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pset07

# pset07 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum...

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ρ MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum Mechanics I Fall , 2002 Professor Robert W. Field Problem Set #7 DUE : At the start of Lecture on Friday , November 1. Reading : CTDL , Pages 290 - 307 , 643 - 662 , 712 - 741 Problems : 1. Consider the two-level problem ( ) V 0  ∆ V H = E B 0 ( ) = E ( ) + V −∆ V E D 0 ( ) + E D 0 , ∆ ≡ E B 0 ( ) E ( ) E B 0 ( ) ( ) E D 0 0 2 2 ( ) ( ) where E B 0 and E D 0 are respectively the zero-order energies of a bright and a dark state. The names bright and dark refer to the ability to absorb and emit a photon. A. Solve for the eigenstates E + , ψ + , and E , ψ in terms of E B (0) , ψ B (0) , and E (0) , ψ (0) , where , by definition E + > E and let V be real and V > 0. D D Use the standard notation for the two level problem ( ) ( ) ψ + = cos( θ /2) ψ B 0 + sin( θ /2) ψ D 0 ( ) ( ) ψ = − sin( θ /2) ψ B 0 + cos( θ /2) ψ D 0 where tan θ = V/ . B . By definition , at the instant of pulsed-excitation by a photon pulse , ( ) Ψ (x , t=0) = ψ B 0 (x). Solve for Ψ (x , t) = a + (t) ψ + (x) + a (t) ψ (x). C. Construct the 2 × 2 density matrix , (t) , that corresponds to Ψ (x , t) in the ψ B (0) , ψ (0) basis set. D

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ρ ρ Chemistry 5.73 Page 2 Problem Set #7 ˆ 0 Ψ ( ) 0 B D. The detection operator , D = Ψ ( ) B , is a projection operator , which projects out the bright state character. Write the matrix representation of D , D , in the ψ B (0) , ψ (0) basis set. D E. Calculate the expectation value of D for the Ψ (x , t) state represented by the (t) from part C , as D = Trace( D ). You have just re-discovered Quantum Beats !
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