Stern-Gerlach-Experiment with j=1 atoms? Suppose you do a Stern-Gerlach experiment with atoms in a $j=1$ state. There would be three separate beams ($m_j = -1, 0, 1$) coming out of the apperatus. But what would be the relative distribution of atoms in these beams?
On first thought I would say that the distribution should be $\frac{1}{3}$, $\frac{1}{3}$, $\frac{1}{3}$. But considering the "classical" distribution of the projections of the angular momentum onto one axis I could imagine that there could be more atoms in the $m_j = 0$ state. This would still be compatible with symmetry considerations.
To state the question differently: how does the density matrix of an unpolarized beam of $j=1$ atoms look like?
 A: The z-projection of the angular momentum of a completely unpolarized system is distributed uniformly both classically and quantum mechanically. 
In quantum mechanics, the density matrix of a three state system
is given by:
$\rho = \frac{1}{3}(I_3+\overrightarrow{n}.\overrightarrow{\lambda})$, where $I_3$ is the three dimensional unit matrix.
$\overrightarrow{n}$ is the polarization vector and $\overrightarrow{\lambda}$ is a vector of Gell-mann matrices. 
In a fully polarized system $\overrightarrow{n}.\overrightarrow{n}= 1$, while in the case of a completely unpolarized system:
$\overrightarrow{n} = 0$. In this case:
$\rho = \frac{1}{3}(I_3)$. Therefore, the three states are equally distributed.
In classical mechanics, the phase space of a spin system is the unit sphere. The z-component of a unit angular momentum system is given by: 
$ J_z = cos(\theta)$.  
Where $\theta$ is the inclination angle. A completely unpolarized classical angular momentum should be distributed uniformly with respect to the surface area of the sphere which is the classical phase space.
The area of a spherical zone of thickness $h$ is $S = 2 \pi R h$, where $R=1$ is the sphere radius (This result was already known to Archimedes). Therefore the z-coordinate is uniformly distributed on the sphere and the z-component of the angular momentum expressed as:
$ J_z = \frac{z}{R}$
is linear in $z$, therefore also uniformly distributed.
