Step 3: At the Boundary B

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 the figure below. The wave function is not smooth, but it has only one value at the boundary.
 

Wave functions with kinds are also not acceptable.

 

This type of wave function also gets a big X. Click here. Schrödinger’s Equation can be used to calculate the probabilities for the speeds of the electron. A wave function such as the one here will give two values for the speed probabilities at the boundary. 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 Wave Function Sketcher in your wave function until the shape at the boundary is acceptable. Print your result.
You have now completed the third step in creating a wave function.
 
Step 3. Adjust the phase and amplitude of each part of the wave function until it is smooth across the boundary.