I do not understand this solution to this Quantum Mechanics Problem

[Xyius]

I know this is very long but it would mean so much if someone could help me understand!

THE PROBLEM
A beam of spin 1/2 atoms goes through a series of Stern-Gerlach-type measurements as follows:
a.)The first measurement accepts $s_z=\hbar/2$ and rejects $s_z=-\hbar/2$
atoms.

b.) The second measurement accepts $s_n=\hbar/2$ atoms and rejects $s_n=-\hbar/2$ atoms, where $s_n$ is the eigenvalue of the operator $S\dot n$ with n making an angle β in the xz-plane with respect to the z-axis.
c.) The third measurement accepts $s_z=-\hbar/2$ atoms and rejects $s_z=\hbar/2$ atoms.

What is the intensity of the final $s_z=-\hbar/2$ beam when the $s_z=\hbar/2$ beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final $s_z=-\hbar/2$ beam?

THE SOLUTION:
The following is the solution to this problem that I do not fully understand. The word from the text will be in italics and my commentary will be in parenthesis.

Choosing the $S_z$ diagonal basis, the first measurement corresponds to the operator M(+)=|+><+|. (This makes sense, not really any problems here.) The second measurement is expressed by the operator M(+;n)=|+;n><+;n| where |+;n>=$cos(β/2)|+>+sin(β/2)|->$ with α=0. (This is my main confusion on this solution, where did they get this from? It kind of makes sense if the angles were just β, but why β/2??) Therefore $M(+;n)=(cos(β/2)|+>+sin(β/2)|->)(cos(β/2)<+|+sin(β/2)<-|)=cos^2(β/2)|+><+|+cos(β/2)sin(β/2)(|+><-|+|-><+|)+sin^2(β/2)|-><-|$. (This makes sense, they are just multiplying it out.

The final measurement corresponds to the operator M(-)=|-><-|, and the total measuement $M_T=M(-)M(+;n)M(+)=|-><-|{cos^2(β/2)|+><+|+cos(β/2)sin(β/2)(|+><-|+|-><+|)+sin^2(β/2)|-><-|}|+><+|=cos(β/2)sin(β/2)|-><+|.$ (I didn't actually do this part since I did not understand the previous part, but it looks like all they did was multiply out all the measurement operators.) The intensity of the final $s_z=-\hbar/2$ beam, when the $s_z=\hbar/2$ beam surviving the first measurment is normalized to unity, is thus $cos^2(β/2)sin^2(β/2)=sin^2(β)/4$. (I do now see where they get this either. I get the trig identity part, but now the step before that. Sorry I know this might be kind of hard to read!) To maximize $s_z=-\hbar/2$ beam, set β=π/2 i.e. along OX, and intensity is 1/4 ( I get the intensity of 1/4 part but along OX? What is OX? The x axis?)

 

[vela]

Are you familiar with the spin rotation matrices?

 

[Xyius]

Are you familiar with the spin rotation matrices?

I know of rotation matrices in general. Such as when rotating a coordinate system.

 

[vela]

You want to look into how the angular momentum operators are generators of rotations.

 

[mathfeel]

I know of rotation matrices in general. Such as when rotating a coordinate system.

Spacial rotation matrices is one representation of the rotation group. Spin operators is another. You need to understand some basic representation theory and how the angular momentum operators are generators of rotation.