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lecture37_38 - Schrodinger equation and standard problems...

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Unformatted text preview: Schrodinger equation and standard problems Standard problems: • Free particle • Quantum particle in a box • Potential wells • Potential barriers and tunneling • Harmonic oscillator ( 29 ( 29 ( 29 ( 29 x E x x U dx x d m ψ ψ ψ = +- 2 2 2 2 0. Free particle ( 29 ( 29 ( 29 ( 29 ( 29 m p m k E Ce x x E dx x d m x U ikx 2 2 2 2 2 2 2 2 = = ⇒ = =- ⇒ = ψ ψ ψ ( 29 x mE i Ce x mE k 2 2 2 = = = ψ λ π 1a. Classical standing waves on a string (review) n = 1,2,3... 1 2 1 λ = L L L 2 1 2 1 = = λ 2 2 2 2 λ λ = = L L L = = 2 2 2 λ 3 2 3 λ = L 3 2 3 L = λ n n L λ 2 = n L n 2 = λ ( 29 n n n n k x k C y λ π 2 sin = = x y 1. Quantum particle in a box ( 29 = n n x C x λ π ψ 2 sin m L h n m p E n n 2 2 2 2 8 2 = = 1b. Quantum particle in a 1-dimensional box ( 29 { } ( 29 ( 29 { } ( 29 ( 29 { } = = ⇒ < ∞ =- ⇒ ≤ ≤ = or for 2 for 2 2 2 L L x x x E dx x d m L x x U ψ ψ ψ ψ x L U(x) ⇒ = 2 n L n λ Quantum numbers: n = 1,2… Wavelength: Momentum and energy are quantized L nh h p n n 2 = = λ Wave function: ( 29 ( 29 L x C x π ψ sin 1 = ( 29 ( 29 L x n C x n π ψ sin = Example: What is the energy difference between the first excited state and the ground state of an electron in the “box” of size L=1nm? ( 29 ( 29 ( 29 ( 29 ( 29 eV J J kg m s J E m L h m L h m L h E E E 1 . 1 10 8 . 1 10 11 . 9 8 63 . 6 3 10 11 . 9 10 8 10 63 . 6 3 8 3 8 1 8 2 19 19 2 31 2 9 2 34 2 2 2 2 2 2 2 2 1 2 = ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ∆ =- =- = ∆----- m L h n E n 2 2 2 8 = 1d. Quantum particle in a 3-dimensional box We have 3 independent standing waves, and 3 independent quantum numbers. ( 29 m L h n n n m p p p E z y x z y x 2 2 2 2 2 2 2 2 8 2 + + = + + = 1c. Probability and normalization ( 29 1 2 = ∫ ∞ ∞- dx x ψ ( 29 L C L C dx L x n C L 2 1 2 1 sin 2 2 2 = ⇒ = ⇒ = ∫ π ( 29 ( 29 L x n L x n π ψ sin 2 = Normalization: 2. Potential wells ( 29 { } { } < ≤ ≤ = L x x U L x x U or for for x L U(x) U { } ( 29 ( 29 x E dx x d m L x ψ ψ =- ≤ ≤ 2 2 2 2 : for { } ( 29 ( 29 ( 29 x E x U dx x d m L x x ψ ψ ψ = +- < 2 2 2 2 : or...
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lecture37_38 - Schrodinger equation and standard problems...

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