Chapter2_11 - = <A | and (A op | >) =...

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PHY4604 R. D. Field Department of Physics Chapter2_11.doc University of Florida Dirac “Bracket” Notation (2) Complex Conjugation: The complex conjugate is as follows () > Ψ Ψ ≡< > Ψ Ψ < 2 1 * 1 2 | | , and () > Ψ Ψ ≡< > Ψ Ψ < 2 1 * 1 2 | | | | op op O O . Schwartz Inequality: One can show that |< Ψ 1 | Ψ 2 >| 2 < Ψ 1 | Ψ 1 >< Ψ 2 | Ψ 2 > Hermitian Conjugation: (O op | Ψ >) = < Ψ |O op (< Ψ |O op ) = O op | Ψ > < Ψ 2 |O op | Ψ 1 > = (< Ψ 1 |O op | Ψ 2 >) * Linear Operators: Linear operators operate on “Ket-vectors” producing other “Ket-vectors”: A op | Ψ > = | A Ψ > and A op | Ψ > = |A Ψ > (operator)(“Ket-vector”) = (“Ket-vector”) (operator)(“Ket-vector”) = (“Ket-vector”) Taking the hermitial conjugate of both sides of both equations gives (A op | Ψ >) = < Ψ |A op
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Unformatted text preview: = &lt;A | and (A op | &gt;) = &lt; |A op = &lt;A | Expectation Value: The average value of the dynamical variable O is given by &lt;O&gt;=&lt; |O op | &gt;=&lt; |O &gt; and &lt;O&gt;=&lt; |O op | &gt;=&lt;O | &gt; . Expectation Value of Hermitian Operators: The expectation values of hermitian operators are real. Proof: The expectation value of the dynamical variable H op is &lt;H&gt; = &lt; |H op | &gt; and taking the complex conjugate of both sides gives &lt;H&gt; * = &lt; |H op | &gt; * = &lt; |H op | &gt; = &lt; |H op | &gt; = &lt;H&gt;, where I used H op = H op . Thus, &lt;H&gt; = &lt;H&gt; * , which means that &lt;H&gt; is real....
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