083DSE - 3D Schrodinger Equation • Simply substitute...

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P460 - 3D S.E. 1 3D Schrodinger Equation • Simply substitute momentum operator • do particle in box and H atom • added dimensions give more quantum numbers. Can have degeneracies (more than 1 state with same energy). Added complexity. • Solve by separating variables t i m x t z y x V t z y x or i p Ψ = Ψ + Ψ h h r h r ) , , , ( ) , , , ( 2 2 2 2 2 2 ψ φ E z y x V t z y x t z y x m = + = Ψ ) , , ( ) ( ) , , ( ) , , , ( 2 2 2 h
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P460 - 3D S.E. 2 • If V well-behaved can separate further: V(r) or V x (x)+V y (y)+V z (z). Looking at second one: • LHS depends on x,y RHS depends on z • S = separation constant. Repeat for x and y z z z y x y x y x y x z y x z z z y x y x y x z y x z y x m V E z V V V E V V z y x z y x assume E z V y V x V + = + + + + = + + + = = + + + 2 2 2 2 ) ( ) ) ( ) ( ) ) ( ) ( ) ( ) , , ( )) ( ) ( ) ( ( 2 2 2 2 2 2 2 2 2 2 2 ψ h S V V S V E y x y x y x z dz d y x z z = + + + = + ) ( ) ( 2 2 2 2 2 2 1
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P460 - 3D S.E. 3 • Example: 2D (~same as 3D) particle in a Square Box • solve 2 differential equations and get • symmetry as square. “broken” if rectangle E S E S S S E E E E S E V E S S V E S V z y x z z dz d y y dy d x x dx d z z y y x x = + + = + + = = + = = + = = + ) ( ) ' ( ' ' ' 2 2 2 2 2 2 ψ ) ( ) ( ) , ( ) ( ) ( 0 , 0 , , 0 y x y x y V x V V satisfies box inside V a y y a x x V y x y x = + = = > < > < = ) ( 2 2 2 2 2 2 y x ma y x n n E E E + = + = π h
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P460 - 3D S.E. 4 • 2D gives 2 quantum numbers. Level nx ny Energy 1-1 1 1 2E0 1-2 1 2 5E0 2-1 2 1 5E0 2-2 2 2 8E0 • for degenerate levels, wave functions can mix (unless “something” breaks degeneracy: external or internal B/E field, deformation….) • this still satisfies S.E. with E=5E0 ion normalizat dxdy n n A y x n n E E E y x a y n a x n y x ma y x y x 1 | | .. 2 , 1 , sin sin ) , ( ) ( 2 2 2 2 2 2 2 = = = + = + = ∫∫ ψ π h 1 sin sin sin sin 2 2 21 12 2 21 2 12 = + + = = = β α βψ αψ mix a y a x a y a x A A
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P460 - 3D S.E. 5 Spherical Coordinates • Can solve S.E. if V(r) function only of radial coordinate • volume element is • solve by separation of variables • multiply each side by ψ φ θ E r V r r r E r V r r r r r M M = + + + = + ) ( ) , , ( ] ) (sin ) ( [ ) , , ( ) ( 2 2 2 2 2 2 2 2 sin 1 sin 1 2 2 2 2 h h ) sin )( ( ) ( d r rd dr vol d = ΘΦ = ΘΦ + + Φ Θ = ΘΦ R R r R r r r r r r r M R V E 2 2 2 2 2 2 2 2 2 sin 1 sin sin 1 2 ) ( ) ( ) ( ) ( ) ( ) , , ( h ΘΦ R r 2 2 sin
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P460 - 3D S.E. 6 Spherical Coordinates-Phi • Look at phi equation first
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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083DSE - 3D Schrodinger Equation • Simply substitute...

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