hw209a - Physics 512 Homework Set #9 Solutions Winter 2003...

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Physics 512 Winter 2003 Homework Set #9 – Solutions Note: The homework set posted on the web had a second page (part c ) of problem #3 and problem #4). I mistakenly omitted the second page on the set I handed out in class, so only the Frst three problems are required. What should have been problem #4 is now part of Homework #10. 1. We consider Frst order time dependent perturbation theory for a Hamiltonian H ( t )= H 0 + V ( t ). Suppose that a system is initially in an eigenstate | s i of H 0 at time t 0 . Let O be some observable of the system. Show that the expectation value of O at time t is given to Frst order in V ( t )by h ψ ( t ) |O| ψ ( t ) i = h s |O| s i− i ¯ h Z t t 0 h s | [ e O ( t ) , e V ( t 0 )] | s i dt 0 where e O ( t ) and e V ( t ) are in the interaction picture. Since we are interested in a system at a later time t , we transform to the interaction picture and use the time evolution operator. Going from the Schr¨ odinger picture to the interaction picture is simple (we just put tilde’s over everything) h ψ ( t ) |O| ψ ( t ) i = h e ψ ( t ) | e O ( t ) | e ψ ( t ) i To Frst order, the time evolution operator in the interaction picture gives | e ψ ( t ) i = e T ( t, t 0 ) | e s i = ± 1 i ¯ h Z t t 0 e V ( t 0 ) dt 0 ² | e s i where | e s i is the initial state, converted to the interaction picture. Since we assume that V is Hermitian (so that ˜ V is also Hermitian), the bra state evolves similarly h e ψ ( t ) | = h e s | ± 1+ i ¯ h Z t t 0 e V ( t 0 ) dt 0 ² We may combine these expressions to obtain h ψ ( t ) |O| ψ ( t ) i = h e s | ± i ¯ h Z t t 0 e V ( t 0 ) dt 0 ² e O ( t ) ± 1 i ¯ h Z t t 0 e V ( t 0 ) dt 0 ² | e s i = h e s | e O ( t ) | e s i + i ¯ h Z t t 0 ³ h e s | e V ( t 0 ) e O ( t ) | e s i−h e s | e O ( t ) e V ( t 0 ) | e s i ´ dt 0 = h s |O| s i ¯ h Z t t 0 h e s | [ e O ( t ) e V ( t 0 )] | e s i dt 0 1
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(where we have ignored the higher order term where e V appears twice). Note that in the frst term we have converted back to the Schr¨odinger picture by removing the tilde’s. Finally, since | s i is an eigenstate o± H 0 , it is easily converted into the interaction picture (at time t 0 ) according to | e s i = e iH 0 t 0 / ¯ h | s i = e iE s t 0 / ¯ h | s i Since this is just a phase relating | s i to e s i , it cancels ±rom the bra-ket h e s |···| e s i . This proves the result h ψ ( t ) |O| ψ ( t ) i = h s |O| s i− i ¯ h Z t t 0 h s | [ e O ( t ) , e V ( t 0 )] | s i dt 0 2. Apply frst-order time dependent perturbation theory to a Forced harmonic oscillator H =( a a + 1 2 + f ( t ) a + f ( t ) a
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hw209a - Physics 512 Homework Set #9 Solutions Winter 2003...

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