The sim sends neutral atoms through a magnet whose field changes with height, and records where each atom lands on a screen. The pattern shows which directions an atom's magnetic moment can take, which is how this experiment first tested whether those directions are quantised.

What's on screen

• Atoms (green) leave the source. Inside the magnet each takes the colour of its Beam Legend row.
• Magnet: N pole above (pointed), S pole below. |B| max and |B| min mark the strongest and weakest field.
• Detection window: each atom that reaches the screen leaves a dot. Its title names the rule running.
• Orange dots: the classical prediction (section 04).
• Sidebar: the Total detected, and the Beam Legend with a count per row.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: the sim sets g_J = 1 (a real silver atom has g_J = 2) and draws the screen in pixels, so compare gaps as ratios, not as millimetres.

The atoms carry no net charge, so the magnetic field exerts no Lorentz force on them. They do carry a magnetic moment μ, locked to their angular momentum J:

In a uniform field the energy U = −μ·B only twists the moment and gives no net force. A force needs the field to change with position, which is why one pole is pointed:

In the sim ∂B/∂z is positive, so an atom with positive m_J is pushed down and one with negative m_J up. The push acts only while the atom is between the poles. After that it drifts in a straight line to the screen.

Suppose the moment could point in any direction with equal chance. Then μ_z would take every value from −μ to +μ with equal weight, and so would the force. The same 1/(2μ) applies everywhere between the two edges, so the screen would fill evenly.

In the sim, the classical mode gives each atom a random value anywhere between −j and +j, so its edges sit where the outermost quantum values would land. The orange dots then fill the space between with no gap.

Quantum theory allows J_z only the values m_Jħ, with m_J = −j, −j + 1, …, +j. That is 2j + 1 values in steps of one. Each value gives its own force, so each gives its own place on the screen.

• Integer j includes m_J = 0, so one band sits on the centre line. Half-integer j does not.
• j = 0 allows only m_J = 0. Then μ_z = 0 and there is no force at any gradient.
• A larger j adds bands outward without moving the ones already there.

An atom is pushed for the time L/v it spends in the magnet, then drifts a distance D. Adding the two stages gives its height on the screen:

• Linear. ζ is proportional to ∂B/∂z, so doubling the gradient doubles every gap. It is not squared.
• Equal gaps. Band positions are −m_Jζ, so neighbours are always ζ apart.
• Slower atoms bend more. ζ goes as 1/v², so the sim's small spread of speeds blurs each band a little.

The magnet ties each atom's height to its m_J, so where an atom lands tells you which m_J it had. Quantum theory gives only the probability of each outcome:

The sim's source has no preferred direction, so every m_J has the same probability, 1/(2j + 1). After N atoms each band holds about N/(2j + 1), give or take √(Np(1 − p)). At j = 1/2 and N = 100 that is 50 ± 5: close, but never forced to be equal.

The sim draws one m_J for each atom at the source and then moves the atom classically. That reproduces these counts. It cannot show an atom in a superposition, a precessing moment, or what a second magnet would do.

• The 1922 experiment. Stern and Gerlach sent silver atoms through a gradient of about 10 T/cm (1000 T/m) and found two bands where classical physics predicts one smear. The two bands are now explained by electron spin ½, proposed three years later.
• Cold-atom experiments. Gradient pulses still separate atoms by spin state so that each state can be counted.
• Beam experiments and clocks. Rabi's molecular-beam resonance method and caesium beam clocks use gradient magnets to select atoms in one state.

Key points

• The force is F_z = μ_z ∂B/∂z. No gradient, no force.
• Classically μ_z takes every value. Quantum mechanics allows only m_J = −j … +j.
• j sets how many bands there are. The gradient sets how far apart they are.
• An unpolarised beam gives each band probability 1/(2j + 1).

Common mix-ups

• "A stronger magnet splits the beam into more bands." The number of bands is fixed by j. The gradient only moves them apart.
• "The classical pattern is just the quantum one, blurred." The classical rule has no gaps, and no gradient creates any.
• "The field pushes the atoms because they are charged." The atoms are neutral. The force comes from the moment in a field gradient.

Things to try on your own

• Set j = 0 and drag the gradient from 1 to 6 T/m. Describe the screen.
• At j = 3/2, find which band sits closest to the centre line and note which m_J it is. Repeat at j = 1.
• Run the classical rule at j = 1 and again at j = 2, and compare each pattern's edges with the outermost bands in quantum mode.