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ME501DensityMatrixNotes

ME501DensityMatrixNotes - 1 ME 501 Supplemental Handout...

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1 1. Basis Wavefunctions for the Two-Level System The Schrödinger Wave Equation (SWE) is given by ( , ) ( , ) op r t i H r t t Ψ = Ψ G G = where the Hamiltonian operator is the sum of a time-independent term H 0 which gives the energy levels in the absence of an external field, and a time-dependent interaction term V ( t ) which accounts for the electric dipole interaction of the electron with the external laser field. 0 ( ) op H H V t = + The wavefunction for the system is expressed as the linear superposition of basis wavefunctions ψ 1 G r ( ) and ψ 2 G r ( ) , ( ) ( ) ( ) ( ) ( ) 1 1 2 2 , r t c t r c t r ψ ψ Ψ = + G G G The basis wavefunctions ψ 1 G r ( ) and ψ 2 G r ( ) are eigenfunction solutions of the time- independent SWE in the absence of an external field. Operating with H 0 on the normalized wavefunctions Ψ 1 G r , t ( ) and Ψ 2 G r , t ( ) gives the level energy, ( ) ( ) ( ) ( ) ( ) 0 1 1 1 1 1 1 , , , , exp / H r t r t r t r i t ε ψ ε Ψ = Ψ Ψ = G G G G = ( ) ( ) ( ) ( ) ( ) 0 2 2 2 2 2 2 , , , , exp / H r t r t r t r i t ε ψ ε Ψ = Ψ Ψ = G G G G = The eigenfunctions ψ 1 G r ( ) and ψ 2 G r ( ) are orthonormal, ( ) ( ) ( ) ( ) * * 1 1 2 2 1 r r d r r d ψ ψ ψ ψ +∞ +∞ −∞ −∞ ∀ = ∀ = G G G G ME 501 Supplemental Handout Quantum Mechanical Analysis Radiative Transitions for a Two-Level System:

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ME501DensityMatrixNotes - 1 ME 501 Supplemental Handout...

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