Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
The Stern-Gerlach experiment sends atoms through a magnetic field, causing them to move up or down based on their magnetic properties. This movement (deflection) tells us something about the angular momentum of the atoms, specifically how their angular momentum behaves along the z-axis (the vertical direction in this case).
Each deflection peak (a spot where atoms hit a detector) represents a different value of how the angular momentum is oriented along the z-axis.
Therefore, the peaks represent different possible values of the z-component of angular momentum.
$$\color{blue}{\bf [b]}$$
In atoms, angular momentum can come from two sources:
1. The orbital angular momentum (how the electrons orbit the nucleus).
2. The spin angular momentum (how the electrons spin on their axis).
Note that:
$\bullet$ The number of deflections or peaks in an experiment depends on how many possible "directions" the angular momentum can point along the z-axis. These directions are determined by a quantum number called $l$, which is always an integer $l=0,1,2,3,\dots$
$\bullet$ The number of z-axis components is given by $2l + 1$. For example, if $l = 1$, there will be 3 possible directions for angular momentum along the z-axis.
Our experiment shows an even number of peaks (instead of an odd number), which suggests that the angular momentum does not come from just the orbital movement.
Atoms also have spin angular momentum. For an electron, the spin quantum number $S$ is $1/2$, meaning it has two possible z-axis values: $+1/2$ and $-1/2$.
If only the spin were involved, you'd see 2 deflection peaks.
In this experiment, there are 4 peaks. This can be explained by the atom having both orbital and spin angular momentum.
If the total angular momentum (a combination of orbital and spin angular momentum) is $3/2$, then the z-axis components would have the values $-3/2$, $-1/2$, $1/2$, and $3/2$, which would produce 4 peaks.
Therefore, the 4 peaks in the experiment suggest that the total angular momentum is made up of both orbital and spin angular momentum, leading to more deflection possibilities than if only one type of angular momentum was involved.
