Consider the following wave functions (where a, b, k, and a are constants): (i) ψ(x) = a sin(kx) (iv) ψ(x) = ae-ikx (i) ψ(x) = b cos(kx) (v) ψ(x) = be+ikx (iii) ψ(x) = a/x (a) determine which of the above func- (vi) ψ(x) be-ax tions are eigenfunctions of the momentum operator, px, and identify the correspond- ing eigenvalue. (b) determine which of the above functions are eigenfunctions of the kinetic energy oper- ator, kx, and determine the corresponding eigenvalue. 4. evaluate the momentum expectation values of the free particle wavefunctions ex and etilx (where k is a positive constant). what do your results tell you about the direction in which free particles described by each of these wavefunctions are moving?

Ablock of mass m = 2.5 kg is attached to a spring with spring constant k = 710 n/m. it is initially at rest on an inclined plane that is at an angle of θ = 23° with respect to the horizontal, and the coefficient of kinetic friction between the block and the plane is μk = 0.19. in the initial position, where the spring is compressed by a distance of d = 0.16 m, the mass is at its lowest position and the spring is compressed the maximum amount. take the initial gravitational energy of the block as zero. a) what is the block's initial mechanical energy? b) if the spring pushes the block up the incline, what distance, l, in meters will the block travel before coming to rest? the spring remains attached to both the block and the fixed wall throughout its motion.

Which example has gravitational potential energy? a.a horse slowing down from a run b.an apple about to fall from a tree branch c.a bicycle being pedaled down a horizontal path d.beaker sitting on the floor

Which one of the following research methods can show cause and effect