Can stationary states describe a free particle incident on a potential step?

[Sirius24]

Homework Statement

dJ(x,t)/dx = -dY^2 / dt , where y is the wave equation, and the d's represent partial derivatives. I want to make an assumption that I can describe the wave equation as a stationary state, so my question is the following:

What is the definition of a stationary state and can it be used to describe a free particle incident on a finite potential step from the left? This is not a specific question for the problem, but I need to know in order to make an assumption to solve it.

 

[Villhelm]

I'm not that clued up on this myself, but stationary states appear to be similar to solving for standing waves on a string.

The assumption is made that the wavefunction y(x,t) = Bsin(kx +/- wt) (B arbitrary, k the wave number, w the angular frequency, x position and t time) and the wave is bouncing back and forth within a potential well "box". The stationary states correspond to the particular circumstances that produce standing wave patterns.