Density matrix homework

The most general density matrix can be constructed from ˙as ˆ= 1 2 (1 + a˙) where a is a real vector. And we see as noted above that we need to measure 3 observables, namely the polarization, to determine the state of the ensemble. The density matrix for a spin 1 system has 8 independent parameters.

Although describing a quantum system with the density matrix is equivalent to using the wavefunction, one gains significant practical advantages using the density matrix for certain time-dependent problems – particularly relaxation and nonlinear spectroscopy in the condensed phase.

Theorem 9.1 The expectation value of an observable A in a state, represented by a density matrix ˆ, is given by hAi ˆ = Tr(ˆA) Proof: Tr(ˆA) = Tr (j ih jA) = X n hnj ih jAjni= = X n h jAjnihnj | {z } 1 i= h jAj i= hAi: q.e.d.(9.8) 9.2 Pure and Mixed States Now we can introduce a broader class of states represented by density matrices, the so-

This expression states that the density matrix elements represent values of the eigenstate coefficients averaged over the mixture: Diagonal elements (nm= ) give the probability of occupying a quantum state n : ρ =cc * =p ≥0 (1.18) nn n n n. For this reason, diagonal elements are referred to as populations.