Tutorial Exercises for
Measurement of Quantum-mechanical Spin-1/2 Systems

 

When performing these virtual experiments it is important to understand the role of probability and statistics in the measurement of quantum systems.  To get an idea of how to properly collect data, double-click the Statistics node and follow the exercises below.  In the simulation you can perform measurements at a rate of 1 per click (the "Step 1" button) or 1,000 per click (the "Step1000" button). 

NOTE: For these exercises, the incident beam contains an ensemble of identically prepared particles all in the same unknown spin superposition state: [c+|z+> + c|z>].  

 

Statistics Exercises: [c+|z+> + c|z>]

~~1.  In these exercises the initial beam of spin-1/2 particles is in an unknown superposition of spin eigenstates: [c+|z+> + c|z>].  Predict the results of the experiments of measuring the z component of spin.  

 

~~2. Click the "Step 1" button a few times to get some data.  Predict the result of doing the experiment many more times.  What do you think c+ and c will turn out to be?

 

~~3. Click the "Step 1000" button a few times to get a lot of data.  What do you think c+ and c are now?  Did your answer change with more data (better statistics)?  How do you think you should perform measurements in the future?

 

~~4. The individual particles in the beam were each in a state c+|z+> + c|z> before measurement, yet after measurement, a definite state (either |z+> or |z>) is detected.  Given the effect of measurement on states (the so-called collapse of the wave function), can you even know what state a single particle was in before measurement?  Explain.