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**Unformatted text preview: **(not in text!!) dτ ≡ <| > Bra: <| = * ; Ket: > = Then: <| >* = * * dτ Normalization: dτ = < > = 1 ; < >* = 1* = 1 3/5 Hermitian Operators & Theorems dτ = * * dτ ; 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