Not quite ... classical orbits over extended time periods (T >> 1 orbital period) tends to require significant precision thanks to the "chaotic" aspects of the diffeqs. See [1] for some additional details.
Many orbital sims use high order methods, predictor-corrector type approaches.
As for "quantum physics of a simple object like a pair of spheres in a vacuum" ... I gotta say, as a (non-practicing) physicist, I haven't the foggiest notion about what you are thinking of. Do you mean the collective wave function of the entire ensemble of atoms? Or the valence/band electrons? Or something else?
Basically, we've been exploiting symmetries for many years to reduce insanely hard problems to less insanely hard problems. For crystalline material, we exploit periodic boundary conditions to reduce the size of the Hamiltonian to something reasonable. We still have to deal with O(N^3) and higher multicenter matrix elements like <phi_i(R1) | Operator_k(R2) | phi_j(R3)>. This is computationally hard for anything but small N (say 1000-10000).
Many orbital sims use high order methods, predictor-corrector type approaches.
As for "quantum physics of a simple object like a pair of spheres in a vacuum" ... I gotta say, as a (non-practicing) physicist, I haven't the foggiest notion about what you are thinking of. Do you mean the collective wave function of the entire ensemble of atoms? Or the valence/band electrons? Or something else?
Basically, we've been exploiting symmetries for many years to reduce insanely hard problems to less insanely hard problems. For crystalline material, we exploit periodic boundary conditions to reduce the size of the Hamiltonian to something reasonable. We still have to deal with O(N^3) and higher multicenter matrix elements like <phi_i(R1) | Operator_k(R2) | phi_j(R3)>. This is computationally hard for anything but small N (say 1000-10000).
[1] http://carma.astro.umd.edu/nemo/pitp/papers/gd2_s3.4.pdf