05_03_02 - [email protected] 63 03/02/2005...

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Unformatted text preview: [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 ] ) ( [ ¡ ∫ ∫ ∞ ∞- ∂ ∂ ∂ ∂ ∞ ∞- 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 ) ( * * 2 1 2 1 δ [email protected] 64 03/02/2005 The harmonic oscillator: ) ( ) ( ) ( 2 2 1 2 2...
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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 University (Engineering School).

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05_03_02 - [email protected] 63 03/02/2005...

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