From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Operator Theory question regarding uncertainty principle Date: 19 Jan 1999 17:55:59 -0500 Newsgroups: sci.math Keywords: Heisenberg's inequality in Operator Theory In article <782sf9$lir$1@nnrp1.dejanews.com>, wrote: >Hi, > >I have a note here stating that when two operators satisfy the following >property: > >PQ-QP=I > >and ||f(t)|| =1 in L^2, that these two facts alone establish > >1/2 < ||Pf(t)||*||Qf(t)|| > >I can prove this in the specific case where Pf = t*f(t) and Qf=df/dt, but >I can't prove it just from the PQ-QP=I statement. > >Can you? Are there other qualifications on the operators necessary? If so, >what are they? > From Dunford and Schwartz: Linear Operators II, Exercise XII.9.40 (edited for ASCII): Let A and B be self-adjoint operators in Hilbert space such that D_0 = dom(AB) intersect dom(BA) is dense. Let x be in D_0. Write E(C) = (Cx, x) and V(C) = norm((C-E(C)*I)x)^2 for every C for which Cx is defined. Then Heisenberg inequality says V(A) * V(B) >=(1/4) * (abs(E(AB-BA)))^2 So, the note you mentioned has to be amended. I've gone through the exercise, and it is an application of Cauchy-Schwarz inequality. The self-adjointness of A and B is important. The subtraction of E(C)*I in the definition of V(C) is not essential since AB-BA = [A,B] = [A-p*I, B-q*I] for scalars p, q. It does make good sense, though, because it minimizes the norm. Good luck, ZVK(Slavek).