Chapter3_5 - The overlap between two arbitrary wave...

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PHY4604 R. D. Field Department of Physics Chapter3_5.doc University of Florida Insert a complete set of states! Summary – Continuous Case Eigenvalue Equation: Consider the eigenkets and eighnvalues of the hermitian operator A op . > >= a a a A op | | , where a is a continuous ( real ) variable. The eigenkets for an orthonormal set ) ' ( | ' a a 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: > >= da a a | ) ( | ψ with 1 | | = >< da a a The expansion coefficients are complex functions and are given by ) ( ' ' | ) ' ( | a da a a a a = > < >= < or > =< | ) ( a a . Probability Interpretation: If we make a measurement the probability of measuring a between a and a+da is 2 | ) ( | ) ( a a ρ = , Overlap:
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Unformatted text preview: The overlap between two arbitrary wave functions is da a a da a a I = &gt; &gt;&lt; &lt; &gt;= &gt;=&lt; &lt; ) ( ) ( | | | | | . Transformation from A-rep to B-rep: Suppose there is another hermitian operator B op . &gt; &gt;= b b b B op | | , where b is a continuous ( real ) variable. We can expand the ket | &gt; in terms of the eigenkets of B op ( the B-representation ) we get &gt; &gt;= db b b | ) ( | with 1 | | = &gt;&lt; db b b . The coefficient functions (a) are related to the coefficient functions (a\b) by &gt; &lt; = &gt; &gt;&lt; &lt; &gt;= &gt;=&lt; =&lt; db b b a db b b a I a a a ) ( | | | | | | ) ( , where &lt;a|b&gt; is the transformation function....
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