Spin Splits a Beam into Exactly Two
Send a beam of neutral atoms between magnet poles where the field is stronger on one side, and the screen shows not a fuzzy band but exactly two sharp dots. Classical physics would smear the atoms into a continuous vertical band, since each tiny magnet could point any way. Instead, exactly two. This is the most direct evidence of quantization: the spin angular momentum component Sz takes only two values, ±½ℏ. In the first figure you toggle classical versus quantum by hand to begin.
Here is the apparatus seen from the side. Atoms leave the source on the left, pass between the magnet poles in the middle, and land on the screen at the right. The top pole is sharp, so the field tilts up and down, creating a force that pushes each magnet up or down. Toggle classical dipole and the screen shows a continuous vertical band smeared up and down. Switch to quantum spin ½ and that band vanishes, leaving up and down, exactly two dots. The Sz you meet first here is the z-component of spin, a different symbol from entropy S. This is the moment quantization is visible to the eye.
How far the two dots separate is set by the field gradient. The force on each magnet is Fz = μz(∂Bz/∂z), the magnetic moment μz times the gradient ∂Bz/∂z along z. Raise the gradient with the slider and the upper dot pushes higher, the lower dot lower, so the gap between them grows together. At zero gradient the two dots merge at the center and the split disappears. The fact that there are two dots stays the same, but how far apart they sit depends on the force the apparatus makes.
Why exactly two dots? Because the spin is ½. In general a particle of spin j splits into 2j+1 dots. Step j up through ½, 1, 3/2 and the dots grow to 2, 3, 4. Read it backward and just counting the dots on the screen tells you the particle's spin. Two dots means spin ½, three means 1, four means 3/2. The number of split lines is the fingerprint of the quantum number.
Let us name the two dots. The upper dot is Sz = +½ℏ, that is ms = +½, the up state |↑⟩, and the lower dot is Sz = −½ℏ, ms = −½, the down state |↓⟩. Here ms is the spin magnetic quantum number, always written as a subscript so it is not confused with mass m. The magnetic moment is μz ≈ ∓μB, so spin up means the moment points down (gs ≈ 2, and μB is the Bohr magneton). Move the prepared-state angle with the slider and the two bars for |↑⟩ and |↓⟩ grow in opposite directions. This two-valued |↑⟩/|↓⟩ is the physical identity of the qubit you met abstractly in the earlier lesson.
Chain two devices and the nature of measurement shows itself. First a z-filter passes only the up state |↑⟩. Feed that clean up state into a second analyzer tilted by angle θ. Then the chance of up is cos²(θ/2) and the chance of down is sin²(θ/2). Raise θ from zero with the slider and the up share shrinks while the down share grows. At θ = 90° both are exactly fifty-fifty. You clearly selected the up state, yet measuring along a new axis erased that information. Measuring a new axis wipes out the old axis's answer.