QMIIPertTheoryNote2011 - Physics 4143: Some remarks on the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 4143: Some remarks on the two-state coupling problem and perturbation theory Brian Kennedy, School of Physics, Georgia Institute of Technology, Atlanta, Georgia, 30332-0430 Consider a two state quantum mechanical system with unperturbed Hamiltonian H and eigenvectors and eigenvalues given by H ψ α = ϵ α ψ α and H ψ β = ϵ β ψ β , respectively. The system is subject to a perturbation V and the total Hamiltonian becomes, in the matrix representation with ψ α = ( 1 ) T and ψ β = ( 1 ) T , H = H + V = ( ϵ α ϵ β ) + ( V αα V αβ V βα V ββ ) . We can rewrite this as follows H = H + V = ( ϵ α + V αα ϵ β + V ββ ) + ( V αβ V βα ) . This is a simple 2 X 2 matrix and I assume everyone can find its eigenvalues and eigenvectors. I want to approach the diagonalization of H in a less direct way that reveals the structure of the problem and which simplifies the explicit calculation of the eigenvectors. The idea is to write H in a standard form, as the sum of two matrices: a multiple of the unit matrix and a traceless matrix. As the eigenvectors of H aresum of two matrices: a multiple of the unit matrix and a traceless matrix....
View Full Document

This note was uploaded on 01/22/2012 for the course PHYS 4143 taught by Professor Kennedy during the Spring '11 term at Georgia Tech.

Ask a homework question - tutors are online