Construct Density Operator from Ensemble Average of Sx, Sy and Sz

[Geocentric]

Homework Statement

I have been given a problem. The density matrix can be constructed if the ensemble average of Sx, Sy and Sz are given. But I have no idea on how to construct the density matrix from these Si's. Any help is most welcome.

Homework Equations

Ensemble average(Si)=Trace($$\rho$$Si)
Density operator($$\rho$$) = wa.Pa + wb.Pb
where Pa & Pb are projection operators

The Attempt at a Solution

 

[fzero]

You should start by constructing the density operator from a suitable pair of eigenstates and arbitrary factors $$w_i$$. You can then compute expressions for the ensemble averages and try to solve for your unknowns.

 

[Dickfore]

How many real parameters does a 2x2 Hermitian matrix have? What is the average of $\langle S_{i} \rangle, \; i = 1,2,3$ in a (mixed) state given by a density matrix?

 

[arkajad]

Notice however that, if you start with Pa,Pb chosen in advance, for spin 1/2 you will have three equations for two unknowns.

A general density matrix for spin 1/2 is

$$\rho=\frac12\left(1+n_1\sigma_1+n_2\sigma_2+n_3\sigma_3\right)$$

where $$n_1^2+n_2^2+n_3^2\leq 1$$

 

[Dickfore]

How many real parameters does a 2x2 Hermitian matrix have? What is the average of $\langle S_{i} \rangle, \; i = 1,2,3$ in a (mixed) state given by a density matrix?

Also, you should impose the normalization condition of unit trace.