Extension of Section 2.1.8 - Trace

There are some missing part between eq. (2.60) and eq. (2.61) of the main text. For pure state \(|\psi\rangle\), it can be written as a linear combination of an orthonormal basis \(\{|j\rangle\}\),

\[ |\psi\rangle = \sum_j c_j |\psi_j\rangle \]

Then, the trace can be written under basis \(\{|j\rangle\}\) as

\[\begin{split} \begin{split} {\rm tr}(A|\psi\rangle\langle \psi|) =& \sum_j \langle j|A|\psi\rangle\langle \psi|j\rangle \\ =& \sum_j \langle j|\psi\rangle^* \langle j|A|\psi\rangle \\ =& \sum_j c_j^*\langle j| A|\psi\rangle = \langle \psi|A|\psi\rangle. \end{split} \end{split}\]

Mathematically, above relation is valid for arbitrary square matrix \(A\). However, in quantum machanics, an observable \(A\) should be a Hermitian matrix.