Stern-Gerlach experiment spin
Send a beam of neutral atoms between a magnetic field that is strong on one side and weak on the other, and the screen shows two sharp dots, not a blurry band. Classically the magnets point every which way, so a band should smear continuously up and down. But there are exactly two dots. The spin component Sz of the atom takes only two values, ±½ℏ, the most direct evidence of quantization. The first picture toggles classical and quantum by hand.
The picture looks at the apparatus from the side. Atoms leave the source on the left, pass between the magnet poles, and hit the screen on the right. The upper pole is sharp, so the field slopes up and down and a force pushes the magnet up or down. Switch to a classical dipole and a vertical band smears on the screen. Switch to quantum spin one half and the band vanishes, leaving exactly two dots, up and down. The Sz met here is the z component of the spin, a different symbol from entropy. This is the moment quantization is visible.
How far the two dots spread is set by the slope of the magnetic field. The force on the magnet is Fz = μz(∂Bz/∂z), the magnetic moment times the slope ∂Bz/∂z along z. Raise the slope and the upper dot is pushed higher and the lower dot lower, so the gap grows. When the slope is 0 the two dots gather to the middle and the split vanishes. That there are two dots stays, and how far they sit depends on the force the apparatus makes.
Why exactly two dots. Because the spin is one half. In general a particle of spin j splits into 2j+1 dots. Raise j through one half, 1, and 3/2 and the dots grow to 2, then 3, then 4. Count the dots on the screen and you know the spin. Two dots means spin one half, three means 1, four means 3/2. The number of split lines is the fingerprint of the quantum number.
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 |↓⟩. ms is the spin magnetic quantum number, always written with a subscript, so it is not the mass. The moment is μz ≈ the other way. When the spin is the up state |↑⟩ the moment points down, and the down state |↓⟩ is its pair. gs ≈ 2 is that ratio. Change the preparation angle and the probabilities of the two bars |↑⟩ and |↓⟩ grow in opposite directions. These two-valued states are the physical identity of a qubit.
Chain two apparatuses and the nature of measurement shows. First pass only the up state |↑⟩, then send that clean up state into a second analyzer tilted by an angle. The probability of up is cos²(θ/2) and the probability of down is sin²(θ/2). Raise the angle from 0 and up shrinks while down grows. At θ = 90° both are exactly 50 to 50. You selected an up state, and the moment you measure along a new axis that information is erased. Measuring the new axis removes the answer of the old axis.