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# hw6 - HOMEWORK ASSIGNMENT 6 PHYS852 Quantum Mechanics II...

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HOMEWORK ASSIGNMENT 6 PHYS852 Quantum Mechanics II, Spring 2008 Topics covered: Applications of time-dependent perturbation theory, Fermi-Golden rule Please feel free to keep only the term closest to resonance when computing amplitudes or probabilities. 1. AC Stark Effect : Consider a two level atom with bare Hamiltonian H 0 = E e | e )( e | + E g | g )( g | . We want to drive this atom with an oscillating electric field, vector E ( vector r, t ) = E 0 cos( ωt ) vectore z . The interaction between the atom and the field is the usual V = vector d · vector E ( t ). Here the dipole moment operator is of course vector d = e vector R . a.) Find the expression for the perturbation operator as a two-by-two matrix in the basis {| e ) , | g )} . Compute the matrix elements explicitly for the hydrogen atom transition 1S to 2S, and also for the 1S to 2P transitions. What can you say about the ability to drive the 1S to 2S transition via the electric-dipole interaction? b.) Consider the 1S and 2P levels as | g ) and | e ) , respectively. If the atom starts in state | g ) , use perturbation theory to compute is the probability as a function of time to make the transition to state | e ) . Your answer should be valid up to second-order in the probabilities. c.) As a function of the detuning, Δ = ω ω a , (where ω a = ( E e E g ) / planckover2pi1 ), approximate the quantum- mechanical average energy as E avg = P g ( t ) E g + P e ( t ) E e . Then compute the time-average of this energy (specifically, averaged over an extremely long time-scale). Show that the average energy is always larger than E g , and is proportional to the field intensity | E 0 | 2 . You can think of the difference between E avg and E g as the AC analog to the stark shift to the ground state energy. d.) Based on the previous results, would you expect an atom to be accelerated or decelerated when moving into a region of increasing field intensity? Another way of looking at it is to see that if the internal energy changes with position, then the atoms center-of-mass motion sees a potential V ( vector r ) = E internal ( vector r ). Then the question becomes is field intensity maximum a potential minimum or a potential maximum. e.) Consider that when the atom is in | e ) , the field has one fewer photon than when the atom is in | g ) , and that the photon carries an energy E = planckover2pi1 ω . Compute the total energy of the atom-field system when the atom is in | e ) . Subtract from this the total energy of the atom-field system when the atom is in | g ) . Multiply this by the probability to be in | e ) at time t , and average the result over time. This is the correct AC stark shift.

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hw6 - HOMEWORK ASSIGNMENT 6 PHYS852 Quantum Mechanics II...

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