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Unformatted text preview: 1 2 The Schrödinger 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 Schrödinger equation in 3 dimensions is V( x,y,z ) is the potential energy function and the wave function Ψ ( 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 Schrödinger 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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This note was uploaded on 02/17/2010 for the course PHY PHY3101 taught by Professor Prosper during the Spring '10 term at FSU.
 Spring '10
 PROSPER

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