91Ó°ÊÓ

(a)

A Hermitian operator is equal to its Hermitian conjugate, that is:

${\stackrel{Áåœ}{Q}}^{†}=\stackrel{Áåœ}{Q}$

which has the result that for inner products

The expectation value of the Hermitian operator is:

…… (1)

An anti-Hermitian operator is equal to the negative of its Hermitian conjugate, that is:

(b)

Consider two Hermitian operators $\stackrel{Áåœ}{Q}$and $\stackrel{Áåœ}{R}$, their commutator is:

role="math" localid="1656321298089" $$\begin{gathered} \left[\stackrel{Áåœ}{Q},\stackrel{Áåœ}{R}\right]=\stackrel{Áåœ}{Q},\stackrel{Áåœ}{R}-\stackrel{Áåœ}{R}\stackrel{Áåœ}{Q} \\ \mathrm{The}\mathrm{conjugate}\mathrm{transpose}\mathrm{of}\mathrm{this}\mathrm{commutator}\mathrm{is}: \\ \left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{\mathrm{R}}\right]={\left(\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{R}\right)}^{†}={\stackrel{Áåœ}{R}}^{†}{\stackrel{Áåœ}{Q}}^{†},so: \\ {\left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{R}\right]}^{†}={\stackrel{Áåœ}{R}}^{†}{\stackrel{Áåœ}{Q}}^{†}-{\stackrel{Áåœ}{Q}}^{†}{\stackrel{Áåœ}{R}}^{†} \\ \mathrm{For}\mathrm{a}\mathrm{Hermitian}\mathrm{operator}{\stackrel{Áåœ}{\mathrm{R}}}^{†}=\stackrel{Áåœ}{\mathrm{R}}\mathrm{and}{\stackrel{Áåœ}{\mathrm{Q}}}^{†}=\stackrel{Áåœ}{\mathrm{Q}},\mathrm{thus}: \\ {\left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{R}\right]}^{†}=\stackrel{Áåœ}{R},\stackrel{Áåœ}{Q}-\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{R} \\ =\left[{\stackrel{Áåœ}{R}}^{†}{\stackrel{Áåœ}{Q}}^{†}\right] \\ =-\left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{R}\right] \\ \mathrm{Note}\mathrm{that}\mathrm{equation}\left(3\right)\mathrm{is}\mathrm{used},\mathrm{in}\mathrm{the}\mathrm{last}\mathrm{two}\mathrm{lines}.{\left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{\mathrm{R}}\right]}^{†}=-\left[\stackrel{Áåœ}{\mathrm{Q}},\stackrel{Áåœ}{\mathrm{R}}\right]. \\ \mathrm{A}\mathrm{Hermitian}\mathrm{operator}\mathrm{would}\mathrm{be}\mathrm{the}\mathrm{result}\mathrm{of}\mathrm{two}\mathrm{anti}-\mathrm{Hermitian}\mathrm{operators}. \end{gathered}$$