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Chapter2_19

# Chapter2_19 - PHY4604 R D Field Momentum-Space Operators...

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PHY4604 R. D. Field Department of Physics Chapter2_19.doc University of Florida Momentum-Space Operators Expectation Value of (p x ) op : We see that + + + + + +∞ = = >= < x x x x x x ip x x x ip x op x dp p p p dx dp e p dx d dp e p i dx dx x d x i p x x ) ( ) )( ( ) ( ' ) ' ( 2 ) ( ) ( | ) ( | / / ' φ φ φ φ π ψ ψ ψ ψ h h h h h Hence, when acting on momentum-space wavefunctions (p x ) op = p x . Expectation Value of (x) op : In momentum-space the (x) op is given by x op p i x = h ) ( . The proof is as follows: + + + + + +∞ = = >= < dx x x x dxdp e x dp d dx e x i dp dp p d p i x x x ip x x ip x x x x op x x ) ( ) )( ( ) ( ' ) ' ( 2 ) ( ) ( | ) ( | / / ' ψ ψ ψ ψ π φ φ φ φ h h h
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