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 rt iH r t t 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 HH V t = + The wavefunction for the system is expressed as the linear superposition of basis wavefunctions ψ 1 G r and 2 G r ( ) , ( ) ( ) ( ) ( ) 11 22 , c t r r ψψ Ψ= + G GG The basis wavefunctions 1 G r ( ) and 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, ( ) ( ) ( )( ) 01 1 1 1 ,, , , e x p / Hr t r t r t r i t εψ ε Ψ GGG G = ( ) ( ) ( ) 02 2 2 2 , , e x p / t r t r t r i t Ψ G = The eigenfunctions 1 G r and 2 G r ( ) are orthonormal, () () () () ** 1 rr d d +∞ +∞ −∞ −∞ =∀ = ∫∫ ME 501 Supplemental Handout Quantum Mechanical Analysis Radiative Transitions for a Two-Level System: Derivation of the Density Matrix Equations
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2 () () () () ** 21 1 2 0 rr d d ψψ +∞ +∞
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This note was uploaded on 12/26/2011 for the course ME 501 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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ME501DensityMatrixNotes - 1 ME 501 Supplemental Handout...

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