Note: In each folder with superposition exercises (labeled Z, X, and Y), there are 4 sub-folders. Each of these subfolders allows you to experiment (using the three different orientations of the Stern-Gerlach apparatus) on beam (ensemble) of identically prepared particles each in the same spin superposition state. Each folder also corresponds to a different set of exercises. These superpositions are made up of eigenstates of the spin operator corresponding to the folder label (Z, X, or Y). Make sure to answer the following questions for all superpositions.
~~1. Calculate the eigenstates of Sx and Sy. Write these eigenstates in terms of |z+> and |z−> and in terms of each other.
~~2. Run the first (1) Z superposition (an ensemble of particles in a superposition of spin eigenstates of Sz). Measure Sz using the Measure z simulation. What are your results? Predict what will happen when you now measure Sx with another identically prepared ensemble of particles. Predict what will happen when you now measure Sy with another identically prepared ensemble of particles. Run the simulations to test your predictions. Use your results to determine the initial superposition state of the particles that make up the ensemble. Report your results in the simplest basis possible (x, y, or z) and also in terms of the z basis (|state> = [c+|z+> + c−|z−>]. Does the phase relationship between c+ and c− matter? In other words, are the results of the experiment affected if c+ = +/-c− or c+ = +/- ic−? Can such a relative phase, then, be detected? Repeat this analysis for the remaining three superpositions.
~~3. Run the first (1) X superposition (an ensemble of particles in a superposition of spin eigenstates of Sx). Measure Sx using the Measure z simulation. What are your results? Predict what will happen when you now measure Sy with another identically prepared ensemble of particles. Predict what will happen when you now measure Sz with another identically prepared ensemble of particles. Run the simulations to test your predictions. Use your results to determine the initial superposition state of the particles that make up the ensemble. Report your results in the simplest basis possible (x, y, or z) and also in terms of the z basis (|state> = [c+|z+> + c−|z−>]. Does the phase relationship between c+ and c− matter? In other words, are the results of the experiment affected if c+ = +/-c− or c+ = +/- ic−? Can such a relative phase, then, be detected? Repeat this analysis for the remaining three superpositions.
~~4. Run the first (1) Y superposition (an ensemble of particles in a superposition of spin eigenstates of Sy). Measure Sy using the Measure z simulation. What are your results? Predict what will happen when you now measure Sz with another identically prepared ensemble of particles. Predict what will happen when you now measure Sx with another identically prepared ensemble of particles. Run the simulations to test your predictions. Use your results to determine the initial superposition state of the particles that make up the ensemble. Report your results in the simplest basis possible (x, y, or z) and also in terms of the z basis (|state> = [c+|z+> + c−|z−>]. Does the phase relationship between c+ and c− matter? In other words, are the results of the experiment affected if c+ = +/-c− or c+ = +/- ic−? Can such a relative phase, then, be detected? Repeat this analysis for the remaining three superpositions.