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Unformatted text preview: 1 2 The Schrdinger Equation in 3D Central Potentials Angular Momentum Summary 3 The symbol means differentiate the function f with respect to x only while treating the remaining variables as constants. Similarly for y and z . Example: 4 The timeindependent Schrdinger equation in 3 dimensions is V( x,y,z ) is the potential energy function and the wave function ( x,y,z ) is, in general, a function of all 3 coordinates. E is the energy associated with the state described by the wave function 5 The 3D Schrdinger equation can be solved for a quantum object confined to a box with impenetrable walls by a wave function of the form L 3 L 1 L 2 Since the walls are assumed to be impenetrable, the wave function must go to zero at the walls. Why? 6 The boundary conditions on the walls implies k i L i = n i p , where i = 1, 2, 3 and n i = 1, 2,, which implies the energy levels L 3 L 1 L 2 E n 1 n 2 n 3 = h 2 8 m n 1 2 L 1 2 + n 2 2 L 2 2 + n 3 2 L 3 2 which, for a cubical box, is E n 1 n 2 n 3 = h 2 8 mL 2 n 1 2 + n 2 2 + n 3 2 ( ) 7 Note that the quantum numbers n 1 n 2 n 3 = 211, 121, 112...
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 Spring '10
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