Density matrix of spin 1 system

[Dishsoap]

Homework Statement

Consider an ensemble of spin 1 systems (a mixed state made of the spin 1 system). The density matrix is now a 3x3 matrix. How many independent parameters are needed to characterize the density matrix? What must we know in addition to S_x, S_y and S_z to characterize the mixed state completely?

2. The attempt at a solution

I know that 8 parameters are needed to characterize the density matrix (9 elements, minus one because we know the trace of the density matrix is one.) However, I'm not sure what else is needed to characterize it... S_x, S_y and S_z are all that is needed for the spin 1/2 state, which is all that is treated in all of the books in which I've found a treatment of the density matrix (Sakurai, Gottfried/Yan, Shankar).

When I Google the problem, I get a nonsensical solution that suggests things to do with polarization and quadripole moments, but no explanation of the answer.

 

[Fightfish]

I know that 8 parameters are needed to characterize the density matrix (9 elements, minus one because we know the trace of the density matrix is one.)

This is actually a problem from Sakurai. While it is correct that 8 parameters are required to parameterise the density matrix, could I check whether you did the counting correctly? (because your 8 = 9 - 1 explanation seems suspicious and you have not mentioned whether you've made use of the fact that the density matrix must be Hermitian)

The hint is to consider what other operators you can construct from $S_x, S_y, S_z$ (and then, if you want to, you can prove that they are sufficient by considering an explicit form of the density matrix in terms of the 8 parameters)