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PH 401 Dr. Cecilia Vogel
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Review Outline Time dependent perturbations integrating out the time oscillatory perturbation stimulated emission (laSEr) Time dependent perturbations approximations perturbation symmetry
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Time-dependent Perturbation Recall last time we derived the following eqn for evolution of the amplitude for a state | f > given that the particle started in state | i > and that a 1 st order approximation was in order c’ f (t) =(-i/ ) e i(E f -E i )t/ The derivative = computable ftn of t, that can in principle be integrated to get c f (t).
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Oscillatory Perturbation Let‘s suppose that the perturbation is a sinusoidal function of time V pert =V o cos( pert t) This would be the case if your particle were being perturbed by the oscillating E-field of a light wave, for example. c’ f (t) =(-i/ ) e i(E f -E i )t/ c’ f (t) =(-i/ ) e i(E f -E i )t/ cos( pert t) The ftn of t is clear in this case, and can be integrated to get c f (t).
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Oscillatory Perturbation c’ f (t) =(-i/ ) e i(E f -E i )t/ cos( pert t) c f (t) =(-i/ ) ∫e i(E f -E i )t/ cos( pert t)dt integral from some initial time, often taken to be t=0, to some final time, tf. Let A= (-i/ ) (time indep) c f (t) =(A/2) ∫e i(E f -E i )t/ (e i pert t + e -i pert t )dt c f (t) =(A/2) ∫(e i[ (E f -E i )/ +i pert ]t + e i[ (E f -E i )/ -i pert ]t )dt do the integral….
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Oscillatory Perturbation c f (t) =(A/2) ∫(e i[ (E f -E i )/ + pert ]t + e i[ (E f -E i )/ - pert ]t )dt do the integral…. If you take |c f | 2, you will have the probability that the perturbation will cause a particle in state | i > to end up in state| f > Notice that cf (and thus the probability) is max if one of the denominators is zero. this is a resonance between the system and the perturbation
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Resonant Perturbation Notice that cf (and thus the probability) is max if one of the denominators is zero. Resonance if pert = (E f -E i )/ If this is a light-wave perturbing your system, then pert =energy of a photon Resonance occurs if the energy of the photon is equal to the energy needed to excite system from initial to final state. Photon is absorbed. Excitation can occur if pert is not exactly right, but less likely
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Resonant Perturbation But also… resonance if pert = (E i -E f )/ ! If this is a light-wave perturbing your system, then pert =energy of a photon Resonance occurs if the energy of the photon is equal to the energy that will be released when the system DE-excite from initial to final state. Photon is NOT absorbed. Photon comes in and stimulates emission of another photon of the same frequency.
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LaSEr This is how a laser works Light interacts with atoms in excited state. Photons stimulate emission of more photons of the same frequency, creating more light, without absorbing the original light. More photons means the light is amplified LASER = Light Amplification through Stimulated Emission of Radiation
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PAL
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