573lec29

573lec29 - 5.73 Lecture #29 29 - 1 Begin Many-e Atoms:...

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29 - 1 5.73 Lecture #29 updated September 19, Begin Many-e Atoms: Quantum Defect Theory See MQDT Primer by Stephen Ross, pages 73-110 in Half Collision Resonance Phenomena in Molecules (AIP Conf. Proc. #225, M. Garciá-Sucre, G. Raseev, and S.C. Ross) 1991. turning points of for Vr e rr rn an n n n l hll l ll l () , / =− + + + ≈± + << ± 22 2 0 2 2 12 0 2 2 1 11 1 1 2 m * * * * Last Time: * μ n l (r) rR n l (r) dominated by small lobe (n-independent nodal position) at inner turning point, amplitude scales as n–3/2, and large lobe at outer turnin point (essentially all of the probability. envelope ur pr nn / EI P n n n n l l l −− 2 1 1 2 nodes: radial nodes angular nodes total nodes gives spacing between radial nodes λ expectation value scaling (see below) (H - atom energy levels) n n σ σ σ σ σ <− ≥+ 1 1 1 3 2 2 geometric mean of expectation values of r for off - diagonal matrix elements when is n n nn n n n n l ( ) [] =′ ++ ′′ 2 2 2 2 // (? ) σ σ σ σ
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29 - 2 5.73 Lecture #29 updated September 19, TODAY 1. Many-e atom treated as core plus outer e that sees shielded core as Z(r). 2. l -dependent energy shifts n-independent quantum defects 3. energy shifts are actually phase shifts in u n l (r) relative to u n l (r) for H-atom 4. Rigorous QDT EI P n n n l l =− −µ () 2 A. regular and irregular Coulomb functions f,g satisfy Hydrogen-like Schr. Eq. OUTSIDE core B. Boundary conditions at r C .πµ l is a phase shift repeated patterns in each integer region of ν D. Multi-channel QDT µ matrices e colliding with core can also transfer energy and angular momentum to core-e noninteger values of require mixture of and find satisfies boundary condition number of members in series of © s with integer spacings, constant quantum defect ν ν ν µ ∞∴ E nr n l l 12 / fg * channels rather than eigenstates * focus on dynamics, but in a “black box” way. Dynamics happens within a restricted region of space. This region of space is always sampled, regardless of E, in the same way. Everything is determined by the boundary conditions for the outgoing wave.
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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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573lec29 - 5.73 Lecture #29 29 - 1 Begin Many-e Atoms:...

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