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 π , 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 , ( φ π ψ and +∞ = dx e x p x ip x x h h / ) ( 2 1 ) ( ψ π φ with ) ( 2 1 / ) ( y x dp e x y x ip x = +∞ δ π h h . Note that h h / ) / ( ) ( x x p f p = φ , where f(k) is the Fourier transform. Also, 1 | ) ( ) ( ' ) ' ( ) ( ) ' ( ' ) ( ) ' ( 2 1 ) ( ' ) ' ( 2 1 ) ( ) ( | / ) ' ( / / ' >= =< = = = = >= < ∫ ∫ + + + + + + + + + + +∞ ψ ψ ψ ψ δ ψ ψ ψ ψ π ψ ψ π φ φ φ φ 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
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