9 - ECE331 Review: Potential Well Problem Considering a...

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1 ECE331 Review: Potential Well Problem V(x) = 0, 0<x<a Considering a particle trapped in a potential well with boundary conditions: V(x)= , x 0, and x L If 0<x<a (reg. II), 2 2 Ψ mE Lu 0 ) ( ) ( 2 2 = Ψ + x x x h ECE331 Lu (a) 4 lowest energy levels; (b) corresponding wave functions; (c) corresponding probability functions.
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2 ECE331 Wave Theory to One-Electron Atom 2 q Consider a single electron atom, in a spherical coordinate system potential and wave function are ) ( ) ( ) ( ) , , ( 4 ) ( 0 ϕ θ πε Φ Θ = Ψ = r R r r r V Schodinger’s wave Equation Lu 0 ) , , ( )) ( ( 2 ) , , ( 2 0 2 = Ψ + Ψ r r V E m r h ECE331 Wave Theory to One-Electron Atom ,... 3 , 2 , 1 ), , ( ) ( n n r R r R n = = Without going to the detailed process, the solutions are ) ( ,..., 2 , 1 , 0 , 2 1 ) ( ) 1 ( ,..., 2 , 1 , 0 ), , ( ) ( 2 1 spin s l m e n l l jm m l h ± = ± ± ± = = Φ = Θ = Θ π Lu 2 2 2 0 4 0 2 ) 4 ( n q m E n h = Where n, l, m, s are quantum numbers. The electron energy,
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3 ECE331 Overlapping Probability Density Functions In Reg. I , In Reg. II, V 0 = 2E 4 2 h ε Lu A q m a 529 . 0 2 0 0 0 = = πε Bohr radius, ECE331 Wave Theory to One-Electron Atom Note: (1). The electron energy is quantized. This is similar to the particle confined in a potential well. (2) The quantum numbers define the number of
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This note was uploaded on 04/09/2010 for the course ECE 331 taught by Professor Rajan during the Spring '09 term at Ohio State.

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9 - ECE331 Review: Potential Well Problem Considering a...

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