to? If not, why not? b) Answer the same question as in part (a) but for the...

Homework Statement http://img508.imageshack.us/img508/7199/46168034nt3.jpg The Attempt at a Solution I'm just totally lost with this question. The theory just eludes me totally. Just how do you determine whether it is/isn't an eighenfunction of the linear momentum operator?

For Schrodinger's equation \frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi Solving to find that \psi = Aexp(ikx)+Bexp(-ikx) I am curious about the physical meanings of the two terms of the solutions. In solving a free particle encountering a potential barrier, In the...

[SOLVED] orthogonal eigenfunctions From Sturm-Louisville eigenvalue theory we know that eigenfunctions corresponding to different eigenvalues are orthogonal. For example, \Phi_{xx} + \lambda \Phi = 0 would be of Sturm-Louisville form (note: \Phi_{xx} represents the second derivative of...

Hi all, How do we find the eigenfunctions if we are given the wavefunction? I have a wave function at time = 0 and it is of a *free* particle and I need to find the wave function at a later time t. I did : \Psi(x,t)=\Psi(x,0)*e^{-iHt/hbar} then \Psi(x,t)=\sum_{n}(<\phi_{n}|\Psi(x,0)>...

Homework Statement Suppose that f(x) and g(x) are two eigenfunctions of an operator Q^{\wedge}, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q^{\wedge}, with eigenvalue q. Homework Equations I know that Q^{\wedge}f(x) = qf(x) shows...

I hope I catched the correct forum. Under http://en.wikipedia.org/wiki/LTI_system_theory e^{s t} is an eigenvalue. I don't really understand that the following is an eigenvalue equation: \quad = e^{s t} \int_{-\infty}^{\infty} h(\tau) \, e^{-s \tau} \, d \tau \quad = e^{s t}...

what is a eigen value and eigen function? i have read a lot abt it...i understand the math behind it.. what is its physical significance of it?

How can i prove that u(x)=exp(-x^{2}/2) is the eigenfunction of \hat{A} = \frac{d^{2}}{dx^{2}}-x^2