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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 13: Solutions Topics covered: Hilbertspace Frame Transformations, TimeDependent Perturbation Theory 1. The Hamiltonian for a driven twolevel system is H = ~  2 ih 2  + ~ cos( t )(  1 ih 2  +  2 ih 1  ) , (1) where is the separation between the bare levels, and is the driving frequency. (a) Make a frame transformation generated by the operator G = ~  2 ih 2  , and determine the equation of motion for the statevector in the new frame, defined by  G ( t ) i = U G ( t )  ( t ) i . First, we note that U GS ( t ) = e iGt/ ~ =  1 ih 1  + e it  2 ih 2  (2) (the  1 ih 1  term must be there in order to satisfy U GS (0) = I ). Following the general theory in the lecture notes, we have H G ( t ) = U GS ( t ) H S ( t ) U SG ( t ) G = (  1 ih 1  + e it  2 ih 2  ) H (  1 ih 1  + e it  2 ih 2  ) ~  2 ih 2  = ~ (  )  2 ih 2  + ~ cos( t ) (  e it  1 ih 2  + e it  2 ih 1  ) = ~  2 ih 2  + ~ 2 (  1 ih 2  +  2 ih 1  ) + ~ 2 ( e i 2 t  1 ih 2  + e i 2 t  2 ih 1  ) . (3) Hence the equation of motion for  G ( t ) i is: d dt  G ( t ) i = i ~ H G ( t )  G ( t ) i (4) (b) Make the rotating wave approximation (RWA) by assuming that , and dropping any terms that oscillate at or near 2 . Write, in terms of the detuning =  , the effective timeindependent Hamiltonian, H G , that then governs the time evolution of  G ( t ) i . Dropping the terms rotating at 2 gives: H G,RWA = ~  2 ih 2  + ~ 2 (  1 ih 2  +  2 ih 1  ) , (5) In terms of the Rabi Hamiltonian, H Rabi = S z + S x , we have H G,RWA = H Rabi + ~ 2 . (6) 1 (c) Assume that the system begins at time t = 0 in the groundstate of H G , and calculate  G ( t ) i . Is this a stationary state in the rotating frame? Now use  S ( t ) i = U SG ( t )  G ( t ) i to see what this state looks like in the Schrodinger picture. Is it a stationary state in the Schr odinger picture? The eigenstates of H G are those of H Rabi , and the eigenvalues of H G , are those of H Rabi plus the constant shift ~ 2 . The ground state of H G is therefore   G i = ( + 2 + 2 )  1 i   2 i q 2 ( 2 + 2 + 2 + 2 ) , (7) which is a stationary state in the rotating frame (Recall that a stationary state evolves as  ( t ) i = e it  (0) i ). In the Schrodinger picture, this state becomes   S i = ( + 2 + 2 )  1 i  e it  2 i q ( + 2 + 2 ) 2 + 2 , (8) which is not a stationary state. The point is the eigenstates of H G are not energy eigen states, but the problem can be solved in the rotating frame, and then transformed at the end to get the solution in the Schr odinger picture. Note that for timedependent V , energy is not a conserved quantity. Up to a tiny correction due to the dropped terms in the RWA, H G,RWA is a constant of motion. This constant is often referred to as the quasi energy....
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.
 Spring '11
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