Chapter6_17 - = <A Ψ | and (A op ↑ | Ψ >) ↑ =...

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PHY3063 R. D. Field Department of Physics Chapter6_17.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: = <A Ψ | and (A op ↑ | Ψ >) ↑ = < Ψ |A op = <A ↑ Ψ | Expectation Value: The average value of the dynamical variable O is given by <O>=< Ψ |O op | Ψ >=< Ψ |O Ψ > and <O>=< Ψ |O op | Ψ >=<O ↑ Ψ | Ψ > . Expectation Value of Hermitian Operators: The expectation values of hermitian operators are real. Proof: The expectation value of the dynamical variable H op is <H> = < Ψ |H op | Ψ > and taking the complex conjugate of both sides gives <H> * = < Ψ |H op | Ψ > * = < Ψ |H op ↑ | Ψ > = < Ψ |H op | Ψ > = <H>, where I used H op = H op ↑ . Thus, <H> = <H> * , which means that <H> is real....
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This note was uploaded on 05/29/2011 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.

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