Quantum Fundamentals 2023
Two electrons are placed in a magnetic field in the \(z\)-direction. The initial state of the first electron is \(\frac{1}{\sqrt{2}}\begin{pmatrix}
1\\ i\\
\end{pmatrix}\) and the initial state of the second electron is \(\frac{1}{2}\begin{pmatrix}
\sqrt{3}\\ 1\\
\end{pmatrix}\).

Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at \(t = 0\).

Find the probabilty of measuring each particle to have spin-up in the \(x\)-, \(y\)-, and \(z\)-directions at some later time \(t\).

Calculate the expectation values for \(S_x\), \(S_y\), and \(S_z\) for each particle as functions of time.

Are there any times when all the probabilities you have calculated are the same as they were at \(t = 0\)?