How Do Boundary Conditions Affect Solutions in a Half-Infinite Square Well?

[hootna]

Hey, I'm considering a square well which is finite on one side (left) and infinite on the other (right).

So the wave function is:

Left-most region: Ae^(ikx) + Be^(-ikx)
Inside the well: Csin(lx) + D(cos(lx))
Right-most region: 0

where k and l are known.

The problem is with boundary conditions: On the left, we have both (continuity and continuity of derivative), but on the right we only have one (continuity). So we have 3 equations to solve for 4 variables... what gives?

If the context helps, I'm trying to figure out the probability that particles entering from the left in region 1 will be reflected back (i.e., the ratio of the probability flux in region 1 traveling to the left to the incident flux in region 1 traveling to the right).

Any help would be greatly appreciated.

Thanks,
Thomas.

 

[jtbell]

Put the infinite wall at x = 0. Now, what can you say about the wave function inside the well?

 

[Dr Transport]

Think about this, make a mirror image of the well and think about the finite well. Draw the first few wave functions and really think hard about what you are drawing and the answer will pop out at you.