QMIIPertTheoryNote2011

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

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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....
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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.

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