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05_03_02 - 63 Orthogonality for different Energies It is a...

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[email protected] 63 03/02/2005 Orthogonality for different Energies It is a feature of the wave functions of any potential V(x) that the stationary wave functions are orthogonal to each other. E E E x m E x V Φ = Φ + Φ - ) ( 2 2 2 2 [ ] - - - - - Φ Φ - Φ Φ = Φ Φ Φ + Φ - Φ = Φ Φ dx o dx dx x V dx E E x E x E x E E x E E E x m E E E 2 1 2 1 2 2 2 1 2 2 2 2 2 1 2 1 * * * 2 * * 2 ] ) ( [ ± ± ² ± ± ³ ´ - - Φ Φ + Φ Φ = Φ Φ dx x V dx E E E E x E x m E E ] ) ( [ 2 1 2 1 2 2 1 * * 2 * 2 - - Φ Φ + Φ Φ = Φ Φ dx x V dx E E E E x E x m E E ] ) ( [ 2 1 2 1 2 2 1 * * 2 * 1 = = = Φ Φ = Φ Φ - - - m n m n nm E E E E E E E E dx dx E E m n if 1 if 0 0 ) ( * * 2 1 2 1 δ
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[email protected] 64 03/02/2005 The harmonic oscillator: ) ( ) ( ) ( 2 2 1 2 2 2 2 x E x Cx x x m Φ = Φ + Φ - 2 0 2 2 max 0 2 2 1 : amplitude n oscillatio Maximum with n oscillatio classical ) ( ϖ ϖ m
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