Chapter3_4 - ∑ ∑ = ∗ = =>><<>=>=<<...

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PHY4604 R. D. Field Department of Physics Chapter3_4.doc University of Florida Insert a complete set of states! Summary – Discrete Case Eigenvalue Equation: Consider the eigenkets and eighnvalues of the hermitian operator A op . > >= n n n op a a a A | | , where n = 1, 2, ..., n max . The constatnts a n are real and the eigenkets for an orthonormal set ij j i a a δ >= < | . Expansion Theorem: The eigenkets of the hermitian operator A op form a complete set of states, in terms of which any “ket” (or wave function) can be expanded ( the A-representation ) as follows: = > >= max 1 | | n n n a a n ψ ψ with 1 | | max 1 = >< = n n n n a a The expansion coefficients are complex numbers and are given by i n a n n n i a i a a a ψ ψ ψ = > < >= < = max 1 | | or > =< ψ ψ | i a a i . Probability Interpretation: If we make a measurement the probability of measuring a n is 2 | | n n a a P ψ = , Overlap: The overlap between two arbitrary wave functions is
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Unformatted text preview: ∑ ∑ = ∗ = = > >< < >= >=< < max max 1 1 | | | | | n n a a n n n n n n a a I φ . Transformation from A-rep to B-rep: Suppose there is another hermitian operator B op . > >= n n n op b b b B | | , where n = 1, 2, . .., n max . We can expand the ket | ψ > in terms of the eigenkets of B op ( the B-representation ) we get ∑ = > >= max 1 | | n n n b b n with 1 | | max 1 = >< ∑ = n n n n b b . The coefficients n a are related to the coefficients n a by ∑ ∑ = = > < = > >< < >= >=< =< max max 1 1 | | | | | | n n b n i n n n n i i i a n i b a b b a I a a . The “transformation matrix” is given by T ij = <a i |b j > ....
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