Is there something similar to time step increments for simulating a system of multiple particles in quantum mechanics?

In Newtonian Physics it is possible to simulate a system of multiple electrically charged bodies, that all exert an electric force on each other, using time step increments. For instance given the initial position vector, velocity vector vector, mass, and electric charge of each body in the system, acceleration vector of each body in the system can be calculated. Next the initial position vector of each body, the initial velocity vector of each body, the initial acceleration vector of each body, and the length of the time step can be used to calculate the new position vector, and velocity vector for each body using the equations of motion for a single body accelerating at a constant rate, but with time passed replaced with the size of the time step. Finally the process can be repeated for an indefinite number of time steps.

For this method of simulating a system of multiple bodies in Newtonian Physics the smaller the time step increment the more accurate the simulation is. This method for simulating a system of multiple particles works in any number of dimensions, and for any force law $f(r)$ with $r$ being distance, and $f(r)$ being a function of distance. Also this method of simulating multiple bodies that interact with each other works for Newtonian Gravity. Using this method it is possible to approximate the shapes of the orbits of bodies in a bounded system, and get an idea as to rather there are any stable orbits in the system.

It is possible to simulate a multiple body system of electric charges in relativity using either time step inrements or proper time step increments. In the case of proper time step increments the four velocity vector and four acceleration vector would be used as well as the rest mass of each object.

As I understand it the schrödinger equations are to quantum mechanics what equations of motion are to classical physics.

Some examples of bounded systems would be a hydrogen atom as the electron is bounded to the nucleus, the nucleus of an atom as the protons and neutrons are bounded to each other, and hadrons as the quarks are bounded to each other.

Is there something similar to using time step increments for systems systems of multiple particles such as atoms, atomic nuclei, and collections of quarks, which can approximate the shape of these systems, and rather or not the systems are stable?

• Google time dependent Hartree Fock – Lewis Miller Sep 10 '18 at 13:12