4.4.2.2 So­lu­tion esdb-b

Ques­tion:

Con­tin­u­ing the pre­vi­ous ques­tion, what are the stan­dard de­vi­a­tions in en­ergy, square an­gu­lar mo­men­tum, and $z$ an­gu­lar mo­men­tum?

An­swer:

Since the ex­pec­ta­tion value in en­ergy is $E_2$, as are the eigen­val­ues of each state, the stan­dard de­vi­a­tion is zero.

$$\sigma_E=\sqrt{\frac 12 (E_2-E_2)^2 + \frac 12 (E_2-E_2)^2} = 0.$$

This is ex­pected since every mea­sure­ment pro­duces $E_2$; there is no de­vi­a­tion from that value.

Sim­i­larly the stan­dard de­vi­a­tion in $L^2$ is zero:

$$\sigma_{L^2}= \sqrt{\frac 12(2\hbar^2-2\hbar^2)^2+\frac 12(2\hbar^2-2\hbar^2)^2} = 0.$$

For the $z$ an­gu­lar mo­men­tum, the ex­pec­ta­tion value is zero but the two states have eigen­val­ues $\hbar$ and $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar$, so

$$\sigma_{L_z}= \sqrt{\frac 12(\hbar -0)^2+\frac 12(-\hbar -0)^2} = \hbar .$$

Whether $\hbar$ or $\vphantom{0}\raisebox{1.5pt}{$-$}$$\hbar$ is mea­sured, the de­vi­a­tion from zero has mag­ni­tude $\hbar$.