- #1
Xyius
- 501
- 4
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?)
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?)