Eigenfunctions Definition and 181 Threads

  1. B

    Find the complete orthonormal set of eigenfunctions of the operators B-hat

    Homework Statement A bound quantum system has a complete set of orthonormal, no-degenerate energy eigenfunctions u(subscript n) with difference energy eigenvalues E(subscript n). The operator B-hat corresponds to some other observable and is such that: B u(subscript 1)=u(subscript 2) B...
  2. B

    Eigenfunctions and hermitian operators

    Hi. I'm just a bit stuck on this question: Write down two equations to represent the fact that a given wavefunction is simultaneously an eiigenfunction of two different hermitian operators. what conclusion can be drawn about these operators? Im not quite sure how to start it. Thanks!
  3. R

    Discrete eigenvalues and their eigenfunctions

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

    Demonstrating Orthonormality of Eigenfunctions for De Cero-Spin Field

    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?
  5. T

    Solving Eigenvalues and Eigenfunctions of Hamiltonian

    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...
  6. C

    Eigenvalues and eigenfunctions

    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...
  7. M

    Dirac Delta from Continous Eigenfunctions

    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...
  8. M

    Orthogonal Eigenfunctions (Landau Lifshitz)

    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...
  9. Z

    Expansion of a wave fuction in energy eigenfunctions

    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...
  10. T

    Common sets of eigenfunctions in angular momentum

    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...
  11. K

    Angular momentum and eigenfunctions

    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...
  12. M

    How Do You Prove the Eigenfunctions of Angular Momentum?

    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.
  13. jegues

    Eigenvalue/Orthogonal Eigenfunctions

    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...
  14. W

    Eigenfunctions from eigenvalues unsure

    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...
  15. I

    How are these eigenfunctions obvious (by inspection)?

    [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
  16. M

    How Do You Transform Eigenvalues into Eigenfunctions in Quantum Mechanics?

    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...
  17. C

    Commutation and Eigenfunctions

    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...
  18. J

    Proof that commuting operators have a shared base of eigenfunctions

    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...
  19. R

    ODE/PDE- eighenvalues+ eigenfunctions

    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...
  20. C

    Energy eigenfunctions in time-independent perturbation theory

    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...
  21. O

    Fourier Series from Eigenfunctions

    Homework Statement "Using the eigenfunctions for the Hamiltonian of an infinite square-well potential defined over[-1,1] in the standard, dimensionless setting, construct Fourier series representation of the following functions..." the functions are e^(-100x^2), e^(-5x^2), e^(-x^2) It also...
  22. O

    Are Wigner Functions eigenfunctions of J^2 and Jz?

    Homework Statement I have a question related to representation of rotation operator R in the basis spanned by the eigenvectors of J2 and Jz. I am studying from Quantum Mechanics by Zettili. The development of Wigner D-matrix and its elements Dj (Wigner functions) is clear. But the book goes on...
  23. M

    Question on orthogonal eigenfunctions

    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)...
  24. snoopies622

    Understanding Momentum Eigenfunctions in Quantum Mechanics

    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} }...
  25. K

    Eigenfunctions and Eigenvalues

    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...
  26. I

    Show eigenfunctions are orthogonal

    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
  27. I

    Angular momentum eigenfunctions

    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...
  28. H

    Bessel Functions - Eigenvalues + Eigenfunctions

    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...
  29. K

    Is sin[(n+1/2)x] also an eigenfunction for this problem?

    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...
  30. K

    Eigenfunctions of an Integral Operator

    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) =...
  31. M

    Eigenfunctions of translation operator and transposed operator property proof

    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...
  32. H

    Is this correct? (eigenfunctions)

    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...
  33. JK423

    Commuting operators => Common eigenfunctions?

    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...
  34. A

    Completeness of Eigenfunctions

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

    What is meant 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...
  36. J

    Finding eigenvalues and eigenfunctions

    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...
  37. J

    Time operator, or Time eigenfunctions

    "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...
  38. P

    Eigenfunctions of Laplace operator on a squere with finite differences

    Homework Statement I want to numerically compute the eigenfunctions and eigenvalues of Laplace operator on a square with Dirichlet boundary conditions (i.e. u|_{\partial}=0). Exact analytical solutions are well known sinusoidal modes: u_{m,n}(x,y)=\sin(k_mx)\sin(k_ny) , where...
  39. T

    Quantum Mechanics and Eigenfunctions Checks

    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...
  40. R

    Problems related to eigenfunctions and eigenvalues

    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
  41. J

    A (probably simple) quantum problem - energy eigenfunctions?

    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...
  42. J

    Orthogonality of eigenfunctions with continuous eigenvalues

    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...
  43. W

    Proof that the eigenfunctions of a self-adjoint operator form a complete set.

    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?
  44. C

    [PhD Qualifier] Spin eigenfunctions

    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...
  45. P

    Bounday-Value Problem: Eigenvalue and Eigenfunctions

    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...
  46. S

    Spin eigenfunctions for two particles

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

    Normalizing the Momentum Eigenfunctions

    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...
  48. E

    Are Sums and Differences of Eigenfunctions Also Eigenfunctions?

    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...
  49. N

    Degenerate Eigenfunctions of Hamiltonian

    Homework Statement I found the following problem in two places.But I doubt the first one is wrong. Let \ u_1(\ x ) and \ u_2(\ x ) are two degenerate eigenfunctions of the hamiltonian \ H =\frac{\ p^2 }{2\ m }\ + \ V (\ x ) Then prove that \int...
  50. A

    Momentum eigenfunctions with periodic boundary conditions

    Homework Statement A particle of mass m is confined to move in one dimension. its wavefunction is periodic with period L\gg 1 - i.e. periodic boundary conditions are imposed. a)Determine the eigenfunctions and eigenvalues of momentum. Normalise the eigenfunctions on the interval [0,L)...
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