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PHY 6645 - Quantum Mechanics I - Fall 2011 Homework set # 4, due September 21 1. Show that Ω e = lim n→∞ n 1 1+ Ω n (0.1) where Ω is a linear operator. 2. Show that if U is a linear operator on a vector space V and if for all vectors V ∈ V < V |V >=< V |V > (0.2) where |V >= U|V >, then U must be a unitary operator on V. 3. If X and P are canonically conjugate observables, we have X = X † , P = P † and [X, P ] = h̄i. Using these equations, show that , where F (P ) is any function of the operator P . a. [X, F (P )] = h̄idF dP n n d h̄ n b. < x|P |Ψ >= i dx n < x|Ψ > where the |x > are the eigenstates of X. ixp/h̄ where the |p > are the eigenstates of P and N is a normalization c. < x|p >= Ne constant. 4. Problem 4.2.1 in Shankar’s book. 1