Russell's Blog

Today, we started out by examining the zeroth and first order perturbational solution to the two-electron problem (H^-, He, Li⁺, and so on). The perturbation is fairly simple (it's just repeated cases of the Coulomb potential), but for multiple electrons, results in a fairly nasty integral over six dimensions. Much of the first half of the class was devoted to a technique for solving this integral. I wrote down the technique, but I don't anticipate needed to learn it. The solution is in the book, and the technique is much too complicated to appear in a test question. It looked useful, so I'll study it later when I have more time.

We then calculated the energies of various two electron systems. It was noted that the perturbational approach seems to converge very slowly, and that the second order calculation would be a nightmare. Onwards to variational methods!

We then rushed through an introduction to the variational method. The variational method can be thought of as a generalization of perturbation theory, but only for the ground state. It works as follows: You assume a priori that there is an eigenstate of the Hamiltonian with a minimum energy, and you make a guess about what it might be (this only works if your guess is "reasonable"). The "true" ground state will be the eigenfunction that minimizes the energy. So, you guess its form, and tweak whatever parameters it has until you find the minimum energy. If you guessed the exact form of the function, and you do enough tweaking, in theory you can find the exact eigenfunction.

A little more formally, you know that each eigenstate spans all of function space. So, the ground state will span all of function space. So, whatever you guess, it must be a linear combination of the eigenstates. So, you write the trial function as a sum over the eigenfunctions with some coefficient for each one (you can always do this). Each eigenstate corresponds to an energy eigenvalue, so you can replace the sum over eigenfunctions with a sum over their corresponding energies (and their coefficients). As you vary your trial functions and evaluate the sum, the lowest possible value you can find will be the case where the coefficient of the ground state is 1, and 0 for all the others.