4.5.1.1 So­lu­tion com­mutea-a

Ques­tion:

The pointer state

$$\mbox{2p$_x$} = \frac 1{\sqrt 2}\left(-\psi_{211}+\psi_{21-1}\right).$$

is one of the eigen­states that $H$, $\L ^2$, and $\L _x$ have in com­mon. Check that it is not an eigen­state that $H$, $\L ^2$, and $\L _z$ have in com­mon.

An­swer:

It is an eigen­state of $H$ and $\L ^2$, but not of $\L _z$. Since the $z$ an­gu­lar mo­men­tum of a $\psi_{nlm}$ state is $m\hbar$, the com­bi­na­tion above has a 50%/50% prob­a­bil­ity that the $z$ an­gu­lar mo­men­tum is $\hbar$ or $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar$.