Quantum Fundamentals 2021
For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).
Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
For \(\hat{Q}\doteq
\begin{pmatrix}
2 & i \\ -i & -2
\end{pmatrix}
\), calculate
\(\langle1|\hat{Q}|2\rangle\), \(\langle2|\hat{Q}|1\rangle\),
\(\langle a|\hat{Q}| b \rangle\) and \(\langle b|\hat{Q}|a \rangle\).
What kind of mathematical object is \(|a\rangle\langle b|\)? What is the result if you multiply a ket (for example, \(| a\rangle\) or \(|1\rangle\)) by this expression? What if you multiply this expression by a bra?