Discussion
An atom with non-zero magnetic moment has an angular momentum of magnitude \(\sqrt{12}\,\hbar\). When a beam of such atoms is passed through a Stern-Gerlach apparatus, how many beams does it split into?
Answer: B
The beam splits into 7 beams.
The magnitude of an angular momentum with quantum number \(j\) is \(\sqrt{j(j+1)}\,\hbar\). Setting
\[ j(j + 1) = 12 \Rightarrow j = 3 \]
In a Stern-Gerlach apparatus the beam splits according to the projection \(m_{j}\), which takes the \(2j + 1 = 7\) values \(-3, -2, \ldots, 3\).
3 is the value of \(j\) itself, 9 is \((j)^{2}\) or the count for \(j = 4\), and 25 is \(2j + 1\) with 12 taken as \(j\); none gives the number of \(m_{j}\) states.
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