Lec+5+BraKet+Th+I+_+II+2015

# Lec+5+BraKet+Th+I+_+II+2015 - F Grieman 158 a Lecture 5 Rvw...

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158 a Lecture 5 Rvw: Time Independent Sch. Eq. Bra Ket Notation Hermitian Operator Theorems Q.M. Treatment of Translational Motion (on board) F. Grieman 1/5

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( - ħ 2 / 2m ) [1/ (x)]  x 2 ) (x) + V(x ) = i ħ [ 1/ (t) ]  t (t) = E Take time equation i ħ d dt = E Separate variables  d    ( E/ i ħ ) dt ln = Et/ i ħ + ln A (t) = Ae Et/ i ħ = Ae i Et/ Take spatial equation ( 2 / 2m ) [1/ ( x )] 2  x 2 ( x ) + V( x ) = E ( 2 / 2m ) 2 (x) + V(x) (x) = E (x)  x 2 Ĥ (x) = E (x) Time Independent Schroedinger Equation !!! Our starting point for our Q.M. Schroedinger Eq. 2/5
Overall Wave Function and Probability (,t) = () (t) =  () Ae i Et/ dP = Ψ*Ψ dτ =  () A e + i Et/ () A e i Et/ dP =  () () ; just spatial part Bra Ket Notation

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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

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