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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 = + 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 = = = = = ( 29 x mE i Ce x mE k 2 2 2 = = = 1a. Classical standing waves on a string (review) n = 1,2,3... 1 2 1 = L L L 2 1 2 1 = = 2 2 2 2 = = L L L = = 2 2 2 3 2 3 = L 3 2 3 L = n n L 2 = n L n 2 = ( 29 n n n n k x k C y 2 sin = = x y 1. Quantum particle in a box ( 29 = n n x C x 2 sin m L h n m p E n n 2 2 2 2 8 2 = = 1b. Quantum particle in a 1dimensional 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 x L U(x) = 2 n L n Quantum numbers: n = 1,2 Wavelength: Momentum and energy are quantized L nh h p n n 2 = = Wave function: ( 29 ( 29 L x C x sin 1 = ( 29 ( 29 L x n C x n 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 3dimensional 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 ( 29 L C L C dx L x n C L 2 1 2 1 sin 2 2 2 = = = ( 29 ( 29 L x n L x n 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 = 2 2 2 2 : for { } ( 29 ( 29 ( 29 x E x U dx x d m L x x = + < 2 2 2 2 : or...
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This note was uploaded on 02/11/2012 for the course PHYSICS 222 taught by Professor Ogilvie during the Fall '05 term at Iowa State.
 Fall '05
 Ogilvie
 Schrodinger Equation

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