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 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 = > < >= < = max 1 | | or > =< | i a a i . Probability Interpretation: If we make a measurement the probability of measuring a n is 2 | | n n a a P = , Overlap: The overlap between two arbitrary wave functions is
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Unformatted text preview: = = = &gt; &gt;&lt; &lt; &gt;= &gt;=&lt; &lt; max max 1 1 | | | | | n n a a n n n n n n a a I . Transformation from A-rep to B-rep: Suppose there is another hermitian operator B op . &gt; &gt;= n n n op b b b B | | , where n = 1, 2, . .., n max . We can expand the ket | &gt; in terms of the eigenkets of B op ( the B-representation ) we get = &gt; &gt;= max 1 | | n n n b b n with 1 | | max 1 = &gt;&lt; = n n n n b b . The coefficients n a are related to the coefficients n a by = = &gt; &lt; = &gt; &gt;&lt; &lt; &gt;= &gt;=&lt; =&lt; 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 = &lt;a i |b j &gt; ....
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