Currently we are working on finding solutions for the one-dimensional time-independent Schrodinger equation where we have a particle, in our case a nucleon, in a finite potential well. We arrived at the following relationship to determine the potential that would give rise to exactly $N$ bound states in a well of width $L$ although we were more given this rather then shown where this comes from.
I have done some reading on this relationships comes from but I am still slightly fuzzy on it so any clarification would be helpful. However, my main question is how to find $L$ given some potential and 7 bound states. If I isolate $L^2$ in the above equation I arrive at the following.
When I plug in our values ($V_0 = 40\space MeV$, $N = 7$, and $m = 938.9265\space MeV$) my solution for $L$ comes out in units of seconds. I was informed that I could arrive at an expression with $mc^2$ in it which would return a solution in units of meters however I am not sure how to go about this. I think my issue arises from the fact that I'm not fully aware of where this relationship comes from but it may just be that I am missing something obvious so any input would be appreciated.