Let \hat{Q} be an operator with a complete set of orthonormal eigenvectors:
\hat{Q}\left|e_{n}\right\rangle=q_{n}\left|e_{n}\right\rangle \quad(n=1,2,3, \ldots) .
(a) Show that \hat{Q} can be written in terms of its spectral decomposition:
\hat{Q}=\sum_{n} q_{n}\left|e_{n}\right\rangle\left\langle e_{n}\right| . (3.103)
Hint: An operator is characterized by its action on all possible vectors, so what you must show is that
\hat{Q}|\alpha\rangle=\left\{\sum_{n} q_{n}\left|e_{n}\right\rangle\left\langle e_{n}\right|\right\}|\alpha\rangle .
for any vector |\alpha\rangle .
(b) Another way to define a function of \hat{Q} is via the spectral decomposition:
f(\hat{Q})=\sum_{n} f\left(q_{n}\right)\left|e_{n}\right\rangle\left\langle e_{n}\right| . (3.104).
Show that this is equivalent to Equation 3.100 in the case of e^{\hat{Q}} .
e^{\hat{Q}} \equiv 1+\hat{Q}+\frac{1}{2} \hat{Q}^{2}+\frac{1}{3 !} \hat{Q}^{3}+\cdots (3.100).