The physics of 2-state systems

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Outline Symmetry Oscillations Density matrices Keywords and ReferencesDensity matrix for 2-state systemsThe density matrix for a 2-state system is ρ = 1/2+m·σ. As aresult, the expectation values 〈σ i 〉 = Trσ i ρ = m i . The eigenvaluesare 1/2±|m|; these need not be positive.Take the Hamiltonian to be H = −B·σ/2. Then[H,ρ] = − 23∑B j m k [σ j ,σ k ] = −ijk=13∑ǫ jkl B j m k σ l .jkl=1This can be reduced by multiplying both sides by σ k and taking atrace, to giveṁ = m×B.This looks like the classical equation for a spin precessing in aneffective magnetic field which can be constructed from theHamiltonian.Sourendu Gupta Quantum Mechanics 1 2013: Lecture 5
Outline Symmetry Oscillations Density matrices Keywords and ReferencesPure states: the Bloch sphereFor the pure state ensemble built from the generic quantum state|ψ〉 = cos θ 2 |E +〉+sin θ 2 e−iφ |E − 〉,one has m = (sinθcosφ,sinθsinφ,cosθ). We have chosen theeigenstates of the Hamiltonian as the basis states. Pure stateensembles correspond to points on a sphere (called the Blochsphere).The time evolution of this state adds a phase ωt to φ (whereω = (E − −E + )/. This is an example of the general evolution ofthe density matrix: the vector m precesses around a certaindirection.What happens to pure state ensembles of the eigenvectors of theHamiltonian?.Sourendu Gupta Quantum Mechanics 1 2013: Lecture 5
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The physics of 2-state systems

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