Exercises for
Single Measurements of Quantum-mechanical Spin-1/2 Systems

 

Note: Each of these subfolders allows you to experiment (using the three different orientations of the ideal Stern-Gerlach apparatus) on beam (ensemble) of identically prepared particles each in  the same spin superposition state.  Each subfolder (labeled 1, 2, 3, 4) corresponds to a different initial superposition.  These superpositions are made up of eigenstates of the spin operator Y.  Make sure to answer all of the questions pertaining to each superposition.

 

Exercises for the Superposition in the y basis: |state> = c+|y+> + c|y>

~~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) Y superposition (an ensemble of particles in a superposition of spin eigenstates of Sy).  Measure Sy using the Measure y simulation.  What are your results? 

 

~~3. Predict what will happen when you now measure Sz with another identically prepared ensemble of particles. 

 

~~4. Predict what will happen when you now measure Sx with another identically prepared ensemble of particles. 

 

~~5. Run the Measure z and  Measure x 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 y basis (|state> = [c+|y+> + c|y−>]).  

 

~~6.  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.