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Lect3_Operators

# Lect3_Operators - Notation Operators In QM an operator is...

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Operators In QM, an operator is an object that acts on a ket, transforming it into another ket – Let A represent a generic operator An operator is a linear map A : H ! H A | "# = | " # Operators are linear: A ( a | " 1 # +b | " 2 # ) = aA | " 1 # + bA | " 2 # a and b are arbitrary c-numbers Notation: Generally, we will follow Cohen-Tannoudji, and use capital letters for operators and lower-case letters for c-numbers. Another common convention is to distinguish operators by giving them a ‘^’ I may use this occasionally a ˆ ! ˆ A ˆ

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Matrix representations Just as kets are vectors, operators are matrices Let the set { | 1 # , | 2 # , | 3 # ,…, | M # } be a set of orthogonal unit vectors which spans an entire M -dimensional Hilbert space The c-number \$ j | "# is thus the j th component of the vector | "# Matrix Representation of an operator: Start from the equation: Hit it from the left with the bra \$ j | : Insert `the identity’ between A and | "# : Use the replacements: c j % \$ j | " # , d j % \$ j | " # and A jk % \$ j | A | k # to get: This is just the standard formula for matrix multiplication: ! ! A = ' ! ! A j j = " ! ! k k A j j M k " = = # 1 k M k jk j c A d ! = = 1 v d = A r c Defining states and operators A state (vector) is specified by giving its components in some physically meaningful basis An operator is defined by giving its matrix elements in some physically meaningful basis Operators and/or states can alternatively be defined as the solution to a particular equation
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