Problems for : Selected Topics in Quantum Theory

A set of problems is offered to accompany the set of lectures with the above title. Topic 1: Coherent States 1.1 Let {|n〉}, n = 0, 1, 2, 3, . . ., denote a complete set of orthonormal vectors in an abstract Hilbert space , with 〈m|n〉 = δmn. Next, let |χ〉 = √ 1 − |χ|2 |0〉 + χ|1〉 , for all complex χ, 0 ≤ |χ| ≤ 1, denote a family of two-dimensional vectors. a) If χ→ χ in the sense of complex numbers, show that |χ〉 → |χ′〉 as vectors, namely that ‖|χ〉 − |χ〉‖ → 0 , where ‖|ψ〉‖ := √ 〈ψ|ψ〉. This exercise shows continuity of the labeling of the vectors. b) Find a weight function ρ(|χ|) ≥ 0 such that