PX262 - G2a - stern-gerlach experiment

image: Theresa Knott

a beam of atoms is passed through an inhomogeneous magnetic field
a force acts on the atom, directing it towards a position with smaller interaction energy
if an atom with $μ_{z}$ is placed in a magnetic field directed along $z$ , such that the field is a function of $z$ , the force that acts on it is:

F = - μ_{z} \frac{\partial B}{\partial z}

the stern-gerlach experiment makes use of this effect to measure atomic magnetic moments
the force exerted by the inhomogeneous field causes the beam to be deflect by an amount proportional to the component of their magnetic moments in the transverse place
for atoms with an electrons with an orbital quantum number, $l$ , a split into $2 l + 1$ parts is expected, and for $l = 0$ , there should be no splitting
experimentally, it is seen that a $l = 1$ beam splits into 4, and $l = 0$ beam splits to 2

$l$	expected $m = 2 l + 1$	actual
$0$	$1$ (no split)	$2$
$1$	$3$	$4$

the splitting of $l = 0$ beam implies that the atom possesses a magnetic moment whose $z$ -component can adopt two opposite orientations with respect to the field
the ground state of hydrogen has no angular momentum: $n = 1 ⟹ l = 0 ⟹ L = \sqrt{l (l + 1)} ℏ = 0$
so, this cannot be associated with the orbital motion of the electron
to resolve this, an additional property of particles, called the spin, was introduced
adding angular momenta:

l_{1} \oplus l_{2} = l_{1} + l_{2}, l_{1} + l_{2} - 2, \dots, | l_{1} - l_{2} |

in the stern-gerlach experiment, the orbital angular momentum and spin should be added
for $l = 1$ , $s = 0 :$

l \oplus s = [l + s, | l - s |] = \frac{3}{2}, \frac{1}{2}

there will be 4 values for that $z$ -projection: $+ 3 / 2, + 1 / 2, - 1 / 2, - 3 / 2$
thus, it can be concluded that an electron as spin $1 / 2$
spin can take integer or half integer values: $n, n / 2$