Lecture_14 - PHYS PHYS342 Fall2011 Lecture Lecture 14:...

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HYS 342 PHYS 342 Fall 2011 ecture 14: Schrödinger’s Equation Lecture 14: Schrödinger s Equation in 3 dimensions – Atomic Hydrogen Ron Reifenberger Birck Nanotechnology Center Purdue University Lecture 14 1
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What we’d like to know 1. Does the Schrödinger equation reproduce Bohr’s theory from 1913? Only certain allowed electron orbits? Quantized radius for each orbit? Quantized angular momentum? Quantized energy states? Discrete emission lines? 2. What really is going on during the light emission process inside an atom? 3. What is the electronic structure of other atoms in the periodic table? 2
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What we’ve learned so far Commonly observed objects article ave Particle Box Wave Box Tightly confined objects with small mass ?? 3 No compelling reason to expect such objects to fit into these two boxes!
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hat is the hydrogen atom? What is the hydrogen atom? One electron bound to one proton What is U(r) for the hydrogen atom? 1840 m +e -e r m e m p e 2 12 11 (, ,) () 44 oo qq e Uxyz Ur rr   electrostatics U(r) r 4 Central Force means U(x,y,z) = U(r)
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The 3-dimensional central force problem z P(x, y, z) or P(r, ϑ , ϕ ) pecifies the Transformation quations: sin sin n cos yr r specifies the electron’s position ϑ cos zr equations: sin cos xr  y ϕ r nucleus x sin sin sin cos What does Schrödinger’s equation look like in spherical- 222 2 3 in D xyz   polar coordinates? 5 2 2 (,,) (,,) (,,) 2 x y zU x y zx y zE x y z m   
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22 You can show that http://planetmath.org/encyclopedia/DerivationOfTheLaplacianFromRectangularToSphericalCoordinates.html#foot1096 222 2 2 2 11 n 1 xyz       2 2 2 2 2 2 2 2 sin n 1 r rr r r r      r r Schrödinger’s Equation becomes 2 2 2 2 2 2 1 ( ) 2s i n s i n e rU r E mr r r r r     The math to solve for Ψ is complicated! 2 1 () 4 o e where U r r   Detailed discussion of the solution can be found by reading Appendix BB at: http://booksite.academicpress.com/Morrison/physics/Appendices_AA.pdf 6
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Math Note 1: Proof that 2   2 2 2 1 ( 1 () ) rf rf r r r rr r r     2 22 2 2 2 2 1 1 2 ) 1 ( ( ) f r f r r r r r r r fr r r  2 2 ) (( r 2 1 ( ) r    2 2 11 ( ) ( ) 1 ) rf r
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.

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Lecture_14 - PHYS PHYS342 Fall2011 Lecture Lecture 14:...

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