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Chapter3_8 - The subspace on which it projects is the...

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PHY4604 R. D. Field Department of Physics Chapter3_8.doc University of Florida Operators Relations (1) Hermitian and AntiHermitian: H op = H op (hermitian) I op = -I op (antihermitian) Any operator can be written as A op = H op + I op where H op =(A op +A op )/2 and I op =(A op -A op )/2. The product of two hermitian operators, A op B op , is hermitian only if they commute since (A op B op ) = B op A op = B op A op = A op B op +[B,A] Inverse Operator: The operator A op -1 is the inverse of A op provided A op A op -1 = A op -1 A op = 1 op Note that (A op B op ) -1 = B op -1 A op -1 Unitary Operator: The operator U op is unitary if its inverse is equal it its hermitian conjugate, U -1 op = U op , and hence U op U op = U op U op = 1 op Unitary operators “conserve” probability since if < Ψ | Ψ > = 1 then if U op | Ψ > = |U Ψ > then <U Ψ |U Ψ > = < Ψ | U op U op | Ψ > = < Ψ | Ψ > = 1 . Projection Operators: Any hermitian operator P op satisfying (P op ) 2 = P op is a projection operator.
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Unformatted text preview: The subspace on which it projects is the subspace with eigenvalue 1 . P op | Ψ > = λ | Ψ > and 0 = (P op 2- P op )| Ψ > = ( λ 2 – λ )| Ψ > Hence λ = 1 or λ = 0 . Any state can be written an | Ψ > = P op | Ψ > + (1-P op )| Ψ > = |P Ψ > + |(1-P) Ψ > where | P Ψ > has eigenvalue 1 and |(1-P) Ψ > has eigenvalue . Note that the sates |P Ψ > and |(1-P) Ψ > are orthogonal. Commuting Operators: Consider an hermitian operator H op with satisfying an eigenvalue equation H op | Ψ a > = a| Ψ a > If [A op ,H op ] = 0 then H op A op | Ψ a > = H op | A Ψ a > = A op H op | Ψ a > = aA op | Ψ a > = a| A Ψ a >. Hence, the state |A Ψ a > = A op | Ψ a > is also an eigenket of H op with the same eigenvalue a ....
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