2.6.5 So­lu­tion herm-e

Ques­tion:

Show that an op­er­a­tor such as ${\widehat{x}}^2$, cor­re­spond­ing to mul­ti­ply­ing by a real func­tion, is an Her­mit­ian op­er­a­tor.

An­swer:

If the op­er­a­tor cor­re­sponds to mul­ti­ply­ing by a real func­tion of $x$, call it $r(x)$, then

$$\langle f\vert\widehat r g\rangle = \int_{\mbox{\scriptsize ... ...ll }i} (r f)^* g{ \rm d}x = \langle\widehat r f\vert g\rangle$$

since the com­plex con­ju­gate does not af­fect a real func­tion.