05_03_30 - [email protected] 110 Projection...

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Unformatted text preview: [email protected] 110 Projection probabilities and ket vectors 2 1 1 | | y probabilit with A E 2 2 1 1 2 1 ) , ( A e A e t x t i t i E E -- Φ + Φ = Ψ ∫ ∞ ∞-- Ψ Φ = Ψ = dx t x x A e t i E ) , ( ) ( 1 * 1 1 1 ¡ 2 1 2 1 2 1 A e A e t i t i E E ¢ ¢-- + = Ψ Ψ = Ψ x t x ) , ( 2 | ) , ( | y probabilit with t x x Ψ n x x n = Φ ) ( Measurement: Projection amplitude: Ket vector: ∑ Ψ = Ψ x x x all 03/30/2005 [email protected] 111 03/30/2005 Fourier Series Any “well behaved” periodic function f(x)=f(x+L) can be written as the sum of elementary periodic functions-2-1 1 2-3-2-1 1 2 (L) f(x) ) sin( ) cos( 2 2 2 x n i x n e L L x in L π π π + = ∑ ∞-∞ = = n x in n L e a x f π 2 ) ( ∑ ∞-∞ =-- = n x m n i n x im L L e a e x f π π 2 2 ) ( ) ( m x im L a dx e x f L L L = ∫-- 2 2 2 ) ( 1 π Real functions f(x): * m m a a =- ∑ ∑ ∑ ∞ = ∞ = ∞ =- + = + = 1 2 1 2 1 ) sin( ] Im[ 2 ) cos( ] Re[ 2 ] Re[ 2 ) ( 2 n L n n L n n x...
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This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell.

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05_03_30 - [email protected] 110 Projection...

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