Unformatted text preview: i.e. orthonormal like the discrete solutions < ψ n  ψ m >= δ nm ). Most General Solution: The most general solution is a superposition of the free particle eigenfunctions as follows ∫ ∫ +∞ ∞ − − +∞ ∞ − = Ψ = Ψ dk e k f dk t x k f t x t k kx i k ) ) ( ( ) ( 2 1 ) , ( ) ( ) , ( , where ) 2 /( ) ( 2 m k k h = . The dependence of ω on k is called the dispersion . The coefficient function f(k) (analogous to the overlap constants c n =< ψ n  Ψ >) is given by ∫ +∞ ∞ − − Ψ = dx e x k f ikx ) , ( 2 1 ) ( . Fourier Transforms: f(k) is the Fourier transform of F(x) and F(x) is the inverse Fourier transform of f(k) as follows: ∫ +∞ ∞ − = dk e k f x F ikx ) ( 2 1 ) ( ↔ ∫ +∞ ∞ − − = dx e x F k f ikx ) ( 2 1 ) (...
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 Spring '07
 FIELDS
 mechanics, Fourier Series, Energy, Fundamental physics concepts, free particle, dk

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