How Does Particle Energy Affect Wave Functions Across a Potential Step?

[omiros]

Homework Statement

A particle with mass m moving in the positive x -direction (i.e. from left to right) is incident on a potential step of height V₀ at x = 0 so that the potential experienced by the particle is;
V(x) = 0 for x < 0 and V(x) = V₀ for x ≥ 0

Homework Equations

Determine the time-independent wave function for the particle in the case where the particle energy, E, is greater than V₀. This case corresponds to the solution for an ‘unbound’ particle (E > V₀). Write your wave functions using complex notation; let the amplitudes of the incident, reflected and transmitted waves be C_I, C_R and
C_T respectively. Define the wavenumber, k, in the region x < 0 and the wavenumber
k' in the region x ≥ 0 .

The Attempt at a Solution

ψ(x) = C_I*e^ikx + C_R*e^-ikx for x < 0 (is probably the first part of the equation).

My main problem is what to do with the second one, as the particle is constantly 'under the influence' of the potential V₀ and at the same time I have to find C_T when the wave has not been exactly transmitted so the equation can't just be C_T*e^ikx(in my point of view)

 

[BruceW]

you are right for the case x<0. And for the case x>0, yes, the particle is essentially in a constant potential V₀. In the region x>0, the particle has been transmitted. True, the equation is not C_T*e^ikx. Try using the Schrödinger equation to find what the wavefunction should look like for x>0.