08_580_hw_4 - Physics 580 Quantum Mechanics I Prof P M...

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Physics 580 Handout 8 21 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 4 Prof. P. M. Goldbart University of Illinois 1) Operators and matrix elements : a) Show that if Λ 1 Λ = I and Ω 1 Ω = I then (ΩΛ) 1 = Λ 1 1 . b) Suppose that {| i ⟩} forms an orthonormal basis. Show that if Ω | i = j ji | j then k | | l = Ω kl . c) Show that if, in addition, Λ | i = j Λ ji | j then k | ΛΩ | l = m Λ km ml . Consider two arbitrary kets | ψ and | ϕ . d) Write down the adjoint of the operator Θ = | ψ ⟩⟨ ψ | ? e) Write down the adjoint of the operator Ξ = | ψ ⟩⟨ ψ | + i | ϕ ⟩⟨ ϕ | ? f) In the basis {| i ⟩} the operator Σ is represented by the matrix ( 0 i 1 + i 2 i ) . Construct the matrix that represents the adjoint operator Σ in the same basis? g) By using the relationship (Γ ) ij = (Γ ji ) show that for two arbitrary kets, | ψ and | ϕ , we have ψ | Θ | ϕ = ϕ | Θ | ψ . 2) Projection operators : Operators are called projection operators if they are idempotent, i.e. , Ω 2 = Ω. Once you have acted on a ket with an idempotent operator, repeated action with such an operator will no longer change the ket; the operator has lost its potency. Consider the operator
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This note was uploaded on 01/22/2012 for the course PHYSICS 850 taught by Professor Staff during the Fall '10 term at University of Illinois, Urbana Champaign.

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08_580_hw_4 - Physics 580 Quantum Mechanics I Prof P M...

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