Virtual laboratory/Physics/University-level physics/The Stern–Gerlach experiment — spin quantisation

The Stern–Gerlach experiment — spin quantisation

Classically a beam of atoms carrying a magnetic moment should blur into a continuous spread. Instead it gives two discrete beams — spin takes only two values. Set the field gradient and the angle of the second analyser, and you will see the cos²(θ/2) probabilities and how a measurement in a new axis "erases" the previous one.

Physics · Atomic physics · university level

Field gradient

\nabla B

0.70

2nd analyser orientation

\theta

45 °

Classically we expect a blurred spread (magnets at arbitrary orientations). The experiment gives:

result

2 spots

2nd axis angle θ

45°

P(↑) = cos²(θ/2)

85%

P(↓)

15%

🎯 Challenge: hit P(↑)

streak: 0

I will give you a target P(↑) — tune the second analyser angle θ to reach it. P(↑) = cos²(θ/2).

What just happened?

Atoms with a magnetic moment fly through an inhomogeneous field. Classically their “magnets" would sit at arbitrary angles, so the screen would show a continuous spread. Instead there appear exactly two spots — the magnetic moment (spin) can take only two projection values.

Quantisation and spin projection:

P(\uparrow) = \cos^2\!\frac{\theta}{2}, \qquad P(\downarrow) = \sin^2\!\frac{\theta}{2}

If the “+½" beam is measured again in an axis rotated by $\theta$ , it again gives two outcomes — with probabilities $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ .

Invariant — measurements do not commute:

Measuring spin in one axis destroys the information about spin in the perpendicular axis. You cannot know the spin projection onto two different axes at once — this is a fundamental feature of the quantum world, not an imperfection of the apparatus.

Why this matters for your exam:

This is a university-level extension of atomic structure and quantisation. The Stern–Gerlach experiment (1922) revealed spin and spatial quantisation — today the foundation of qubits and quantum computing.

Remember: the beam splits into two spots — spin is quantised ( $\pm\tfrac12$ ). Measurements in different axes do not commute.

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Theory & formulas

S_z = \pm\tfrac12\hbar

Frequently asked questions

›

Why only two beams?

Because the spin projection onto the measurement axis is quantised — it can take only two values (+½ and −½). A classical magnet could sit at any angle and give a continuous spread; quantum spin gives exactly two beams.

›

What is spin?

The intrinsic magnetic moment of a particle, the "quantum analogue" of rotation — even though the particle does not physically spin. For the electron and many atoms it takes two projection values. Spin is responsible for magnetism, among other things, and is the basis of qubits.

›

Why does the second measurement "spoil" the first?

Because spin measurements along different axes do not commute — you cannot know the projection onto two perpendicular axes at once. Measuring spin in a new axis destroys the information about the previous one, which is why the "+½" beam splits again.

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