5210-chap3

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

This preview shows pages 1–12. Sign up to view the full content.

Particle-in-Box Models

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 = =
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ± = ±
ψ - = ψ 2 2 2 2 a mE dx d

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 64

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

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online