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Unformatted text preview: Chem120A Notes 9/3/08 1-Dimensional Particle in a Box Particle confined in infinite potential well of length a Schrdinger Equation: ) ( ) ( ) ( ) ( 2 2 x E x x V x m = + - Particle freely moves between 0 and a < = a x x a x x V ; ) ( ) ( = x for < x and a x continuity of wavefunction requires ) ( = and ) ( = a , so that wavefunction is smooth and continuous these are the boundary conditions consequences of imposing boundary conditions are quantization and quantum numbers Inside box, ) ( = x V so the Schrdinger Equation is ) ( ) ( 2 2 x E x m = - This is for a x , and is similar to what we saw for a free particle ) ( 2 ) ( 2 x mE x - = ) ( 2 ) ( 2 = + x mE x ) ( ) ( 2 = + x k x (1) Which is similar to a free particle equation where 2 2 2 mE k = (2) (2) can be rearranged to give m k E 2 2 2 = (3) The solutions for (1) are in the form...
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