To what atomic number have they been able to model accurately in the Standard Model? I'm an electrical engineer.  I took only a little bit of modern physics.  One sophomore level intro course and one course in solid state physics, I guess I took a course in astrophysics also.  But no physics major course in QM or particle physics.  In solid state, we did the particle in a box and a particle with a finite potential energy barrier.
Besides the particle in a box, and the sorta pendulum model (oscillator), and the particle and a barrier, I got to see Schrodinger's equation applied to the hydrogen atom and that was bitchy enough, mathematically.
So what atoms can the Standard Model, with its 25 fundamental constants (according to John Baez), accurately model?  i presume we've done helium and maybe lithium, i dunno.
What is the highest atomic number of atoms have they quantitatively modeled really good?
If this list includes carbon and oxygen (I wouldn't expect it to), then I have a question to ask regarding this Triple alpha process thing.
So, in addition, may I ask if anyone using the Standard Model, has been able to simulate in a computer the "cooking of elements" done in stars?  like a simulation of beryllium and helium getting cooked up into carbon (with an excited state)?
 A: A very sweeping overview of an answer - the standard model does not really describe any nuclear interactions. When you talk about the particle in the box and Schrodinger's equations, those are not what most people would describe as "the standard model". The Standard Model describes fundamental interactions - between quarks, gluons, leptons (electrons), and force-carrying bosons (W, Z, and Higgs). To describe the nucleus of an atom from first principles, we would need to at least be able to do so with the proton and neutron - which until recently was not possible, because we didn't know their exact particle content. Now, using lattice QCD (a particular approximate numerical technique) it is possible to model both protons, neutrons, and other simple baryons (See the paper link in the comments below). "Is lattice QCD part of the standard model?" is a question beyond the scope of my answer.
So that's a literal interpretation of your question, taking "The Standard Model" to be first principles + constants (25 of them, by one accounting). But if we add to this some particular approximations we can get nuclei, and if we add to this effective field theory + particle content which forms the basis for nuclear models (liquid drop and shell, for instance), we can accurately describe a significant portion of the periodic table.
What does "significant portion" mean? Well, that also depends what you mean by "accurately", but for instance, there is a package called "FLYCHK" (which I just learned about from Google) which can model plasmas of atoms up to Z=26 (Iron).
