Chem120A+Notes+9-3-08

Chem120A+Notes+9-3-08 - Chem120A Notes 1-Dimensional...

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Chem120A Notes 9/3/08 1-Dimensional Particle in a Box →Particle confined in infinite potential well of length a Schrödinger Equation: ) ( ) ( ) ( ) ( 2 2 x E x x V x m Ψ = Ψ + Ψ - →Particle freely moves between 0 and a < = a x x a x x V ; 0 0 0 ) ( 0 ) ( = Ψ x for 0 < x and a x →continuity of wavefunction requires 0 ) 0 ( = Ψ and 0 ) ( = Ψ 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, 0 ) ( = x V so the Schrödinger Equation is ) ( ) ( 2 2 x E x m Ψ = Ψ - →This is for a x 0 , and is similar to what we saw for a free particle ) ( 2 ) ( 2 x mE x Ψ - = Ψ 0 ) ( 2 ) ( 2 = Ψ + Ψ x mE x 0 ) ( ) ( 2 = Ψ + Ψ 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 kx B kx A x cos sin ) ( + = Ψ A and B are constants to be determined.
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