Is the function x2e−ax^(2) an eigenfunction of operator d2/dx2 − 4a2x2. If it is then what is the corresponding eigenvalue?
Is the function x2e−ax^(2) an eigenfunction of operator d2/dx2 − 4a2x2. If it is then what...
Convince yourself that function exp(-x2/2) is an eigenfunction of the operator (1/2)(-d2/dx2 + x2). Compute the corresponding eigenvalue. (We will see in class that this operator is the Hamiltonian for the harmonic oscillator, if one sets the mass, frequency, and the Planck's constant at 1.)
1. Show y = sin ax is not an eigenfunction of the operator d/dx, but is an eigenfunction of the operator da/dx. 2. Show that the function 0 = Aeimo , where i, m, and A are constants, is an eigenfunction of the angular momentum operator is the z-direction: M =; 2i ap' and what are the eigenvalues? 3. Show the the function y = Jź sin MA where n and L are constants, is an eigenfunction of the Hamiltonian...
6. Is the operator Hermitian? d2 dx2
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?
For each pair, determine if the wavefunction is an eigenfunction of the operator listed. If the wavefunction is an eigenfunction of the operator, clearly identify the corresponding eigenvalue. (a) = + + = Aerky a pika w piks z (b) 3 = tan(ka) 4 = sin(km) Note: tan(x) = 3
Circle each of the three listed functions that is an eigenfunction of the specified operator For each eigenfunction, write the eigenvalue below the function.
Consider a wave function that is an eigenfunction of L2 with the eigenvalue 42h2. What are the possible outcomes of a measurement where we measure the z-projection of the angular momentum operator?
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...
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
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...