5210-chap3 - Particle-in-Box Models Outline The Classical...

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Particle-in-Box Models
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Outline The Classical Particle in a Box (PIB) The 1-D Quantum Mechanical PIB PIB and the Color of Vegetables PIB Properties Multidimensional Systems: The 2D PIB and 3D PIB Other PIB Models The Free Electron Molecular Orbital (FEMO) Model Statistical Thermodynamics: Translational contributions to the thermodynamic properties of gases.
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x P ( x ) 0 L 0 E = ½ mv 2 = p 2 /2 m = any value, including 0 i.e. the energy is not quantized and there is no minimum. P ( x ) = Const. = C 0 ≤ x ≤ L P ( x ) = 0 0 < x , x > L The Classical PIB
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P ( x ) = Const. = C 0 ≤ x ≤ L P ( x ) = 0 0 < x , x > L Normalization dx dx C dx dx x P L L - - + + = = 0 0 1 ) ( 0 0 L C x P 1 ) ( = = < x > dx x xP x - = < ) ( < x 2 > dx x P x x - = < ) ( 2 2 Note: 0 12 2 3 2 2 2 2 2 2 = - = < - = < L L L x x x σ = L dx L x 0 1 L x L 0 2 2 1 = 2 1 2 L L = 2 L = dx L x L = 0 2 1 L x L 0 3 3 1 = 3 1 3 L L = 3 2 L = [ ] L x C 0 = CL = The Classical PIB
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Outline The Classical Particle in a Box (PIB) The 1-D Quantum Mechanical PIB PIB and the Color of Vegetables PIB Properties Multidimensional Systems: The 2D PIB and 3D PIB Other PIB Models The Free Electron Molecular Orbital (FEMO) Model Statistical Thermodynamics: Translational contributions to the thermodynamic properties of gases.
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Quantum Mechanical Particle-in-Box ψ = ψ + ψ - E x V dx d m ) ( 2 2 2 2 a Schrödinger Equation x V ( x ) 0 L 0 V ( x ) = 0 0 x L V ( x ) x < 0 , x > L Outside the box: x < 0, x > L P ( x ) = ψ *( x ) ψ ( x ) = 0 ψ ( x ) = 0 Inside the box: 0 x L ψ = ψ - E dx d m 2 2 2 2 a ψ - = ψ 2 2 2 2 a mE dx d or
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Supplementary Math – The Form of a Wave Equation Consider the 2 nd -order linear differential equation: 0 2 2 = + + by dx dy a dx y d Lets see what happens if we let y = e λ x . In this case: x x dx d λ e e = x x dx d e e 2 2 2 = Substitution back into the original equation now gives us: 0 e e e 2 = + + x x x b a Or: 0 ) ( e 2 = + + b a x
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Supplementary Math – The Form of a Wave Equation 0 2 = + + b a λ 0 ) ( e 2 = + + b a x So this is a solution of our differential equation if: We call this the characteristic equation The possible values of the λ are the roots of the characteristic equation: ( 29 b a a 4 2 1 2 1 - + - = ( 29 b a a 4 2 1 2 2 - - - = Solutions to our differential equation are: x x y y 2 1 e , e 2 1 = =
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Supplementary Math – The Form of a Wave Equation Complex roots: b a a - ± - = 2 2 2 λ Where in this case ( a /2) 2 b < 0. Let ( a /2)2 – b = - ϖ 2 , where ϖ is real. Then: ϖ i b a = - = - = - 1 2 2 2 And the roots are i a i a - - = + - = 2 , 2 2 1
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Supplementary Math – The Form of a Wave Equation ϖ λ i a i a - - = + - = 2 , 2 2 1 x x y y 2 1 e e 2 1 = = Our roots: Our solutions have the form: x i a x i a x y x y - - + - = = 2 2 2 1 e ) ( , e ) ( Because these are linearly independent functions we may write the general solution: ) ( ) ( ) ( 2 2 1 1 x y c x y c x y + = ( 29 x i x i ax e c e c x y - - + = 2 1 2 / e ) ( Using Euler’s formula this may be expressed in trigonometric form: ( 29 x d x d x y ax sin cos e ) ( 2 1 2 / + = - x i x ix sin cos e ± = ±
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ψ - = ψ 2 2 2 2 a mE dx d
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This note was uploaded on 04/11/2011 for the course CHEM 5210 taught by Professor Staff during the Spring '08 term at North Texas.

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5210-chap3 - Particle-in-Box Models Outline The Classical...

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