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Unformatted text preview: PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 13: Solutions Topics covered: Hilbert-space Frame Transformations, Time-Dependent Perturbation Theory 1. The Hamiltonian for a driven two-level 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 state-vector 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 time-independent 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 ground-state 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 time-dependent 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