Lect3_Operators - Operators In QM an operator is an object...

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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 | ψ〉 = | ψ Operators are linear: A ( a | ψ 1 +b | ψ 2 ) = aA | ψ 1 + bA | ψ 2 a and b are arbitrary c-numbers
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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 ψ 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 | ψ〉 is thus the j th component of the vector | ψ〉 Matrix Representation of an operator: Start from the equation: Hit it from the left with the bra j | : Insert `the identity’ between A and | ψ〉 : Use the replacements: c j j | ψ , d j j | ψ and A jk j | A | k to get: This is just the standard formula for matrix multiplication: ψ ψ A = ' ψ ψ A j j = ψ ψ k k A j j M k = = 1 k M k jk j c A d = = 1 v d = A r c
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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 Gives components implicitly instead of explicitly
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