5210-chap3_2

# 5210-chap3_2 - Multidimensional Systems The 2D PIB The 2D...

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Unformatted text preview: Multidimensional Systems: The 2D PIB The 2D Hamiltonian: ) , ( 2 2 2 2 2 2 y x V y x m H + ∂ ∂ + ∂ ∂- = The 2D Schr. Eqn.: ) , ( ) , ( ) , ( ) , ( 2 2 2 2 2 2 y x E y x y x V y x y x m ψ ψ ψ = + ∂ ∂ + ∂ ∂- The Potential Energy a b x y V ( x,y ) = 0 0 ≤ x ≤ a and 0 ≤ y ≤ b V ( x,y ) →∞ x < 0 or x > a or y < 0 or y > b Outside of the box i.e. x < 0 or x > a or y < 0 or y > b ψ ( x,y ) = 0 Solution: Separation of Variables Inside the box: y x H H y m x m H + = ∂ ∂- ∂ ∂- = 2 2 2 2 2 2 2 2 u u Assume: ) ( ) ( ) , ( y Y x X y x = ψ ) , ( ) , ( ) , ( y x E y x H y x H y x ψ ψ ψ = + H x H y ) ( ) ( ) ( ) ( ) ( ) ( y Y x EX y Y x X H y Y x X H y x = + ) ( ) ( ) ( ) ( ) ( ) ( y Y x EX y Y H x X x X H y Y y x = + E y Y H y Y x X H x X y x = + ) ( ) ( 1 ) ( ) ( 1 Solution: Separation of Variables E y Y H y Y x X H x X y x = + ) ( ) ( 1 ) ( ) ( 1 Inside the box: y x H H y m x m H + = ∂ ∂- ∂ ∂- = 2 2 2 2 2 2 2 2 u u If: f ( x ) + g ( y ) = C then: f ( x ) = C 1 and g ( y ) = C 2 C 1 + C 2 = C f ( x ) g ( y ) C = E x = E y E dy Y d Y m dx X d X m =-- 2 2 2 2 2 2 1 2 1 2 u u E = E x + E y E dy Y d Y m dx X d X m =-- 2 2 2 2 2 2 1 2 1 2 u u = E x = E y X E dx X d m x =- 2 2 2 2 u Range 0 ≤ x ≤ a ,... 3 , 2 , 1 8 2 2 2 = = x x x n ma h n E a A a x n A x X x x x 2 sin ) ( = = π E = E x + E y x E dx X d X m =- 2 2 2 1 2 u y E dy Y d Y m =- 2 2 2 1 2 u Y E dy Y d m y =- 2 2 2 2 u Range 0 ≤ y ≤ b ,... 3 , 2 , 1 8 2 2 2 = = y y y n mb h n E b A b y n A y Y y y y 2 sin ) ( = = π ) ( ) ( ) , ( y Y x X y x ⋅ = ψ = b y n a x n ab y x y x π π ψ sin sin 4 ) , ( n x = 1, 2, 3,... n y = 1, 2, 3,... Range 0 ≤ x ≤ a 0 ≤ y ≤ b ψ 2 n x = 1 n y = 1 ψ 2 ψ 2 n x = 2 n y = 1 ψ 2 ψ 2 n x = 2 n y = 2 ψ 2 The Wavefunctions = ⋅ = b y n A a x n A y Y x X y x y y x x π π ψ sin sin ) ( ) ( ) , ( b A a A y x 2 2 = = The Wavefunctions = ⋅ = b y n A a x n A y Y x X y x y y x x π π ψ sin sin ) ( ) ( ) , ( b A a A y x 2 2 = = ψ ψ 2 http://www.falstad.com/mathphysics.html#qm The Energies: Wavefunction Degeneracy 2 2 2 2 2 2 8 8 mb h n ma h n E E E y x y x + = + = + = 2 2 2 2 2 8 b n a n m h y x n x = 1, 2, 3,... n y = 1, 2, 3,... Square Box a = b ( 29 2 2 2 2 8 y x n n ma h E + = 2 2 8 ma h E n x n y 1 1 2 5 1 2 2 1 g = 1 8 2 2 g = 1 10 1 3 3 1 g = 2 13 2 3 3 2 g = 2 g = 2 Application: π →π * Absorption in Benzene 1 1 n x n y 2 5 8 10 13 2 1 1 2 2 2 1 3 3 1 2 3 3 2 E [h 2 /8ma 2 ] The six π electrons in benzene can be approximated as particles in a square box....
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5210-chap3_2 - Multidimensional Systems The 2D PIB The 2D...

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