Show that the classical cross section for elastic scattering of point particles from an infinitely massive sphere of radius R is isotropic.
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Show that the classical cross section for elastic scattering of point particles from an infinitely massive sphere of radius R is isotropic.
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- Suppose we had a classical particle in a frictionless box, bouncing back and forth at constant speed. The probability density of the position of the particle in soma box of length L is given by: 0 ans-fawr (7) p(x)= 0 x L a. Sketch the probability density as a function of position b. What must A be in order for p(x) to be normalized? Remember that you are welcome to use resources to solve integrals such as Wolfram Alpha, a table of integrals etc.P-4 Please help me with this problem very clearly with step by step explanation.Book: Classical Dynamics of Particles and Systems Topic: Calculus of Variations Please answer in a detailed solution. For study purposes. Thanks.
- 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…consider an infinite square well with sides at x= -L/2 and x = L/2 (centered at the origin). Then the potential energy is 0 for [x] L/2 Let E be the total energy of the particle. =0 (a) Solve the one-dimensional time-independent Schrodinger equation to find y(x) in each region. (b) Apply the boundary condition that must be continuous. (c) Apply the normalization condition. (d) Find the allowed values of E. (e) Sketch w(x) for the three lowest energy states. (f) Compare your results for (d) and (e) to the infinite square well (with sides at x=0 and x=L)In the pair production process, photon energy gets converted to particle/antiparticle pairs. Imagine a single photon in free space, turning into one electron and one positron (antielectron), each with mass mec?. Assume for simplicity both particles move together with equal momenta in the same direction as the original photon as shown, and the photon disappears. Prove this can't happen!
- A particle with the velocity v and the probability current density J is incident from the left on a potential step of height Uo, that is, U (x) = Uo at r > 0 and U(x) = 0 at r 0.(d) Prove that for a classical particle moving from left to right in a box with constant speed v, the average position = (1/T) ff x(t) dt = L/2, where T L/v is the time taken to move from left to right. And = : (1/T) S²x² (t) dt L²/3. Hint: Only consider a particle moving from left x = 0 to right x = L = and do not include the bouncing motion from right to left. The results for left to right are independent of the sense of motion and therefore the same results apply to all the bounces, so that we can prove it for just one sense of motion. Thus, the classical result is obtained from the Quantum solution when n >> 1. That is, for large energies compared to the minimum energy of the wave-particle system. This is usually referred to as the Classical Limit for Large Quantum Numbers.