The physics of 2-state systems

Outline Symmetry Oscillations Density matrices Keywords and ReferencesDensity matrices: general considerations1 A density matrix, ρ, describes an ensemble of quantumsystems. The expectation value of a measurement, A, overthe ensemble is 〈A〉 = TrAρ.2 Since the overlap of any state with itself is unity, Trρ = 1.3 The density matrix for a pure state, |ψ〉 is ρ = |ψ〉〈ψ|. In thiscase Trρ = 1 and Trρ 2 = 1.4 In terms of a complete set of states, |i〉, one can writeρ = ∑ ijρ ij |i〉〈j|.5 Since |j(t)〉 = U(t)|j〉, ρ(t) is obtained by the adjointaction— U(t)ρ(0)U † (t). As a result,i d ρ(t) = [H,ρ(t)].dtThis is called von Neumann’s equation.Sourendu Gupta Quantum Mechanics 1 2013: Lecture 5