Chapter3_6 - p | | where p x is a continuous real variable...

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PHY4604 R. D. Field Department of Physics Chapter3_6.doc University of Florida Insert a complete set of states! Insert a complete set of states! Example: Position and Momentum (1) Position-Space Representation: Consider the eigenkets and eighnvalues of the hermitian operator x op . > >= x x x x op | | , where x is a continuous ( real ) variable. The eigenkets for an orthonormal set ) ' ( | ' x x x x >= < δ . The eigenkets of the position operator form a complete set of states, in terms of which any “ket” can be expanded ( the x-representation ) as follows: > >= dx x x | ) ( | ψ ψ with 1 | | = >< dx x x The expansion coefficients are complex functions and are given by ) ( ' ' | ) ' ( | x dx x x x x ψ ψ ψ = > < >= < or > =< ψ ψ | ) ( x x , where ψ (x) is the position-space wave function. If we make a measurement the probability of measuring x between x and x+dx is 2 | ) ( | ) ( x x ψ ρ = . The overlap between two arbitrary wave functions is dx x x dx x x I = > >< < >= >=< < ) ( ) ( | | | | | ψ φ ψ φ ψ φ ψ φ . Momentum-Space Representation: Consider the eigenkets and eighnvalues of the hermitian operator (p
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Unformatted text preview: p | | ) ( , where p x is a continuous ( real ) variable. The eigenkets for an orthonormal set ) ' ( | ' x x x x p p p p − >= < . The eigenkets of the momentum operator form a complete set of states, in terms of which any “ket” can be expanded ( the p x-representation ) as follows: ∫ > >= x x x dp p p | ) ( | with 1 | | = >< ∫ dx p p x x The expansion coefficients are complex functions and are given by ) ( ' ' | ) ' ( | x x x x x x p dp p p p p = > < >= < ∫ or > =< | ) ( x x p p , where ψ (p x ) is the momentum-space wave function. If we make a measurement the probability of measuring p x between p x and p x +dp x is 2 | ) ( | ) ( x x p p = , The overlap between two arbitrary wave functions is x x x x x x dp p p dp p p I ∫ ∫ ∗ = > >< < >= >=< < ) ( ) ( | | | | | ....
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