Chapter2_18 - PHY4604 R. D. Field Momentum-Space...

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PHY4604 R. D. Field Department of Physics Chapter2_18.doc University of Florida Momentum-Space Wavefunctions Change Variable: Since k p x h = we can express the position-space wave function as follows: +∞ +∞ Φ = = Ψ x x ip x x t p E x p i x dp e t p dp e p t x x x x h h h h / / ) ) ( ( ) , ( 2 1 ) ( 2 1 ) , ( π φ , where ) 2 /( ) ( 2 m p p E x x = . For now it is sufficient to consider t = 0 where +∞ = = Ψ x x ip x dp e p x x x h h / ) ( 2 1 ) ( ) 0 , ( ψ and +∞ = dx e x p x ip x x h h / ) ( 2 1 ) ( with ) ( 2 1 / ) ( y x dp e x y x ip x = +∞ δ h h . Note that h h / ) / ( ) ( x x p f p = , where f(k) is the Fourier transform. Also, 1 | ) ( ) ( ' ) ' ( ) ( ) ' ( ' ) ( ) ' ( 2 1 ) ( ' ) ' ( 2 1 ) ( ) ( | / ) ' ( / / ' >= =< = = = = >= < ∫∫ + + + + + + + + + + +∞ dx x x dxdx x x x x dxdx dp e x x dp dx e x dx e x dp p p x x x ip x x ip x ip x x x x x x h h h h h Probability Density: In position-space 2 ) ( ) ( x x ρ = = “position-space probability density” such that
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