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1 Introduction 2 The Density Operator

Physics 125cCourse NotesDensity matrix Formalism040511 Frank Porter1 IntroductionIn this note we develop an elegant and powerful formulation of quantum me-chanics, the Density matrix formalism. This formalism provides a structurein which we can address such matters as: We typically assume that it is permissible to work within an appropriatesubspace of the Hilbert space for the universe. Is this all right? In practice we often have situations involving statistical ensembles ofstates. We have not yet addressed how we might deal with The Density OperatorSuppose that we have a state space, with a denumerable orthonormal basis{|un ,n=1,2,..}. If the system is in state| (t) at timet, we have theexpansion in this basis:| (t) = nan(t)|un .(1)We ll assume that| (t) is normalized, and hence: (t)| (t) =1 = n man(t)a m(t) um|un = n|an(t)|2(2)Suppose that we have an observable (self-adjoint Operator )Q. The matrixelements ofQin this basis are:Qmn= um|Qun = Qum|un = um|Q|un.

Density Matrix Formalism 040511 Frank Porter 1 Introduction ... states. We have not yet addressed how we might deal with this. 2 The Density Operator Suppose that we have a state space, with a denumerable orthonormal basis ... Now ρis the density operator of the system, ...

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