(Solved): EXERCISES 37 2.6. Photon horizontal and vertical polarization states are written as \( |h\rangle \) ...

EXERCISES 37 2.6. Photon horizontal and vertical polarization states are written as \( |h\rangle \) and \( |v\rangle \), respectively. Suppose \[ \begin{array}{l} \left|\psi_{1}\right\rangle=\frac{1}{2}|h\rangle+\frac{\sqrt{3}}{2}|v\rangle \\ \left|\psi_{2}\right\rangle=\frac{1}{2}|h\rangle-\frac{\sqrt{3}}{2}|v\rangle \\ \left|\psi_{3}\right\rangle=|h\rangle \end{array} \] Find \[ \left|\left\langle\psi_{1} \mid \psi_{2}\right\rangle\right|^{2}, \quad\left|\left\langle\psi_{1} \mid \psi_{3}\right\rangle\right|^{2}, \quad\left|\left\langle\psi_{3} \mid \psi_{2}\right\rangle\right|^{2} \] 2.1. A quantum system is in the state \[ \frac{(1-i)}{\sqrt{3}}|0\rangle+\frac{1}{\sqrt{3}}|1\rangle \] If a measurement is made, what is the probability the system is in state \( |0\rangle \) or in state \( |1\rangle \) ? 2.2. Two quantum states are given by \[ |a\rangle=\left(\begin{array}{c} -4 i \\ 2 \end{array}\right), \quad|b\rangle=\left(\begin{array}{c} 1 \\ -1+i \end{array}\right) \] (A) Find \( |a+b\rangle \). (B) Calculate \( 3|a\rangle-2|b\rangle \). (C) Normalize \( |a\rangle,|b\rangle \).