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Chapter2_17

# Chapter2_17 - i.e orthonormal like the discrete...

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PHY4604 R. D. Field Department of Physics Chapter2_17.doc University of Florida The Free Particle A stationary state free particle ( i.e. V(x) = 0) with energy E must satisfy ) ( ) ( 2 2 2 2 x E dx x d m ψ ψ = h and hence ikx Ae x ± = ) ( ψ where A is a constant and E m k = ) 2 /( 2 2 h . But ψ (x) is also an eigenstate of (p x ) op as follows: > ± >= ψ ψ | | ) ( k p op x h . Thus, ) ) ( ( ) , ( t k kx i k Ae t x ω = Ψ corresponds free particle with definite momentum k p x h = (all values of k allowed) and definite energy ) 2 /( ) ( 2 2 m k k E h h = = ω (k > 0 corresponds to the wave moving to the right and k < 0 corresponds to the wave moving to the left). However, these solutions are not normalizable and hence not allowed. A free particle cannot exist in a stationary state ; or put another way, there is no such thing as a free particle with definite energy . The overlap of one solution with another is given by ) ' ( ) ' ( | | 2 | | | 2 ) ' ( )) ' ( ) ( ( 2 ' k k k k A dx e e A x k k i t k k i k k = = >= Ψ Ψ < +∞ δ δ π ω ω , with π 2 / 1 = A . If k = k' we get infinite and if k
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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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