I know this is very long but it would mean so much if someone could help me understand!(adsbygoogle = window.adsbygoogle || []).push({});

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 [itex]s_z=\hbar/2[/itex] and rejects [itex]s_z=-\hbar/2[/itex]

atoms.

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

c.) The third measurement accepts [itex]s_z=-\hbar/2[/itex] atoms and rejects [itex]s_z=\hbar/2[/itex] atoms.

What is the intensity of the final [itex]s_z=-\hbar/2[/itex] beam when the [itex]s_z=\hbar/2[/itex] 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 [itex]s_z=-\hbar/2[/itex] 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 [itex]S_z[/itex] 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>=[itex]cos(β/2)|+>+sin(β/2)|->[/itex] 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 [itex]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)|-><-|[/itex].(This makes sense, they are just multiplying it out.

The final measurement corresponds to the operator M(-)=|-><-|, and the total measuement [itex]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)|-><+|.[/itex](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 [itex]s_z=-\hbar/2[/itex] beam, when the [itex]s_z=\hbar/2[/itex] beam surviving the first measurment is normalized to unity, is thus [itex]cos^2(β/2)sin^2(β/2)=sin^2(β)/4[/itex].(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 [itex]s_z=-\hbar/2[/itex] 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?)

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# I do not understand this solution to this Quantum Mechanics Problem

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