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Unformatted text preview: (not in text!!) dÏ„ â‰¡ < > Bra: < = * ; Ket: > = Then: < >* = * * dÏ„ Normalization: dÏ„ = < > = 1 ; < >* = 1* = 1 3/5 Hermitian Operators & Theorems dÏ„ = * * dÏ„ ; Hermitian Definition < > = < >* ; in Bra Ket notation Theorem I. Hermitian Operators have Real Eigenvalues If is Hermitian & is an eigenfunction of : = o i & * * = o i ** Does o i = o i * ??? <  > = <  >* because is Hermitian So: o i < > = o i *< >* but, < > = < >* ; Why? So, o i = o i * eigenvalue (observable) is real! 4/5 Theorem II. Eigenfunctions of the same Hermitian Operator are orthogonal (Proven in Problem 4.29 & handout) If is Hermitian & & are both eigenfunctions of : < > = 0 (orthogonal) If also Normalized < > = 1 (orthonormal) Similar to unit vectors: ; ; ; ; Now!!! Actually do Q.M. on our first system!!!! 5/5...
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 Fall '15
 Grieman
 Physical chemistry, pH, Hilbert space, Hermitian, bra ket notation, Hermitian Operator Theorems, Hermitian Operators & Theorems

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