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Chapter6_13

# Chapter6_13 - conjugate” Namely “hermitian” operators...

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PHY3063 R. D. Field Department of Physics Chapter6_13.doc University of Florida Hermitian Operators Operators: Operators acting on a function maps it into another function . The following are examples of operators: [ ] ) ( ) ( ) ( 2 ) ( ) ( ) 1 3 ( ) ( ) ( ) ( ) ( ) ( 2 2 2 x cf x f O x f dx x df x f O x f x f O x f x f O x x f x f O op op op op op = = + = = + = Linear Operators: Linear operators are the subset of operators with the following properties (a and b are constants): [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 x f bL x f aL x bf x af L x f aL x af L op op op op op + = + = Hermitian Operators: Let O op be an operator and define the “hermitian conjugate” operator O op to be the operator such that [ ] Ψ Ψ = Ψ Ψ dx t x O t x dx t x t x O op op ) , ( ) , ( ) , ( ) , ( 1 * 2 1 * 2 . An operator is said to be “hermitian” if it is equal to its “hermitian
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Unformatted text preview: conjugate”. Namely “hermitian” operators have the property that O op = O ↑ op Properties of Hermitian Operators: 1. The expectation value of an hermitian operator is a real number: O op = O ↑ op ⇒ <O op > = <O op > * 2. If the expectation value of the operator O op is a real number then O op is hermitian: <O op > = <O op > * ⇒ O op = O ↑ op Note that (A op B op ) ↑ = B ↑ op A ↑ op We will prove this later after we learn “Dirac Notation”....
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