"But with the outcome of the experiment, I really did not
understand anything..." -Otto Stern(2)
Short answer by Otto:
We sent a beam of silver atoms through a very inhomogeneous magnetic
field. In such a field the magnets are deflected because the field
strength on the place of one pole of the magnet is a little different
from the field strength acting on the other pole. So in the balance a
force is acting on the atom and it is deflected.(9)
well done asking this question! That's the spirit.
A foreword: SGE was performed in times when quantum- and nuclear physics was deriving and refining a lot of thinking and results that where coming from very prominent names. Further, vacuum tech was just evolving to allow for such experimentation.
Dirac in 1929:
The general theory of quantum mechanics is now almost complete...
I try to provide you a slightly different (set of) answer, since I'm not fully pleased with the answers given, because there is more to this important "benchmark"(2) Nobelprize-experiment.
The background is this: Classical physics expected a Larmor precession of the magnetic moments.(2) From (2), let me also quote: "We know today that the directional quantization of angular momentum and magnetic moments in magnetic fields as observed in the Zeeman Effect (Zeeman, 1896, 1897) as well as in the Stern-Gerlach experiment are closely related processes."
Key to understand is that electrons have angular momentum and magnetic momentum.(1) This concept was proposed by Uhlenbeck and Goudsmit.(3)
The highly simplified images of the SG-Experiment crusading the internet avoid taking a closer look at what is really going on with regards to the H-field and atomics. Take a look at the real setup provided by Uni Göttingen and you will notice the magnets arrangement is highly specific in design for creating an inhomogeneous field for this experiment.
This peculiar "gradient field" makes the density of the field a function of location, chiefly ∂Bz/∂z, hence a force is acting. Otherwise, there would be just orientation of the atoms. Note that "m" (= g_s * m_s) takes ~ +-1, so Fz(-m) = -Fz(m) or in words: the expectation that the beam is split into two, not just broadening. The yellow curve (i. e. ionization) as a function of detector position (small coil currents and strong currents).
The yellow curve in the picture was not just broadening, but splitting or from side-view developing two maxima.
Further, this magnetic setup is modelled as two wires with current flow in opposite directions. A near ∂Bz/∂z = const is seeked, which is requisite for later calculating the magnetic momentum: The exact position of the beam in z-axis (z = 0) is calculated (approximated).
Advanced topics: Sebens(4) for example is asking discreteness vs. uniqueness of the spin, and he concludes in "...we need quantum physics to explain the results of the Stern-Gerlach experiment."
I hope this answer leaves your question-answer ratio balanced not split.
- Quantenmechanik der Atome. F. Hund. Handbuch der Physik (https://link.springer.com/chapter/10.1007/978-3-642-85687-7_1)
- The Stern-Gerlach Experiment Revisited (https://arxiv.org/abs/1609.09311)
- George Uhlenbeck and the Discovery of Electron Spin (https://doi.org/10.1063/1.881186)
- Particles, Fields, and the Measurement of
Electron Spin (https://arxiv.org/abs/2007.00619)
- Stern-Gerlach: conceptually clean or acceptably vague? (https://arxiv.org/abs/1911.00546)
- Feynman Lectures III - Spin One (https://www.feynmanlectures.caltech.edu/III_05.html)
- Foundations of Potential Theory (https://books.google.de/books?id=AqHyCAAAQBAJ&source=gbs_book_other_versions)
- Molecular Beams in Physics and Chemistry (https://link.springer.com/book/10.1007/978-3-030-63963-1)
- The method of molecular rays, Nobel Lecture, Otto Stern 1946 (https://www.nobelprize.org/uploads/2018/06/stern-lecture.pdf)