class42 - Todays class: Schrdinger in 3D Schrdinger in 3D b...

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Unformatted text preview: Todays class: Schrdinger in 3D Schrdinger in 3D b Hydrogen atom Chapter 8. The 3D Schrodinger Equation In 1D: ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V x x m = + h In 2D: ) , ( ) , ( ) , ( ) , ( 2 2 2 2 2 y x E y x y x V y x = + + h ) , , ( ) , , ( ) , , ( ) , , ( 2 2 2 2 2 2 2 2 z y x E z y x z y x V z y x z y x m = + + + h In 3D: 2 2 2 y x m ) , , ( ) , , ( ) , , ( ) , , ( 2 2 2 2 2 2 2 2 z y x E z y x z y x V z y x z y x m = + + + h Simplest case: 3D box, infinite wall strength V(x,y,z) = 0 inside, = infinite outside. 3D example: Particle in a rigid box c Use mathematics of separation of variables (does not always work, but it works here): Assume we could write the solution as: (x,y,z) = X(x)Y(y)Z(z) Plug it in the Schrdinger eqn. and see what happens! "separated function" a b c (x,y,z) = X(x)Y(y)Z(z) Now, calculate the derivatives for each coordinate: ) ( ) ( ) ( ' ' ) ( ) ( ) ( X ) , , ( 2 2 2 2 z Z y Y x X z Z y Y x x z y x x = = ) , , ( ) , , ( ) , , ( ) , , ( 2 2 2 2 2 2 2 2 z y x E z y x z y x V z y x z y x m = + + + h (Do the same for y and z parts) Now put in 3D Schrdinger and see what happens: 2 z y x m ( ) EXYZ VXYZ m = + + + XYZ" Z XY" YZ X" 2 2 h Divide both sides by XYZ E V Z Y m = + + + Z" Y" X X" 2 2 h (For simplicity I wrote X instead of X(x) and X" instead of ) 2 2 ) ( X x x E V Z Y m = + + + Z" Y" X X" 2 2 h So we re-wrote the Schrdinger equation as: For the particle in the box we said that V=0 inside and V= outside the box. Therefore, we can write with: (x,y,z) = X(x)Y(y)Z(z) outside the box. Therefore, we can write: 2 2 Z" Y" X X" h mE Z Y = + + for the particle inside the box....
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This note was uploaded on 08/24/2010 for the course PHYS 2130 taught by Professor Staff during the Spring '08 term at Colorado.

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class42 - Todays class: Schrdinger in 3D Schrdinger in 3D b...

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