A particle of mass m moves inside a potential energy landscape U (2) = X|2| along the z axis. Part (a) What are the units of the constant X? Part (b) If the particle has kinetic energy me at the origin at z = 0, where are the classical turning points of the motion?
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- 2) Consider a particle in a three-dimensional harmonic oscillator potential V (r, y, z) = 5mw²(r² + y² + z®). The stationary states of such a system are given by ntm(r, y, z) = vn(x)¢r(y)v'm(2) (where the functions on the right are the single-particle harmonic oscillator stationary states) with energies Entm = hw(n +l+m+ ). Calculate the lifetime of the state 201.The difference of the scalar potential squared and the modulus of the vector potential squared, Φ2 - |A|2, is Lorentz invariant (a Lorentz scalar). why the statement true?A potential function is shown in the following with incident particles coming from -0 with a total energy E>V2. The constants k are defined as k₁ = 2mE h? h? k₂ = √√2m (E - V₁) h² k3 = √√2m (E - V₂) Assume a special case for which k₂a = 2nπ, n = 1, 2, 3,.... Derive the expression, in terms of the constants, k₁, k2, and k3, for the transmission coefficient. The transmis- sion coefficient is defined as the ratio of the flux of particles in region III to the inci- dent flux in region I. Incident particles E>V₂ I V₁ II V2 III x = 0 x = a
- Consider 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…c) How does the classical kinetic energy of the free electron compare in magnitude with the result you obtained in the previous part?Consider a particle of mass m moving in 1-dimension under a piecewise-constant po- tential. In region I, that corresponds to x 0. In region II, that corresponds to x > 0 the potential energy is V1(x) = 0. The particle is shot from = -∞ in the positive direction with energy E > Vo > 0. See the figure in the next page for a representation of V(x) as a function of x. Also shown in the graph (green dashed line) the energy E of the particle. (a) Which of the following functions corresponds to the wavefunction 1(x) in region I? (a1) Aeikiæ + Be-iki¤ ; (а2) Ае\1 + Bе-кӕ (a3) Aeikræ (а4) Ве- кта (b) Which of the following functions corresponds to the wavefunction 1(x) in region II? (b1) Сеkп* + De-ikr (62) C'e*I1* + De-*1¤