01_lecnotes_rwf

01_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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1-1 MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lectures #1 & #2 Point of View Isolated (gas phase) molecules: no coherences (intra- or inter-molecular). At t = 0 sudden perturbation: “photon pluck” ρ (0) [need matrix elements of µ( Q )] visualization of dynamics ρ (t) [need H for evolution] H = H (0) + H (1) intramolecular coupling terms (not t-dependent) initially localized, nonstationary state What do we need? pre-pluck initial state: simple, localized in both physical and state space nature of pluck: usually very simple single orbital single oscillator single conformer (even if excitation is to energy above the isomerization barrier) post-pluck dynamics nature of pluck determines best choice of H (0) need H = H (0) + H (1) to describe dynamics * Reduce H to H eff good for a “short” time * transformations between basis sets * evaluate matrix elements of H eff and µ ( Q ) Visualize dynamics in reduced dimensionality ψ ( Q ) * ψ ( Q ) contains too much information * develop tools to look at individual parts of system — in coordinate and state space Detection? * how to describe various detection schemes? * devise optimal detection schemes 1st essential tool is Angular Momentum Algebra define basis sets for coupled sub-systems electronic ψ — symmetry in molecular frame, orbitals rotational ψ — relationship between molecular and lab frame alternative choices of complete sets of commuting operators eg. J 2 L 2 S 2 J vs. L 2 L z S 2 S z z coupled uncoupled
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1-2 5.74 RWF Lectures #1 & #2 reasons for choices of basis set nature of pluck hierarchy of terms in transformations between basis sets needed to evaluate matrix elements of different operators effects of coordinate rotation on basis functions and operators spherical tensor operators Wigner-Eckart Theorem Let’s begin with a fast review of Angular Momentum Angular Momentum JM 2 J + 1 M values for each J 2 ,, J = J ± i J y JJ z ± x 1 J = 2 ( J + + J ) real x i J =− 2 ( J + J ) imaginary y i , [ JJ j ] = ∑ i h ε ijk J k k J ± JM J J + 1) M M ± 1 )] 12 ( = h [( 1243 / JM ± 1 4 product of M values If [ AB ] = 0 then A , ab = a i ab ij B = b j (if [ A,B ] 0, then it is impossible to define an | a i b basis set. e.g. J x , J y ) j 2 angular momentum sub-systems, e.g. L and S J = L + S two choices of basis set 2 2 2 JL , S 2 , J ] [, [, LL , S 2 , S ] z z z LSJM J LM L SM S coupled uncoupled
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1-3 5.74 RWF Lectures #1 & #2 trade J for M L (M S = M J – M L ) operators L z , L ± , S z , S ± destroy J quantum number (coupled basis destroyed) (not commute with J 2 ) operators L ± , S ± , J ± destroy M L , M S , M J (both bases destroyed) example of incompatible terms in special case valid only = ∑ ξ () l ⋅→ ζ ( NLS ) L S H SO r i i s i for L = S = 0 H i matrix elements Zeeman = B µ ( L + 2 S ) z 0 z z 1.4 MHz/Gauss note that H SO and H Zeeman are incompatible because [, 2 +− −+ ) LL S ] = h ( L S L S z 2 −+ +− ) SL S ] = h ( L S L S z (note that if H z were ( L + S
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01_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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