I've been working on trying to understand part of Quantum Mechanics. There are only certain simple systems for which it is possible to solve the maths of Schrodinger's Equation. Ordinary maths won't solve more complex problems so the idea of Perturbation Theory is to change the energy by a small amount - give it a nudge - do some maths on that and come up with a solution written in terms of the original eigenstates of the system, because it won't have changed much. To work out the Brillouin-Wigner version of it (as opposed to the Rayleigh-Schrodinger version that we have looked at in the past) I've been using this document
http://www.phys.ufl.edu/~kevin/teaching/6646/04spring/bw.pdf What I decided to do was to write the ket-vectors as if they were 3-dimensional real vectors and do matrix work based on them to get a feel for what the symbols actually mean. The ket vectors |n> are orthonormal eigenstates - in other words, if you do the inner product of one with another, the answer is zero. Each has its dual, the bra
The inner product of a ket with a bra turns out to be a matrix. The document calls this one P1 and calls it a PROJECTION OPERATOR. (Remember that operators are matrices)
The second projection operator is Q1. It is the identity matrix - P1. It says that P1 and Q1 are complementary.
I have also proved that Q1=P2+P3 so for any number of n, Pn=sum of Qm, provided you miss m=n.
Then I tackle the idea that the Hamiltonian is also the sum of these projection operators multiplied by a factor. He calls this the SPECTRAL REPRESENTATION of the Hamiltonian - in other words, breaking the whole down into the bits that it is made of like white light is broken down into the component wavelength colours.
I've got as far as the commutator proof. The Hamiltonian commutes with the other projection operator Q.
