Quantum Fundamentals 2021
Using the SPINS simulation for a spin-\(\tfrac{1}{2}\) system
(Spins simulation),
for the unknown initial state \(\left|{\psi_4}\right\rangle \), perform measurements of
\(S_x\), \(S_y\), and \(S_z\) separately, and for each measurement determine the probability of obtaining each possible measurement value.(We carried out the same procedure for \(\left|{\psi_3}\right\rangle \) in class; you may
refer to that example.)
Use the probabilities you observed from the measurements to express \(\left|{\psi_4}\right\rangle \) as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from probabilities obtained from spin
measurements.
Compare Theory with Experiment: Design an experiment that will
allow you to test whether the state you identified in Question (a) for the
unknown state \(\left|{\psi_4}\right\rangle \) is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
Make a Conceptual Connection: In general, are the probabilities
obtained from spin-component measurements along only two spin directions (for example, the \(z\) direction and the \(y\) direction) sufficient to determine a spin-\(\tfrac{1}{2}\) quantum state? Why or why not?