4.2.3.3 So­lu­tion an­guc-c

Ques­tion:

What is the min­i­mum amount that the to­tal square an­gu­lar mo­men­tum is larger than just the square an­gu­lar mo­men­tum in the $z$-​di­rec­tion for a given value of $l$?

An­swer:

The to­tal square an­gu­lar mo­men­tum is $l(l+1)\hbar^2$ and the square an­gu­lar $z$-​mo­men­tum is $m^2\hbar^2$. Since for a given value of $l$, the largest that $\vert m\vert$ can be is $l$, the dif­fer­ence is at least

$$l(l+1)\hbar^2-l^2\hbar^2 = l\hbar^2.$$