(5) Find the results of applying the operator 0 = 24 - x? on the function...
A function Ψ(x) is an eigenfunction of an operator A with an eigenvalue λ if Ay(x)-AW(x) where λ is some number. Show that the function ψ(x)-xe-rn is an eigenfunction of the operator A--x2. What is the eigenvalue?
Consider the following second order linear operator: 82 with Notice, that if instead of 3 we had 2 there, we would get a Legendre operator (whose eigenfunctions are Legendre polynomials). But nothing can be further from it than the operator above. The eigenvalue/eigenfunction problem, emerged in the analysis of vibrations of a particular quant urn liquid. An eigenvalue λ corresponds to an excitation mode of frequency Ω = V The eigenfunction ψ(r) would give a spatial profile of the deviation...
Consider the finite difference matrix operator for the 1D model problem u(/d2- f(x) on domain [0, 1] with boundary conditions u(0) = 0 and u(1) = 0, given by [-2 1 1-2 1 E RnXn h2 1 -2 1 This matrix can be considered a discrete version of the continuous operator d/da2 that acts upon a function(r). (a) Show that the n eigenvectors of A are given by the vectors ) (p-1,... , n) with components and with eigenvalues h2...
Consider the following wave function: y(x, t) = cos(kx - omega t). a. Show that the above function is an eigenfunction of the operator partialdifferential^2/partialdifferential x^2[...] and determine its eigenvalue. b. Show that the above function is a solution of the wave equation expressed as partialdifferential^2 y(x, t)/partialdifferential x^2 = 1/v^2 partialdifferential^2 y(x, t)/partialdifferential t^2, given the wave velocity is v = omega/k (where omega = 2 pi V and k = 2pi/lambda).
Find a differential operator that annihilates the given function. xex -x sin 8x +x A differential operator that annihilates xe*- xsin 8x + x4 is (Type the lowest-order annihilator that contains the minimum number of terms. Type your answer in factored or expanded form.)
x < n with BCs y(0)= 0 and y(z) 0. (1 point) Find the eigenvalues and eigenfunctions for y" = Ay on 0 Note that any constant times an eigenfunction is also an eigenfunction. In order to obtain a unique solution find (x) so that x) dx 1 First find the eigenvalues and orthonormal eigenfunctions for n 1, i.e., An, >,(x). For n 0 there may or may not be an eigenpair. Give all these as a comma separated list....
5. The function x< 0 0 < x < a ψ(x)-Ax(1-(x/a)] is an acceptable wavefunction for a particle in a one-dimensional space (x can take values between -oo and +oo) (a) Give two reasons why this is an acceptable wave function. (b) Calculate the normalization constant A. (c) Using the definition for the average of an observable "o" described by the operator "o": and to)
Give the result of operating on the function ЧС y ) g( [(-4 y )/2] ) with the operator [^(A)] -[(d2)/(dy2)]+16 y2 Submit Answer Tries 0/3 Is the function wC y )-e( [(-4 r2)/2) an eigenfunction of the operator [(A)]- -[(d2)/(dy2)]+16 y2? ("yes", "no") Submit Answer Tries 0/3 What is the eigenvalue of [^(A)] -[(d2)/(dy2)1+16 y2 operating on џ( y )-e( 1-4) /21) Submit Answer Tries 0/3
dc Let an operator K = et + x2 - 5 + log(x), that is, dy K[y] = et y - 5y + log(x) y. This K is a linear operator. + x2 dx O True O False
1. Using the method of images, find a Green's function for the Laplace operator in the quadrant r > 0, y > 0 which satisfies G(x, xo) on the boundaries 0 and y 0. 1. Using the method of images, find a Green's function for the Laplace operator in the quadrant r > 0, y > 0 which satisfies G(x, xo) on the boundaries 0 and y 0.