First we add a perturbation to the original Hamiltonian Ho. Most other methods I've seen use V instead of H1 because the perturbation is thought to be a potential energy from some external field.
Thursday 1 November 2018
Brillouin-Wigner perturbation theory part 2
What I'm doing here is trying to do the workings to prove the results in the document linked to yesterday's post -seeing if I can follow the workings and understand it.
First we add a perturbation to the original Hamiltonian Ho. Most other methods I've seen use V instead of H1 because the perturbation is thought to be a potential energy from some external field.
Now here I have one problem. I thought En was a number - an eigenvalue that tells us the measurable energy - but it is having the original Hamiltonian Ho subtracted from it. Perhaps it is En x identity matrix. We had something similar in yesterday's workings where the Hamiltonian was given as the sum of eigenvalues En x Pn. Pn is a matrix. He does say at the top of the document that the curly small Es are eigenvalues and En is distinct so my hunch may well be correct.
Again below I'm not quite sure where the Pn in the summation for Ho on the bottom disappears to.
I can follow this bit. It assumes that perturbed and unperturbed eigenkets are in the same direction, but I think that is the idea with the small perturbation. The energy values change but not the ket direction.
So we finally get to write perturbed state in terms of the unperturbed, which confirms my last comment.
More maths to follow to work out the perturbed energies En.
First we add a perturbation to the original Hamiltonian Ho. Most other methods I've seen use V instead of H1 because the perturbation is thought to be a potential energy from some external field.
