notes09 - 5.73 Lecture#9 9-1 Numerov-Cooley Method 1-D Schr...

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9 - 1 5.73 Lecture #9 updated 9/18/02 8:54 AM wp x t a e ii iE t i i (,) / = ψ h Numerov-Cooley Method : 1-D Schr. Eq. Last time: Rydberg, Klein, Rees Method and Long-Range Model G(v), B(v) rotation-vibration constants V J (x) potential energy curve x = R – R e E v,J , ψ v,J , all conceivable experiments a i i * wp(x,0) [] dx determined by V J (x) Method: A(E,J) = area of V(x) below E: used WKB QC obtained x ± (E,J) Today: What do we do when we have V J (x) (especially when V(x) is not suited for WKB)? Solve Schr. Eq. numerically! No models 15 digit reproducibility cheap V(x) E free evolution of wp initial preparation of wp: This is the final tool we will develop for use in the Schrödinger representation. To summarize the classes of 1–D problem we have solved: * piecewise constant potentials (matrix approach for joining at boundaries) * Airy functions (linear potential and joining JWKB across turning point) * JWKB (quantization condition and semi-classical wavefunctions) * numerical integration (today)
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9 - 2 5.73 Lecture #9 updated 9/18/02 8:54 AM Numerical Integration of the 1-D Schrödinger Equation widely used incredibly accurate no restrictions on V(x) or on E–V(x) [e.g. nonclassical region, near turning points, double minimum potential, kinks in V(x).] For most 1-D problems, where all one cares about is a set of {E i , ψ i }, where ψ i is defined on a grid of points x i , one uses Numerov-Cooley See 1. Cooley, Math. Comput. 15, 363 (1961). 2. Press et. al., Numerical Recipes, Chapters 16 and 17 Handouts 1. Classic unpublished paper by Zare and Cashion with listing of Fortran program (now see LeRoy web site) 2. Tests of N-C vs. other methods by Tellinghuisen Basic Idea: grid method * solve differential equation by starting at some x i and propagating trial solution from one grid point to the next * apply ψ (x) = 0 BCs at x = 0 and by two different tricks and then force agreement at some intermediate point by adjusting E.
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9 - 3 5.73 Lecture #9 updated 9/18/02 8:54 AM fx n , ψ n () = d ψ dx x n ψ n + 1 −ψ n x n + 1 x n = ψ n + 1 n h Euler’s Method want ψ (x) at a series of grid points x 0 , x 1 , …x n =x 0 +nh call these ψ i = ψ (x i ) Need a generating function f( x n , ψ n ) ψ n + 1 n + hf x n , ψ n increment in x x n+1 – x n = h [NOT Planck’s constant]
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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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notes09 - 5.73 Lecture#9 9-1 Numerov-Cooley Method 1-D Schr...

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