Chem120A+Notes+9-8-08

Chem120A+Notes+9-8-08 - Chem120A Notes 3-D Particle in a...

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Chem120A Notes 9/8/08 3-D Particle in a Box →Degeneracy arises when box has sides of equal length →Total energy is a sum of the energy from each coordinate i.e. z y x E E E E + + = (0) →Box has three sides (x, y, z directions) with lengths a , b , and c respectively →We know from 1-D particle in a box that 2 2 2 8 ma n h E = (1) →Therefore, if we take this contribution from each coordinate, we obtain + + = 2 2 2 2 2 2 2 8 c n b n a n m h E z y x (2) →The potential inside the box is: = c z b y a x z y x V 0 for 0 0 for 0 0 for 0 ) , , ( (3)
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→We can also write the Schrödinger equation for inside the box, where ) , , ( ) , , ( 2 2 2 z y x E z y x m Ψ = Ψ - ) , , ( ) , , ( ) , , ( ) , , ( 2 2 2 2 2 2 2 2 z y x E z z y x y z y x x z y x m Ψ = Ψ + Ψ + Ψ - (4) →Separation of Variables gives: ) ( ) ( ) ( ) , , ( z Z y Y x X
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Chem120A+Notes+9-8-08 - Chem120A Notes 3-D Particle in a...

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