Step 3: At the Boundary A

For the next step we need to consider how to connect the wave function in region 1 to the one in region 2. We could imagine several different possibilities. Two are shown below.

(a)


(b)

 

In (a) the wave functions connect smoothly, while in (b) they do not connect. Most people immediately think that (a) looks better. We will now discuss why “looks better” is correct in terms of the physics. To understand why, we need to look at the probability interpretation of the wave function and how that interpretation affects the wave function when the potential is changing.
 
Can you determine the probability of finding the electron at the boundary in Figure (b)?
 
You should find this question difficult. At the boundary the wave function has two different values. Thus, we cannot uniquely state the probability of an electron being detected there. Therefore, we must reject this type of wave function because we cannot use it for determining probabilities. (We recommend that you draw a big X through (b). Click ing here will remind you that it is not acceptable.