Established physics · Quantum Mechanics 201 · Lesson 6 of 15

Spin and the order of measurements

Spin is intrinsic angular momentum; changing which component is measured can change the statistics of a later measurement.

Free science education from TRQC™ · Timeline-Relative Quantum Collapse™.

Picture it

Prepare spin-½ systems in the +z state. Measure z again immediately with an ideal repeatable apparatus: the answer is +z. Now insert an x measurement before that last z measurement and keep every run. The final z answers become 50/50. The intermediate physical operation matters even when you ignore which x outcome it recorded.

A two-state system without a tiny spinning ball

An electron has intrinsic spin, with two outcomes for any chosen spin-component measurement. The measured component is +ℏ/2 or −ℏ/2. “Up” and “down” name those outcomes relative to an axis. They do not mean an electron is a miniature charged sphere whose ordinary surface is mechanically rotating. The spatial axis belongs to the measurement question.

Different axes ask incompatible questions

A +z state is an equal superposition of +x and −x. After an ideal x measurement, its outcome is recorded and the conditional state supports that recorded x value. A subsequent z measurement has equal odds. This sequence shows why the order of operations can matter; it is not explained by assigning a permanently stored definite answer to every axis.

Keep all runs or identify a subset

Keeping every x result differs from retaining only the +x subset. With both x outcomes included, add the probabilities of the mutually exclusive recorded branches. With a selected subset, renormalize to describe that subset. Never quietly discard trials while quoting the statistics as though they describe the entire original ensemble.

The angle law is specific and testable

For a +z preparation measured along a direction tilted by θ, the probability of the positive result is cos²(θ/2). At 0° it is one, at 90° it is one half, and at 180° it is zero. The half-angle distinguishes this spin-½ example from the linear-polarization analyzer in the previous lesson.

A physical operation changes the sequence
  1. Prepare +z
  2. Optionally measure another axis
  3. Measure z again

The demonstration retains both intermediate outcomes; it does not silently postselect one branch.

Try the example on paper

No middle measurement gives final +z with probability 1. A middle x measurement gives final +z with probability 1/2. A middle measurement along z or −z leaves final +z certain when every outcome is retained.

Make it concrete

At θ = 90°, half the systems produce +x and half produce −x. Each subset has a 50% chance of +z in the final measurement. The total is (1/2)(1/2) + (1/2)(1/2) = 1/2. At θ = 0° or 180°, the middle apparatus measures the same spin component with possibly reversed labels, so the final +z result remains certain.

The precise version

A spin-½ component along n is Sₙ = (ℏ/2)σ·n. For a +z input and intermediate axis θ, p = cos²(θ/2). If that ideal intermediate measurement is performed and both outcomes are retained, a final z measurement gives P(+z) = p² + (1 − p)² = [1 + cos²θ]/2. With no intermediate measurement, immediate z repetition gives P(+z) = 1.

Watch the trap

Ignoring an intermediate outcome does not undo the physical measurement. Conversely, merely rewriting a state in the x basis does not perform an x measurement.

Keep this

State preparation, axis choice, measurement order, and trial selection all matter. Say which operations actually happened.

Words for this lesson: spin · qubit · eigenstate · observable · basis · conditional probability

Check your understanding

Reason through each question before opening its answer and explanation.

1. What does electron spin describe?

  1. A classical arrow with simultaneously definite components along every axis
  2. Intrinsic angular momentum with quantized component outcomes
  3. The speed of an ordinary material surface spinning
Show answer and explanation

Intrinsic angular momentum with quantized component outcomes

Spin is an intrinsic quantum angular momentum. The small rotating-ball picture does not capture its measurement structure.

2. A +z state is immediately measured along z with an ideal repeatable apparatus. What is P(+z)?

  1. 1
  2. 1/2
  3. 0
Show answer and explanation

1

The preparation is an eigenstate of this same component measurement.

3. Insert an ideal x measurement, retain all outcomes, then measure z. What is P(+z)?

  1. 1/2
  2. 1/4
  3. 1
Show answer and explanation

1/2

Both intermediate x branches contribute: 1/4 + 1/4 = 1/2.

4. Does ignoring the x result restore the earlier +z preparation?

  1. Only if the final detector is nearby
  2. Yes, unread means no interaction
  3. No, the intermediate interaction still occurred
Show answer and explanation

No, the intermediate interaction still occurred

Discarding a record from your analysis does not reverse the measurement interaction.

5. For a +z state measured along an axis 180° from z, what is the positive-axis outcome probability?

  1. 0
  2. 1/2
  3. 1
Show answer and explanation

0

Cos²(180°/2) = cos²90° = 0. The axis label has reversed.

6. If only a selected middle-measurement subset is retained, what must the analysis do?

  1. Treat it as the unchanged full ensemble
  2. Add its probabilities without renormalizing
  3. Use its conditional probabilities and state which runs were kept
Show answer and explanation

Use its conditional probabilities and state which runs were kept

Selection changes the ensemble being described. Its probabilities must sum to one within that selected subset.

Further reading