mikfig said:
Your comment that our current math can't solve the wavefunctions of higher atoms really piqued my interest. What is the current bleeding edge of mathematical physics/math right now?
Well, there's a difference between not being able to solve something
analytically, which basically means having a mathematically exact expression (with no consideration of whether or not that expression can be evaluated easily or not), and being able to solve something
numerically, i.e. an approximate solution, although one that can usually be made arbitrarily exact.
In practice it's not necessarily very important, which is well-illustrated by Sundman's solution to the classical three-body problem (which is related but a bit different from the quantum-mechanical three-body problem, as in Helium, which is still unsolved) The thing is, although Sundman's solution was analytic, it took the form of an infinite sum. Which means that any real calculation would (in this case) be approximate. Worse, this sum converges very slowly, which means you have to calculate very many terms to get a certain level of accuracy. Solving the same problem
numerically is very simple, however. So from the standpoint of calculating something in practice, the analytical solution isn't useful.
We've been solving the equations involved numerically since the start of quantum mechanics. The ground-state energy of Helium was calculated to three decimal places by Hylleraas (using a hand-cranked mechanical desk calculator) in 1927, and to six places in 1929. And the development of better ways to do this has continued since; the Nobel prize in 1998 was given to Kohn and Pople for having developed such methods. Today we can calculate Helium's energy to within experimental error. Small molecules with light atoms can be calculated to within "chemical accuracy" as well. The largest systems we can solve to useful level of accuracy are about 100-200 (first and second-row) atoms.
In short, that's what quantum chemistry is about: Solving the equations, and developing better ways of doing it. So it's at the intersection of applied math, theoretical physics, computer science and chemistry.
Oh, and if we did solve these equations, would we have to change our current system of electron orbitals? Like each atom has its own "kind" of s, p, d, f, g orbitals?
Nope. Orbitals come from making an approximation that makes the equations soluble (although you still have to do it numerially). In fact, I already mentioned it - by assuming that electrons only see the average repulsion from the other electrons. As you can see from the helium example it's a good approximation (2% error in energy). It's not good enough to be able to calculate specifics of chemistry, but it's good enough to get a qualitative understanding of what's going on, which is what your average chemist uses orbitals for. It's not
exact in the strictest sense, but that's not a problem.
What other kind of major improvements would solving these equations bring?
Being able to solve these equations quickly and exactly would means we'd be able to know every single chemical property of anything we'd want to calculate. Chemists wouldn't need to do experiments anymore, we'd just be able to calculate everything. That's not about to happen though, although quantum chem is becoming more important and useful by the day.
From the pure-math standpoint, the many-body problem isn't really
a problem; it's a
source of problems!