To have exact five bound states in a finite well, the width and depth of the well need to satisfy certain conditions, including symmetric wave functions and normalizability.
The possible combinations of depth and width required for a finite well to have exact five bound states can be determined by solving the time-independent Schrodinger equation for the infinite square well. To find the allowed energy states, the wave functions need to be symmetric about the bottom of the potential well and normalizable. These conditions ensure that the probability density is finite when integrated. Therefore, the width and depth of the well need to satisfy these conditions for the system to have five bound states.