Question

FIGURE Q$41.5$shows the outcome of a Stern-Gerlach experiment with atoms of element $X$.

a. Do the peaks represent different values of the atom’s total angular momentum or different values of the $z-$component of its angular momentum? Explain.

b. What angular momentum quantum numbers characterize these four peaks?

Short answer

a. The peaks represent different values of the atom’s total angular momentum or different values of the $z‐$component of its angular momentum is given below in step.

b. The angular momentum quantum numbers characterize these four peaks are $\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2,{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2,{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$}\right.}$}\right.$and$\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

Step-by-step solution

Part (a) step 1: Given Information

We need to find the peaks represent different values of the atom’s total angular momentum or different values of the $z‐$component.

Part (a) step 2: Explanation

The up-down deflection of atoms in a Stern-Gerlach experiment is along the $z‐axis$, so the deflection depends on the magnetic moment ${\mu }_z$which provides information about the $z‐$component of the angular momentum. Each peak represents a different value of the$z‐$component of total angular momentum.

Part (b) step 1: Given Information

We need to find angular momentum quantum numbers to characterize these four peaks.

Part (b) step 2: Explanation

Orbital angular momentum numbers l are integers, and there are an odd number $\left(2l+1\right)$of $z‐$components. One component has $m=0$and would not be deflected at all. However, we see an even number of peaks, all of which are deflected. The total angular momentum can't be an integer. This is more like an atom whose total angular momentum consists only of its spin,$S=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$with and with two $z‐$components $m_s=\pm \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. In that case, the deflection had $2$peaks. An experiment with $4$deflection peaks can be explained if the total angular momentum quantum number is $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, because then the $z‐$component would have the possible values localid="1650179124044" $\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2,{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$}\right.$, and $\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. Total angular momentum can arise if an atom has both orbital and spin angular momentum.