19_lecnotes_rwf

19_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 #19 19 – 1 From Quantum Mechanical H eff to Classical Mechanical H eff Readings : Chapter 9.4.14 and 9.4.15, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre- Brion and R. Field, 2 nd Ed., Academic Press, 2004. Last time: eff Transformation H LOCAL eff H NORMAL analytical transformation of basis states analytical transformation of H eff numerical transformation of H eff by setting limit preserving parameter to zero limit-preserving and limit-destroying terms in H eff : reversal of role in opposite representations 2-level illustration Two identical coupled Morse oscillators 3-parameter model (some inconsistencies wrt the Darling-Dennison term) 6-parameter model basis for today’s QM CM conversion. Today: Convert from Quantum Mechanical H eff to equivalent Classical Mechanical H eff . Why? ±The QM H eff is vastly simpler than the exact QM H . In addition, the parameters in H eff are determined by experimental observations. So we have a QM representation of reality and we would like to obtain an equivalent CM representation. It is easier to gain insight from the form of trajectories than from attempting to discern the nodal structure of a QM wavefunction. Begin with Heisenberg’s Correspondence Principle But first an aside about classical mechanics H is a function of pairs of conjugate variables i.e. Q i and P i (coordinate and momentum) Hamilton’s equations of motion: H H Q ˙ , P ˙ i =− = i P i Q i I i and φ i (action and angle) H H = φ ˙ , I ˙ i i I i φ i PQ has dimension of action. The action-angle representation is most convenient for going from QM to CM (or vice versa).
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5.74 RWF Lecture #19 19 – 2 replace a a jj i j i j j i j i ve I e I e / / / / / / →+ () = = −− 12 φφ Prescription : 1. express H eff entirely in terms of constants times products of aa , v j is replaced by . 2. replace a j and a j everywhere by Ie j i j / ±φ . 3. simplify by algebra. There are no commutation restrictions. Ha a a a aa aa LOCAL eff h c vv RL R L L R LR =+ + ++ [] + + + + ′ + ωβ ε αα δ 0 1 2 2 2 22 1 1 2 21 41 4 †† This is the same as H LOCAL eff from Lecture #18, where the 6 independent parameters are: ω ωω α α β ε δ 0 1 2 2 4 33 8 2 4 31 6 4 2 2 4 2 32 4 = + = = + = = ′ = sa ss aa sa ssaa ssaa ss aa sa ss
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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19_lecnotes_rwf - MIT Department of Chemistry 5.74 Spring...

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