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14_lecnotes_rwf - MIT Department of Chemistry 5.74 Spring...

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MIT Department of Chemistry� 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #14 14 – 1 Dynamics in State Space and in Q, P Space Readings : Chapter 9.4.7, 9.4.8, and 9.4.11, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre-Brion and R. Field, 2 nd Ed., Academic Press, 2004. Last time: Polyatomic Molecule Vibrations± Polyads — example: ω 1 ≈ 2ω 2 ≈ ω 3 with two anharmonic resonances± interconnected manifold of resonances± rapid increase in number of near-degenerate zero-order states± Today causal dynamics based on Heisenberg equation of motion rather than A t = Trace( A ρ ( t )) We are able to compute the time evolution of any observable quantity using A t = Trace( A ρ ( t )). Need to know ρ ( t ). But this merely tells us what happens , not why it happens . NO CAUSALITY. One of the most interesting classes of information will be the early time behavior of precisely specifiable d 2 d pluck of the system. So we are interested in A and A at t = 0. dt dt 2 Intrapolyad Dynamics 1. number operator N j a a j j t 2. Q or P j coordinate and momentum operators j 3. 4. 5. + 2 ], 1 + Ω , 1 1 − Ω , 2 + Ω , 2 − Ω [recall 1 = k , 2 + 2 , 2 = k 11 33 a a 3 1 22 a a 2 1 , 1 resonance and transfer rate operators. Fractional importance of resonance 1 vs. 2 for various plucks. Find a new set of a , a , N for local rather than normal modes. a L = 2 –1/2 ( a 1 + a 3 ), a R = 2 –1/2 ( a 1 a 3 ) 1 2 2
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5.74 RWF Lecture #14 14 – 2 We start with the Heisenberg Equation of Motion A A = [ A H ] + , dt i t If we want to know about (the number of quanta in mode j ), we need to compute [ a a , j j H ] Useful simplifications: number operators, N i , commute with other number operators all operators commute with constants a number operator for oscillator j commutes with resonance operators that do not involve oscillator j . d So, for the of N 1 , N 2 , and N 3 we need to work out dt [ a a , 3 3 1 ] †± [ a a , 1 ] 3 3 [ a a , 2 2 2 ] †± [ a a , 2 ] 2 2 Since all of the resonance operators operate within a polyad, the polyad quantum number is conserved P = 2 a a 1 + a a 2 + 2 a a 3 1 2 d d P = 0 also P = 0 t t dt dt 2 Therefore the expectation value of the more difficult of the number operators ( a a can be obtained form 1 1 ) the two easy ones.± a a P a a a a
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