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05_02_16 - 39 The stationary Schrdinger equation i x = k...

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[email protected] 39 The stationary Schrödinger equation ) ( ) ( ) ( ) ( ) ( ) , ( 2 2 x E x x V x e x t x x x m t i Φ = Φ + Φ - Φ = Ψ - ϖ 02/16/2005 Conclusion: whenever needs to be computed, one can use Ψ - x i Ψ k Ψ = Ψ - k i x Ψ + Ψ = Ψ x x x ik k i ) ( 2 2 A wave function changes significantly over a wavelength. Whenever the potential does not change much over a wavelength, Ψ Ψ Ψ ∆Ψ Ψ ∆Ψ ) ( k k x k x k k k x x Ψ - Ψ + Ψ = Ψ 2 ) ( 2 2 k ik k i x x x The correspondence principle has to hold in the classical limit where the potential always changes little over the very small wavelength.
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[email protected] 40 The time dependent Schrödinger equation Linearity for equal energies : If < and < are solutions of for a common energy E , then also <± ±<²< is a solution. ϖ ϖ ϖ ϖ ϖ d e x e x t x t i n t i n n - - = - Φ Φ = Ψ ) , ( ) ( ) , ( 0 ) , ( ) , ( ) ( ) , ( 2 2 t x i t x x V t x t x x m Ψ = Ψ + Ψ -
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