Chapter6_15 - x i x x x i x x x i x i x x x i x x x i x p x...

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PHY3063 R. D. Field Department of Physics Chapter6_15.doc University of Florida Commutator and AntiCommutator Commutator: The commutator of the operator A op and the operator B op is defined as follows: Commutator [A op ,B op ] A op B op - B op A op AntiCommutator: The anti-commutator of the operator A op and the operator B op is defined as follows: AntiCommutator {A op ,B op } A op B op + B op A op Commuting Operators: The operator A op and the operator B op are said to “commute” if their commutator is zero. For example, the operators (p x ) op and (p y ) op commute since 0 ) , ( ) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) , ( ] ) ( , ) [( 2 2 2 2 = Ψ + Ψ = Ψ Ψ = Ψ x y y x y x y x y x p p y x p p y x p p op x op y op y op x op y op x h h Thus, 0 ] ) ( , ) [( = op y op x p p . Non-Commuting Operators: Not all operators commute. For example, the operators (x) op and (p x ) op do not commute since ) ( ) ( ) ( ) ( ) ( ))
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Unformatted text preview: ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ] ) ( , ) [( x i x x x i x x x i x i x x x i x x x i x p x x x p x x p op x op op op x op op x Ψ − = ∂ Ψ ∂ + ∂ Ψ ∂ − Ψ − = ∂ Ψ ∂ + ∂ Ψ ∂ − = Ψ − Ψ = Ψ h h h h h h Thus, h i x p op op x − = ] ) ( , ) [( Canonical Commutation Relations: It is easy to see that ] ) ( , ) [( ] ) ( , ) [( ] ) ( , ) [( = = − = op j op i op j op i ij op j op i x x p p i x p δ h where δ ij is the Kroenecker delta function (i = 1,2,3 j = 1,2,3 δ ij = 0 if i ≠ j and δ ij = 1 if i = j, p 1 = p x , p 2 = p y , p 3 = p z , x 1 = x, x 2 = y, x 3 = z) and i op i x i p ∂ ∂ − = h ) ( ....
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