# University of Amsterdam Polarization Vector and Rotation Matrix Paper

November 9, 2021
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November 9, 2021

For the spinor $\chi =\begin{pmatrix} e^{i\alpha }cos\delta \\ e^{i\beta}sin\delta \end{pmatrix}$ , calculate the polarization vector P and construct the matrix $U_R$, which rotates the state into $\begin{pmatrix} 1\\ 0 \end{pmatrix}$ . Prove that the probability $P_\hat{n}$ of finding this particle to be in a state represented by the polarization vector $\hat{n}$ is $P{_\hat{n}}= 1/2 trace[\rho (I+\hat{n}\cdot\sigma )]=1/2(1+P\cdot\hat{n})$ and show that this result agrees with expectations for $\mathbf{\hat{n}}=\mathbf{P},\mathbf{\hat{n}}=-\mathbf{P}, and\ \mathbf{\hat{n}}\perp \mathbf{P}$ .