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

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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 V r e r r r n a n n a n n n ( ) , / = − + + ( ) ( ) = ± + ( ) ± + ( ) << ± 2 2 2 0 2 2 1 2 0 2 2 1 1 1 1 1 1 1 2 m * * * * Last Time: point, amplitude scales as n–3/2, and large lobe at outer turnin point (essentially all of the probability. envelope u r p r n n ( ) ( ) / 1 2 E IP n n n n = 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) r n 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 r n r r n n nn n r r n n n n n l l l ( ) ( ) [ ] ( ) ( ) [ ] = ( ) + + ′ ′ 2 1 2 1 2 2 1 2 2 1 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. -dependent energy shifts n-independent quantum defects 3. energy shifts are actually phase shifts in u n (r) relative to u n (r) for H-atom 4. Rigorous QDT E IP 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 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 n r n l l 1 2 / f g * 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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