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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.

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Obviously, the classical simulation will be an approximation to the quantum-mechanical expectation value; for a harmonic potential it becomes in fact exact as shown by the Ehrenfest theorem. Would you like to know some upper bound on the deviations, some pertubative way of making it more precise, or...? – leftaroundabout Oct 11 '13 at 0:18

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