# Deriving an ODE for the potential energy surface (curve) of a particle system in the classical and quantum framework

I would imagine if I have a system of $$N$$ atoms, each having atom specific parameters, given the position of each atom at a time $$t$$, I could calculate the potential energy of the system (only a single point though) P(t). However if I assume an initial configuration of the atoms at time $$t$$, can I write an ODE for the potential energy surface (curve) P(t)?

If I took a quantum perspective, lets say, where the position of these atoms could be inferred by the wave function, can I also create the random ODE which governs the potential energy curve P(t)? Given a particular wave function of course. Then I would expect I can look at E(P(t)) to have the expected energy surface?

I feel like these formulations have already been posed, but I can't find it in the literature. References would be greatly appreciated if these two problems have been posed before.

Thank you.

There is no simple ODE for $$P(t)$$. The discipline that deals with calculating your $$P(t)$$ is called molecular dynamics. A book on that topic will probably answer most of your questions. Here are the basics:
First some notation and convention: The potential energy is usually denoted $$V$$, so I will write $$V(t)$$ instead of $$P(t)$$. The expectation value is usually denoted $$\langle \rangle$$ so I will write $$\langle V(t)\rangle$$ instead of $$E(P(t))$$.
$$V(t)$$ is not the potential energy surface. The potential energy surface is defined as $$V(R_1,R_2,...,R_{3N})$$ where $$R_i$$ are the positions of the nuclei of the atoms (x,y and z; so there are $$3N$$). If you can calculate $$V(R_1,R_2,...R_{3N})$$ you can solve differential equations to get $$R_i(t)$$, which then give you $$V(t)$$.
In the classical case, the nuclei (mass $$M_i$$) follow Newton's equation $$\frac{\partial^2R_i}{\partial t^2} = \frac{F_i}{M_i}$$ with $$F_i = -\frac{\partial}{\partial R_i}V(R_1,R_2,...,R_{3N})$$. Solving this differential equation gives you all $$R_i(t)$$ which you can insert into $$V(R_1,R_2,...R_{3N})$$ to get $$V(t)$$.
In the quantum case, the differential equation you need to solve is the time-dependent Schrödinger equation $$i\frac{\partial \Psi(R_1,R_2,...,R_{3N},t)}{\partial t} = \left( \sum_i-\frac{1}{2M_i}\frac{\partial^2}{\partial R_i^2} + V(R_1,R_2,...,R_{3N}) \right) \Psi(R_1,R_2,...,R_{3N},t)$$. Solving this gives you $$\Psi(R_1,R_2,...,R_{3N},t)$$ and then you calculate the expectation value as $$\langle V(t) \rangle = \int \int \cdots \int \Psi^* V \Psi dR_1 dR_2 \cdots dR_N$$ .