07_580_hw_3 - Physics 580 Quantum Mechanics I Prof. P. M....

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Physics 580 Handout 7 14 September 2010 Quantum Mechanics I webusers.physics.illinois.edu/ goldbart/ Homework 3 Prof. P. M. Goldbart University of Illinois 1) Schwarz inequality; triangle inequality : Let | S and | T be two vectors. a) By considering the positivity of the norm of the vector | S ⟩ − T | S T | T | T establish the validity of the Schwarz inequality, i.e., S | S ⟩⟨ T | T ⟩ ≥ |⟨ S | T ⟩| 2 . b) Give, and describe the origin of, the condition under which the inequality becomes an equality. c) By considering the norm of the vector | S + | T , and then using the Schwarz inequality, prove the triangle inequality {⟨ S | + T |}{| S + | T ⟩} ≡ | S + T | ≤ | S | + | T | ≡ S | S + T | T . d) Give, and describe the origin of, the condition under which the inequality becomes an equality. 2) Bras as linear functionals : The two-dimensional linear vector space over the real numbers V (2) ( R ) is spanned by the two orthonormal basis vectors | 1 and | 2 . a) What are the values of the elements of the matrix of inner products ( 1 | 1 ⟩ ⟨ 1 | 2 2 | 1 ⟩ ⟨ 2 | 2 ) ? An arbitrary ket | ψ is a linear combination of these basis vectors: | ψ = | 1 a 1 + | 2 a 2 . b) Show that the appropriate expansion coefficients, a 1 and a 2 , are given by a 1 = 1 | ψ and a 2 = 2 | ψ . Recall that a linear functional is an object ϕ | , called a bra, that acts on a ket | θ and returns the scalar ϕ | θ . It is natural to choose as a basis for the set of linear functionals on V (2) ( R ) the bras 1 | and 2 | . For example, when we act with 1 | on | θ the result is 1 | θ . A general bra may be expanded in terms of the basis as
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07_580_hw_3 - Physics 580 Quantum Mechanics I Prof. P. M....

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