Suppose I wanted to develop a potential describing the interaction between two lithium atoms. One way to do this is to calculate the energy between the two lithium atoms for various distances and make a plot of $E$ vs $x$. Using a classical approximation, the force between the two as a function of $x$ is then $F = -\frac{d E}{d x}$. I could then use this in a molecular dynamics simulation (well, of only two atoms anyways).
However, if I really wanted to accurately characterize the system, instead of numerically solving $\hat{H}\psi = E\psi$ for various Li-Li distances, I would use $\hat{H}\psi = i\hbar \frac{\partial}{\partial t}\psi$ and evolve the Li-Li system through time. I could then create an operator $\hat{A}$ whose expectation value is the distance between the nuclei of the two lithium nuclei as a function of time.
My question: if I were to compare the plot of $x$ vs $t$ for my potential using the first method with the plot using the second method, how much would they differ?
I suppose I can always try it out, but that will take a lot of time, and this is more of just a curiosity thing.
