For the next step we need to consider how to connect the wave function in region 1 to the one region 2. We could imagine several different possibilities. Two are shown in Figure 5.
Figure 5: You must consider how to join segments to form a wave function.
In Figure 5a the wave functions connect smoothly, while in Figure 5b 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.
First, we take a short side step and consider a very different wave function. Suppose that you had the wave function shown in Figure 5b.
Can you determine the probability of finding the electron at the boundary? Describe the electron's probability below:
You probably 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
Interpreting the wave function as a measure of the probability of finding an electron at a particular location forces a condition on the wave function. At boundaries where the potential energy changes, the wave function must make a smooth connection between its segments in the two regions. If the segments did not meet at the boundary, then each would give a different probability of detecting the electron there. Since both values of the wave function represent the electron at the same location, two different values for the probability at the same point would be meaningless.
Another possibility which will give only one value for the boundary is shown in Figure 6. The wave function is not smooth, but it has only one value at the boundary.
This type of wave function is also unacceptable. Schrödinger's Equation can be used to calculate the probabilities for different speeds of the electron. A wave function such as the one in Figure 6 will give two values for the speed probabilities. So we must reject it.
The only acceptable wave functions are smoothly connected at all boundaries. When creating wave functions we must adjust parameters to create a smooth connection at all boundaries. However, we cannot mess with the wavelength; it is determined by the kinetic energies. We can change the amplitude and the phase. Adjust these parameters in your wave function until the shape at the boundary is acceptable. Make a sketch of your result.
You have now completed step 3 in drawing wave functions, you adjusted the phase and amplitude on the wavefunction until it is smooth across the boundary.