05_04_15 - [email protected] 141 Spherically...

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Unformatted text preview: [email protected] 141 04/15/2005 Spherically symmetric potentials and wave functions Φ = Φ + Φ + +- ∂ ∂ ∂ ∂ ∂ ∂ E x V z y x m ) ( ) ( 2 2 2 2 2 2 2 2 & Φ + Φ + Φ = Φ ∇ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2 2 2 2 2 2 2 sin 1 sin 1 1 2 ) (sin ) ( ) , , ( ϕ ϑ ϑ ϑ ϑ ϑ ϕ ϑ r r r r r r & ) ( ) ( ) ( )] ( [ 2 2 2 1 2 r E r r V r r r r m Φ = Φ + Φ- ∂ ∂ ¡ Search for only those wave functions for a spherically symmetric potential that are spherically symmetric. The other wave functions will be found later. ) ( ] ) ( [ ) ( 2 2 2 2 r u E r V r u m r- = ∂ ∂ ¡ ) ( ) ( r r r u Φ = Φ + Φ + Φ = Φ ∇ ∂ ∂ ∂ ∂ ∂ ∂ ϕ ϑ ϕ ϑ ϑ sin 1 1 r r r r e e e & & & & Like a 1D Schrödinger equation Φ + Φ + Φ = Φ ∇ ⋅ + + = Φ ∇ ⋅ ∂ ∂ ∂ ∂ ∂ ∂ ϕ ϑ ϕ ϑ ϕ ϑ ϕ ϑ ϑ d d dr e d r e rd e dr x d r r & & & & & & ) sin ( 3D Schrödinger equation: ϕ ϑ ϕ ϕ ϕ ϑ ϑ ϑ ϑ e e e e e e e e r r k k k & & & & & & & & cos , sin...
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05_04_15 - [email protected] 141 Spherically...

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