Sequential Stern Gerlech experiment

[Rayan]

Homework Statement

A beam of atoms with $l=1$ ($s= 0$) is traveling along the y-axis and passes through a Stern-Gerlach magnet A with its (mean) magnetic field along the x-axis. The emerging beam with $m_x= 1$ is separated from the other two beams. (The eigenvalue of $L_x$ for the atoms in this beam is $\hbar m_x = \hbar$). The beam is then passed through a second Stern-Gerlach magnet with the magnetic field along the z-axis. Into how many beams is the beam further split and what the relative number of atoms in each beam? What would be the result if the $m_x= 0$ beam instead passed through a second magnet with the magnetic field along the z_axis?

Relevant Equations

.

So I thought that when the $m_l = 1$ beam passes through the second SG-magnet, it should split into 3 different beams with equal probability corresponding to $ m_l = -1 , 0 , 1 $ since the field here is aligned along z-axis and hence independent of the x-axis splitting.
And I thought that the same should happen if the $m_x=0$ beam passes through the second magnet? but I'm not as sure here!
and then there is a hint that says I should determine the eigenstates of the $L_x$ operator first! But I don't get why? Any advice?

 

[PeroK]

And I thought that the same should happen if the $m_x=0$ beam passes through the second magnet? but I'm not as sure here!
and then there is a hint that says I should determine the eigenstates of the $L_x$ operator first! But I don't get why? Any advice?

That's effectively a guess. It's a good hint to check your guess by looking precisely at how the eigenstates for a spin-1 particle about the x-y-z axes relate to each other.

 

[vela]

and then there is a hint that says I should determine the eigenstates of the $L_x$ operator first! But I don't get why? Any advice?

So you're not just guessing what the answers are. It's okay use your intuition to make an educated guess what the answer should be, but you still need to do the actual math to make a convincing argument.

 

[TSny]

and then there is a hint that says I should determine the eigenstates of the $L_x$ operator first! But I don't get why? Any advice?

Let $| L_z, m_z \rangle$ denote an eigenstate of the $L_z$ operator which has eigenvalue $m_z \hbar$. What are the possible values of $m_z$?

Let $| L_x, m_x \rangle$ denote an eigenstate of the $L_x$ operator which has eigenvalue $m_x \hbar$. What are the possible values of $m_x$?

Suppose the angular momentum state of the particle is known to be $| L_x, m_x \rangle$. If a measurement of the z-component of angular momentum is made on this particle, what are the possible outcomes of the measurement?

Using the notation $| L_x, m_x \rangle$ and $\langle L_z, m_z |$, how would you construct an expression for the probability amplitude that the measurement will yield the outcome $m_z \hbar$?

How do you obtain the probability that the outcome will be $m_z \hbar$?