573lec28

573lec28 - 5.73 Lecture #28 28 - 1 Hydrogen Radial...

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28 - 1 5.73 Lecture #28 updated September 19, Hydrogen Radial Wavefunctions The Hydrogen atom is special because it has electronic states and properties that scale with n and l in a simple and global way. This is “structure” that is more than a collection of unrelated facts. H serves as our model for “electronic structure” of many- electron atoms, molecules, and possibly solids. By showing how E, r σ ⟩ ( size and shapes), n l |r|n l ′⟩ (general matrix element) scale with n and l , it tells us the kind of behavior to look for in more complex systems. * as a perturbation on H (quantum defects) * as a hint of relationships useful for extrapolation, assignment, for recognizing when something behaves differently from naive expectations. TODAY 1. Simplified Radial Equation 2. Boundary conditions at r 0 and r 3. qualitative features of R n l (r) 4. n-scaling of r σ 5. mathematical form of R n l (r) 6. regular and irregular Coulomb functions
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28 - 2 5.73 Lecture #28 updated September 19, For any central force problem so we can take Y m l (,) θφ out of the Schrödinger Equation and we are left with a 1-D radial equation where the only trace of the angular part is the l -dependence of V l (r), the effective potential energy function. Since the differential equation depends on l , R(r) must also depend on l , thus R n l (r) is the radial part of ψ , and it will generally be an explicit function of tw o quantum numbers, n and l . Usually n specifies the number of radial nodes and l the number of angular nodes, but a special numbering convention for Hydrogen (and hydrogenic ions) causes a slight distortion of this rule. H r Vr r = µ + µ + ˆ ˆ () p 22 2 l We know that commute, so spherical harmonics, , are eigenfunctions of H with eigenvalues r, , trial form for separation of Operate on the , , , , . , , HY RrY H r Vr Y Rr E Y z m m rm m l l l 2 2 2 1 2 2 l l l ψθ φ θ φ ψ φ ψ + = = µ + µ + () ( ) = hll p l angular wavefunction and move it through to left. , r Vr Rr E mr φ ψ = µ + + µ () = l 12 44 43 444 l p 2 2 1 2
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28 - 3 5.73 Lecture #28 updated September 19, * equation looks simpler * volume element looks simpler * behavior as r 0 seems more familiar looks like ordinary 1-D Schrödinger Equation.
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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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573lec28 - 5.73 Lecture #28 28 - 1 Hydrogen Radial...

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