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?
6.2(a) Complex functions of the form eikx can be used to model the wavefunctions of particles in a linear accelerator. Show that any linear combination of the complex functions e2ix and e−2ix is an eigenfunction of the operator d2/dx2 and identify its eigenvalue. 6.2(b) Functions of the form sin nx can be used to model the wavefunctions of electrons in a carbon nanotube. Show that any linear combination of the functions sin 3x and cos 3x is an eigenfunction of...
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