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 continuous across x = 0). Then, use these solutions to find the currents. Note that the solution for x > 0 changes character depending on whether the particle energy E is smaller or greater than V0.]
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 continuous across x = 0). Then, use these solutions to find the currents. Note that the solution for x > 0 changes character depending on whether the particle energy E is smaller or greater than V0.]
Trending now
This is a popular solution!
Step by step
Solved in 10 steps with 11 images
