Questions: Stern-Gerlach Experiment: Spin Quantization and Measurement

Question 1

A beam of silver atoms passes through a z-aligned Stern-Gerlach device and the spin-up-z output is selected. That beam passes through an x-aligned device and the spin-up-x output is selected. That beam then enters a second z-aligned device. What does the second z-device produce?

• A. 100% spin-up-z, because the first device already selected for spin-up-z atoms
• B. 100% spin-down-z, because measuring S_x inverts the z-component
• C. 50% spin-up-z and 50% spin-down-z
• D. No output, because sequential incompatible measurements cancel each other

Answer: (C)

Measuring S_x on the spin-up-z beam yields 50/50 spin-up-x and spin-down-x, and completely destroys the definite S_z value. The spin-up-x state is an equal superposition of spin-up-z and spin-down-z, so the final z-device gives 50/50. Option A is the classic misconception: treating the first measurement as revealing a pre-existing property that persists through subsequent measurements. S_x and S_z are incompatible observables — having a definite S_x value means maximal uncertainty in S_z.

Question 2

What did classical physics predict for the Stern-Gerlach experiment, and what was actually observed?

• A. Classical physics predicted two discrete spots; experiment showed a continuous smear of deflections
• B. Classical physics predicted a continuous smear; experiment showed exactly two discrete spots
• C. Classical physics predicted no deflection; experiment showed deflection in a continuous range
• D. Classical physics predicted four discrete spots; experiment showed only two

Answer: (B)

Classically, a magnetic dipole can point in any direction, so the deflection force should vary continuously — producing a smeared stripe on the detector. Instead, only two discrete spots appeared. This directly demonstrated that the z-component of spin is quantized to exactly two values (±ℏ/2), not a classical continuum. The discreteness was not assumed; it was forced on physics by the experimental result.

Question 3

If you take the spin-up output of a z-aligned Stern-Gerlach device and send it through an identical second z-aligned device, 100% of atoms will emerge from the spin-up port.

• T. True
• F. False

Answer: True

This is projective measurement in action. The first device prepares a definite spin-up-z state. A second z-aligned device simply confirms that state — there is no probability of spin-down because the state is already an eigenstate of S_z. This is fundamentally different from measurement revealing a pre-existing classical property: the first measurement prepared the state, and the second confirms it.

Question 4

A Stern-Gerlach device generally splits an incoming beam into two beams of equal intensity, regardless of how the input beam was prepared.

• T. True
• F. False

Answer: False

Equal intensity (50/50 split) only occurs when the input beam is unpolarized — when atoms have random spin orientations. If the input beam has already been selected for a definite spin direction, the intensities will be unequal. For example, a pure spin-up-z beam sent through a z-device gives 100% spin-up and 0% spin-down. Equal splitting along z only arises for beams in eigenstates of S_x or S_y (or any axis perpendicular to z).

Question 5

Why does the sequential Stern-Gerlach experiment (z → x → z) demonstrate that spin components are incompatible observables, rather than simply showing that the x-measurement physically disturbs a pre-existing spin state?

Model answer: If S_z had a pre-existing definite value that the x-measurement merely disturbs, we might expect most atoms to retain their S_z value with occasional disturbances — not a perfect 50/50 split. But the result is exactly 50/50, which is precisely what quantum mechanics predicts: S_x and S_z do not commute, so an eigenstate of S_x is an equal superposition of S_z eigenstates. This is a structural feature of the algebra of spin operators, not a consequence of imprecision. No matter how gently one imagines performing the x-measurement, the incompatibility is fundamental — it reflects that S_x and S_z cannot simultaneously have definite values, as demanded by the Heisenberg uncertainty principle for non-commuting operators.

The key distinction is between 'measurement disturbs a real value' (classical disturbance) and 'there was no definite value to disturb' (quantum incompatibility). The perfect 50/50 outcome rules out any model where S_z has a hidden pre-existing value that the x-measurement merely scrambles.