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Particles with energy E, are incident from the left, on the step-potential of height V0 =...

Particles with energy E, are incident from the left, on the step-potential of height V0 = 2E as shown: a. What are the wave numbers in the two regions, 1 k and 2 k , in terms of E? b. Write down the most general solutions for the Schrodinger Equation in both regions? Identify, with justification, if any of the coefficients are zero. c. Write down the equations that result for applying the boundary conditions for the wave functions at the step. Solve these equations to express each of the coefficients in terms of one of them. d. Derive expressions for the probability density in each region. e. Sketch a plot of the probability density, and identify maximum and minimum values for the probability density and their locations on the x-axis. f. What is the average value of the position for the particle considering only the region to the right of the step?

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Answer #1

Let us consider the left side with no potential as region 1 and the right side with the potential step as region 2.

a. The wave numbers are given respectively as

b.The most general solution in both regions respectively are given as

As the wave function in region 2 must vanish when x is very large, so we have C = 0.

c. Boundary condition states that (1.) wave function must be same at the boundary and (2.) wave function must be continuous at the boundary, i.e., its first derivative must be same.

Solving the two equations we got from the two boundary conditions, we get,

d.

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