Find the Z Transform for the following function: f(t)=e^(-ct) cos(2πt/3)u(t)
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Find the Z Transform for the following function:
f(t)=e^(-ct) cos(2πt/3)u(t)
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- Consider a one-dimensional particle which is confined within the region 0≤x≤ a and whose wave function is u(x, t) = sin (x/a) exp (-iwt). (a) Find the potential V(x). (b) Calculate the probability of finding the particle in the interval a/4 ≤ x ≤3a/4. fictr fidt t 2it 92.niz Isinn: = n fidar 1). (b(6\xh; fingen-like wan-like ction nitConsider a particle of mass m moving in one dimension with wavefunction $(x) 2 2πα sin for VI and zero otherwise. Is the wave function an eigenfunction of p? If so, what is the eigenvalue?It's a quantum mechanics problem.
- 40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow theConsider the one-dimensional step-potential V (x) = {0 , x < 0; V0 , x > 0}(a) Calculate the probability R that an incoming particle propagating from the x < 0 region to the right will reflect from the step.(b) Calculate the probability T that the particle will be transmitted across the step.(c) Discuss the dependence of R and T on the energy E of the particle, and show that always R+T = 1.[Hints: Use the expression J = (-i*hbar / 2m)*(ψ*(x)ψ′(x) − ψ*'(x)ψ(x)) for the particle current to define current carried by the incoming wave Ji, reflected wave Jr, and transmitted wave Jt across the step.For a simple plane wave ψ(x) = eikx, the current J = hbar*k/m = p/m = v equals the classical particle velocity v. The reflection probability is R = |Jr/Ji|, and the transmission probability is T = |Jt/Ji|. You need to write and solve the Schrodinger equation in regions x < 0 and x > 0 separately, and connect the solutions via boundary conditions at x = 0 (ψ(x) and ψ′(x) must be…3